# On the Rouquier dimension of wrapped Fukaya categories

Shaoyun Bai  Department of Mathematics, Princeton University, New Jersey
08544, USA<EMAIL_ADDRESS>and  Laurent Côté  Department of
Mathematics, Harvard University, 1 Oxford St, Cambridge, MA 02138
<EMAIL_ADDRESS>
###### Abstract.

We study the Rouquier dimension of wrapped Fukaya categories of Liouville
manifolds and pairs. We describe upper and lower bounds for this invariant in
terms of various quantities of interest in symplectic topology. By combining
these bounds, we get information about concrete geometric questions such as:
(1) given a Weinstein manifold, what is the minimal number of intersection
points between the skeleton and its image under a generic compactly-supported
Hamiltonian diffeomorphism? (2) what is the minimal number of critical points
of a Lefschetz fibration on a Liouville manifold with Weinstein fibers?

Using homological mirror symmetry, we also obtain improved upper bounds for
the Rouquier dimension of derived categories of coherent sheaves on certain
complex algebraic varieties and stacks. These bounds are sharp in dimension at
most $3$. As a result, we resolve a well-known conjecture of Orlov for a large
class of new examples, including all toric $3$-folds and certain log
Calabi–Yau surfaces. The proof of these upper bounds crucially relies on the
arborealization theorem of Álvarez-Gavela–Eliashberg–Nadler and is therefore
an application of symplectic flexibility to a problem in algebraic geometry.

###### Contents

  1. 1 Introduction
  2. 2 Category theory
  3. 3 Wrapped Fukaya categories
  4. 4 Lower bounds: central actions on the wrapped Fukaya category
  5. 5 Upper bounds: constructing arboreal covers
  6. 6 Applications to symplectic geometry
  7. 7 Appendix

##  1\. Introduction

###  1.1. Overview and motivation

In his seminal work [rouquier], Rouquier introduced a notion of dimension for
triangulated categories. Here is a sketch of the definition: if $\mathcal{T}$
is a triangulated category and $G$ is a (split-)generator, then $G$ is said to
generate $\mathcal{T}$ _in time $t$_ if every object of $\mathcal{T}$ can be
built from $G$ by taking finite direct sums, shifts, summands, and _at most
$t$_ cones. The Rouquier dimension of $\mathcal{T}$ is by definition the
minimal generation time over all generators of $\mathcal{T}$ shifted down by
$1$.

In symplectic topology, an important class of triangulated categories are
derived Fukaya categories in their various guises. We will be mainly
interested in wrapped Fukaya categories of Liouville manifolds. More
precisely, given a Liouville manifold $(X,\lambda)$, its wrapped Fukaya
category $\mathcal{W}(X)$ is an $A_{\infty}$ category and as such, it
canonically determines a triangulated category
$H^{0}(\operatorname{Perf}\mathcal{W}(X))$. The Rouquier dimension of this
triangulated category is a numerical invariant of $(X,\lambda)$ which will be
the focus of this paper.

As motivation for the geometrically-inclined readers, we begin by stating some
questions of rather classical flavor. We will see later that the Rouquier
dimension provides new information towards answering these questions.

######  Question 1.1.

Let $(X,\lambda)$ be a Weinstein manifold with skeleton $\mathfrak{c}_{X}$. If
$\phi:X\to X$ is a generic compactly-supported Hamiltonian diffeomorphism,
what is the minimal possible number of intersection points of
$\phi(\mathfrak{c}_{X})$ with $\mathfrak{c}_{X}$?

(Recall that a Liouville manifold is said to be _Weinstein_ if the Liouville
vector field is gradient-like with respect to a proper Morse function.)

If $(X,\lambda)=(T^{*}M,pdq)$ for a closed manifold $M$, then
$\mathfrak{c}_{X}=0_{M}$ and 1.1 is classical. As is now well-known, it
follows from the existence of Floer cohomology (or finite dimensional analogs)
that $|\mathfrak{c}_{X}\cap\phi(\mathfrak{c}_{X})|$ is bounded from below by
the sum of the Betti numbers of $M$.

In general, however, the skeleton of a Weinstein manifold is a highly singular
object for which there is no available notion of Floer cohomology (and it
seems rather unlikely that such a notion exists). Note that _even for the
cotangent bundle of a closed manifold_ , equipped with a Weinstein structure
homotopic to the standard Morse–Bott one, the skeleton is typically singular.

To the best of our knowledge, the only result in the literature addressing 1.1
is the following fact: if $\mathcal{W}(X)$ is not the zero category, then
$|\mathfrak{c}_{X}\cap\phi(\mathfrak{c}_{X})|\geq 1$. The original proof of
this fact goes through Rabinowitz–Floer homology, see [ritter2013topological,
Sec. 1.6].

######  Question 1.2.

Let $X$ be a Liouville manifold, considered up to homotopy. What is the
minimal possible number of critical points of a Lefschetz fibration
$f:X\to\mathbb{C}$ with Weinstein fibers?

Let us denote this number by
$\operatorname{Lef}_{w}(X)\in\mathbb{N}\cup\\{\infty\\}$. According to work of
Giroux–Pardon [giroux-pardon], any Weinstein manifold admits a Lefschetz
fibration with Weinstein fibers, so $\operatorname{Lef}_{w}(X)<\infty$ iff $X$
is Weinstein (up to homotopy). For topological reasons, one has the lower
bound $\operatorname{Lef}_{w}(X)\geq\operatorname{rk}H_{n}(X;\mathbb{Z})$ when
$X$ has real dimension $2n$. One also has the lower bound
$\operatorname{Lef}_{w}(X)\geq 1$ if $\mathcal{W}(X)$ is not the zero
category.

The real analog of $\operatorname{Lef}_{w}(X)$ was considered by
Abouzaid–Seidel [AScomplexity] under the name “complexity”. By definition, the
complexity of a Weinstein manifold $X$, which we denote by
$\operatorname{WMor}(X)$, is the minimal number of critical points of a
Weinstein Morse function on $X$, where $X$ is considered up to Weinstein
homotopy.111A _Weinstein Morse_ function on $(X,\lambda)$ is a proper Morse
function with respect to which the Liouville vector field is gradient-like.
Here $\lambda$ is a particular choice of Liouville structure on $X$ of the
given homotopy class. Surprisingly, Lazarev [lazarev-gt] showed that for
Weinstein manifolds having non-zero middle homology, one has
$\operatorname{WMor}(X)=\operatorname{Mor}(X)$, where $\operatorname{Mor}(X)$
denotes the minimal number of critical points of a _smooth_ Morse function on
$X$. Does a similar phenomenon hold for $\operatorname{Lef}_{w}(X)$?

######  Problem 1.3.

Let $(X,\lambda)$ be a Liouville manifold and let $(X_{0},\lambda_{0})$ be a
Liouville domain. Find conditions on this data which obstruct the existence of
a Liouville embedding

(1.1) $(X_{0},\lambda_{0})\hookrightarrow(X,\lambda).$

1.3 is of course classical. A standard source of embedding obstructions is the
Viterbo restriction map on symplectic cohomology. The fact that this map is a
morphism of algebras already implies, for instance, that the embedding (1.1)
cannot not exist if $SH^{\bullet}(\widehat{X}_{0})\neq 0$ while
$SH^{\bullet}(X)=0$, where $\widehat{X}_{0}$ is the Liouville completion of
$X_{0}$. One can get more obstructions by studying higher algebraic structures
which are also preserved under Viterbo restriction. For instance, an important
source of embedding obstructions are dilations [seidel-solomon]. In a somewhat
different direction, we also mention the recent work of Ganatra–Siegel
[ganatra-siegel] and Moreno–Zhou [moreno-zhou] extracting many embedding
obstructions from rational symplectic field theory.

###  1.2. The Rouquier dimension of wrapped Fukaya categories

Let us now explain how the Rouquier dimension of wrapped Fukaya categories
provides new insight into the above questions. The following theorem makes
progress on 1.1.

######  Theorem 1.4 (= Theorem 6.9).

Let $(X,\lambda)$ be a Weinstein manifold with properly embedded cocores (this
is a generic condition) and nonzero wrapped Fukaya category. If $\phi:X\to X$
is a generic compactly-supported Hamiltonian symplectomorphism, then:

(1.2)
$|\mathfrak{c}_{X}\cap\phi(\mathfrak{c}_{X})|\geq\operatorname{Rdim}\mathcal{W}(X)+1.$

(The result we prove in Theorem 6.9 is stated slightly differently and is more
general.) Theorem 1.4 is only useful when combined with a lower bound on
$\operatorname{Rdim}\mathcal{W}(X)$. The most important lower bound for us is
Theorem 4.2, which will be discussed later in the introduction. By combining
Theorem 1.4 and Theorem 4.2, one obtains the following corollary (in the
following statement, both $\mathcal{W}(X)$ and $SH^{\bullet}(X)$ are defined
over a field $k$ and are, say, $\mathbb{Z}/2$-graded):

######  Corollary 1.5 (= Theorem 6.9 \+ Theorem 4.2).

Under the assumptions of Theorem 1.4, let $R\subseteq SH^{0}(X)$ be a
subalgebra which is of finite type over $k$. Suppose that there exists a
(split-)generating Lagrangian $K\in\mathcal{W}(X)$ such that
$HW^{\bullet}(K,K)$ is a Noetherian $R$-module, where $R$ acts via the closed-
open string map. Then:

(1.3)
$|\mathfrak{c}_{X}\cap\phi(\mathfrak{c}_{X})|\geq\dim_{R}HW^{\bullet}(K,K)+1.$

Here $\dim_{R}(-)$ denotes the _Krull_ dimension of an $R$-module.

Let us sketch the proof of Theorem 1.4. To begin with, it can be shown that
the Rouquier dimension of any $A_{\infty}$ category is bounded above by the
length of the shortest resolution of the diagonal bimodule by Yoneda bimodules
(minus $1$). This is a purely algebraic fact which has nothing to do with
Fukaya categories.

We now set $r=|\mathfrak{c}_{X}\cap\phi(\mathfrak{c}_{X})|$. Our goal is to
construct a resolution of the diagonal bimodule $\Delta_{\mathcal{W}(X)}$ of
length $r$. We do this by the following geometric argument. First, observe
that the intersection points of $\mathfrak{c}_{X}$ with
$\phi(\mathfrak{c}_{X})$ are in bijection with the intersection points of the
skeleton of $(\overline{X}\times X,-\lambda\oplus\phi_{*}\lambda)$ with the
diagonal Lagrangian. Since $\mathfrak{c}_{X}\pitchfork\phi(\mathfrak{c}_{X})$,
the diagonal intersects the skeleton transversally in $r$ points. By applying
deep work of Chantraine–Dimitroglou-Rizell–Ghiggini–Golovko [cdgg] or
Ganatra–Pardon–Shende [gps2], one can then construct a resolution of the
diagonal by $r$ cocores of $\overline{X}\times X$. Morally, the idea is that
as one scales the diagonal Lagrangian by the Liouville flow, it converges to
cocores at each point where it intersects the skeleton. The upshot is that we
have resolved the diagonal Lagrangian by $r$ cocores in $\overline{X}\times
X$, which are just products of cocores in $X$. Finally, following [ganatra]
and [gps2], we can turn this resolution of the diagonal Lagrangian into a
resolution of the diagonal bimodule.

######  Example 1.6.

For $(T^{*}S^{n},\lambda)$, where $n\geq 2$ and $\lambda$ is homotopic to the
standard Morse–Bott form $pdq$, we will see that
$\operatorname{Rdim}\mathcal{W}(T^{*}S^{n})=1$ (say with
$\mathbb{Z}/2$-gradings and $\mathbb{Q}$-coefficients) and hence
$|\mathfrak{c}_{X}\cap\phi(\mathfrak{c}_{X})|\geq 2$. This lower bound is of
course sharp.

######  Example 1.7.

Let $G$ be a compact, simply-connected Lie group. Suppose that
$(T^{*}G,\lambda)$ is a Weinstein structure which is homotopic to
$(T^{*}G,pdq)$. Then we will see that
$\operatorname{Rdim}\mathcal{W}(T^{*}G)\geq\operatorname{rank}G$ and hence
$|\mathfrak{c}_{X}\cap\phi(\mathfrak{c}_{X})|\geq\operatorname{rank}G+1$.

######  Example 1.8.

Let us consider the case where $2c_{1}(X)=0$ so that both $SH^{\bullet}(X)$
and $\mathcal{W}(X)$ are $\mathbb{Z}$-graded. Following
[pomerleano2021intrinsic], an object $L_{0}\in\mathcal{W}(X)$ is called a
_homological section_ if the restriction of the closed-open map $\mathcal{CO}$
to the degree $0$ part of $SH^{\bullet}(X)$

(1.4) $\mathcal{CO}:SH^{0}(X)\to HW^{\bullet}(L_{0},L_{0})$

defines an isomorphism of algebras. If $\operatorname{Spec}(SH^{0}(X))$ is
smooth, [pomerleano2021intrinsic, Cor. 1.4] tells us $L_{0}$ is a split-
generator. Then we can conclude

(1.5)
$|\mathfrak{c}_{X}\cap\phi(\mathfrak{c}_{X})|\geq\operatorname{Rdim}\mathcal{W}(X)+1\geq\dim
SH^{0}(X)+1.$

In practice, a large class of such Weinstein manifolds arise from the
complement of the anti-canonical divisor of log Calabi–Yau varieties.

We now turn to 1.2. We have the following proposition:

######  Proposition 1.9 (= Proposition 6.6).

Let $X$ be a Liouville manifold with nonzero wrapped Fukaya category. Then we
have

(1.6) $\operatorname{Lef}_{w}(X)\geq\operatorname{Rdim}\mathcal{W}(X)+1.$

######  Corollary 1.10 (= Proposition 6.6 \+ Theorem 4.2).

If $R\subseteq SH^{0}(X)$ and $K\in\mathcal{W}(X)$ are as in Corollary 1.5,
then

(1.7) $\operatorname{Lef}_{w}(X)\geq\dim_{R}HW^{\bullet}(K,K)+1.$

The proof of Proposition 1.9 is straightforward and (at least implicitly)
well-known to experts: the Fukaya–Seidel category associated to a Lefschetz
fibration admits a semi-orthogonal decomposition of length equal to the number
of critical points of the fibration, which immediately implies the
corresponding upper bound for the Rouquier dimension. Nevertheless, from the
perspective of making progress on 1.2, Proposition 1.9 is quite useful when
combined with lower bounds for the Rouquier dimension obtained by other means,
such as Theorem 4.2.

######  Example 1.11.

We have $\operatorname{Lef}_{w}(T^{*}S^{n})=2$ for $n\geq 2$. If $G$ is a
simply-connected compact Lie group, then
$\operatorname{Lef}_{w}(T^{*}G)\geq\operatorname{rank}G+1$. If $X$ admits a
homological section as in Example 1.8, then $\operatorname{Lef}_{w}(X)\geq\dim
SH^{0}(X)+1$.

######  Example 1.12.

If $\mathcal{W}(X)$ is $\mathbb{Z}$-graded and $L\in\mathcal{W}(X)$ is
diffeomorphic to a $n$-torus, then $L$ is a _point-like object_. By work of
Elagin–Lunts [elagin-lunts-koszul], the existence of such an $L$ implies that
$\operatorname{Rdim}\mathcal{W}(X)\geq n$ (see Example 3.12). Hence
$\operatorname{Lef}_{w}(X)\geq n+1$ (similarly, Theorem 1.4 implies that
$|\mathfrak{c}_{X}\cap\phi(\mathfrak{c}_{X})|\geq n+1$).

As an application of Proposition 1.9, we prove that the natural complex analog
of Lazarev’s previously mentioned result
$``\operatorname{WMor}(X)=\operatorname{Mor}(X)"$ _fails_. This is a
consequence of the following corollary.

######  Corollary 1.13 (= Corollary 6.8).

There exists a Liouville manifold
$T^{*}S^{3}_{\operatorname{exotic}}=(T^{*}S^{3},\lambda)$ which is formally
isotopic to $(T^{*}S^{3},pdq)$, but where

(1.8)
$\operatorname{Lef}_{w}(T^{*}S^{3})\neq\operatorname{Lef}_{w}(T^{*}S^{3}_{\operatorname{exotic}}).$

The term “formally isotopic” should be interpreted in the context of the
h-principle: it means that $pdq$ and $\lambda$ are indistinguishable from the
perspective of differential topology. Corollary 1.13 provides the first
example of a pair of formally isotopic Weinstein manifolds with non-zero
middle homology which can be distinguished by
$\operatorname{Lef}_{w}(X)$.222For Weinstein manifolds with vanishing middle
homology, $\operatorname{Lef}_{w}(-)$ distinguishes exotic examples for
trivial reasons: for example, if $X=\mathbb{C}^{n}$ and $X^{\prime}$ is an
exotic $\mathbb{C}^{n}$, then $\operatorname{Lef}_{w}(X)=0$ but
$\operatorname{Lef}_{w}(X^{\prime})>0$ since otherwise $X^{\prime}$ would be
subcritical and hence standard. In particular, Corollary 1.13 implies that, in
contrast to $\operatorname{WMor}(-)$, the number $\operatorname{Lef}_{w}(-)$
is a truly symplectic invariant.

The proof of Corollary 1.13 uses a construction of Eliashberg–Ganatra–Lazarev
[e-g-l], who exhibited an exotic $T^{*}S^{3}_{\operatorname{exotic}}$ which
contains a regular Lagrangian $3$-torus. We will see that this implies that
$\operatorname{Rdim}\mathcal{W}(T^{*}S^{3}_{\operatorname{exotic}})=3$, and
hence $\operatorname{Lef}_{w}(T^{*}S^{3}_{\operatorname{exotic}})\geq 4$. In
contrast, it is known that $T^{*}S^{3}$ admits a Lefschetz fibration with $2$
critical points.

We turn finally to 1.3. We have the following proposition.

######  Proposition 1.14 (= Proposition 6.2).

Let
$i:(X_{0}^{\operatorname{in}},\lambda^{\operatorname{in}})\subseteq(X_{0},\lambda)$
be a Liouville embedding and suppose that
$(X_{0}^{\operatorname{in}},\lambda^{\operatorname{in}})$ and
$(X_{0}-i(X_{0}^{\operatorname{in}}),\lambda)$ are Weinstein up to
deformation. Then

(1.9)
$\operatorname{Rdim}\mathcal{W}(X^{\operatorname{in}})\leq\operatorname{Rdim}\mathcal{W}(X)$

where $X^{\operatorname{in}},X$ are the completions of
$X_{0}^{\operatorname{in}},X_{0}$ respectively.

Proposition 1.14 is a direct consequence of deep work of Ganatra–Pardon–Shende
[gps2, Sec. 8.3]. More precisely, they constructed a Viterbo restriction
functor between the wrapped Fukaya categories, and proved that it is a
localization for Weinstein cobordisms. Proposition 1.14 then follows from the
basic (and purely algebraic) fact that the Rouquier dimension cannot increase
under localization.

By combining various upper and lower bounds for
$\operatorname{Rdim}\mathcal{W}(-)$ discussed previously, it is easy to write
down examples of putative embeddings $X_{0}^{\operatorname{in}}\hookrightarrow
X_{0}$ which are obstructed by $\operatorname{Rdim}\mathcal{W}(-)$. For
instance:

######  Example 1.15.

Let $X$ be a Weinstein manifold of real dimension $2(n^{2}-1)$ admitting a
Lefschetz fibration with Weinstein fibers having $<n-2$ critical points. Then
$X$ does not contain $T^{*}SU(n)$ as a Weinstein subdomain.

There is of course nothing special about Example 1.15, and there is a plethora
of similar examples. In any case, it seems highly unlikely to the authors that
examples such as these could be handled using invariants already in the
literature. From this perspective, $\operatorname{Rdim}\mathcal{W}(-)$ is a
useful addition to the symplectic topologist’s toolkit of embedding
obstructions.

On the other hand, $\operatorname{Rdim}\mathcal{W}(-)$ has many blind spots.
The most obvious defect of $\operatorname{Rdim}\mathcal{W}(-)$ as an embedding
obstruction is that it outputs a very limited range of values: indeed, it
seems plausible that $\operatorname{Rdim}\mathcal{W}(X^{2n})\in[0,n]$ (see
Corollary 5.17 for a somewhat weaker statement).

######  Remark 1.16.

The _Orlov spectrum_ of a category is the set of all generation times shifted
down by $1$, and the Rouquier dimension is by definition the smallest element
in this set. It is tempting to view the elements of the Orlov spectrum of
$\mathcal{W}(X)$ as a hierarchy of “categorical capacities”, where the
Rouquier dimension corresponds to the lowest capacity. From this perspective,
one could hope to obtain more sensitive embedding obstructions by taking into
account the higher capacities. Unfortunately, this speculation comes with a
serious drawback: beyond a few exceptions, the Orlov spectrum seems fiendishly
difficult to compute!

###  1.3. Applications to algebraic geometry

Let $Y$ be a smooth $n$-dimensional quasi-projective scheme over a field and
let $D^{b}\operatorname{Coh}(Y)$ be the bounded derived category of coherent
sheaves on $Y$. Rouquier proved that one has
$n\leq\operatorname{Rdim}D^{b}\operatorname{Coh}(Y)\leq 2n.$ Orlov conjectured
that the lower bound is always an equality. More precisely:

######  Conjecture 1.17 (Orlov; see Conjecture $10$ in [orlov2009remarks]).

Let $Y$ be a smooth quasi-projective scheme of dimension $n$. Then

(1.10) $\operatorname{Rdim}D^{b}\operatorname{Coh}(Y)=n.$

Orlov’s conjecture has been studied by many authors and is now established for
the following list of examples. To the best of our knowledge, this list is
exhaustive.

  1. (1)

Smooth affine varieties of finite type, projective spaces and smooth
projective quadrics [rouquier, Prop. 7.18, Ex. 7.7, 7.8];

  2. (2)

All smooth projective curves [orlov2009remarks, Thm. 6];

  3. (3)

The following schemes defined over an algebraically-closed field of
characteristic $0$: del Pezzo surfaces with
$\operatorname{rk}\operatorname{Pic}(X)\leq 7$, Fano threefolds of type
$V_{5}$ and $V_{22}$, toric surfaces with nef anti-canonical divisor, toric
Deligne–Mumford stacks defined over $\mathbb{C}$ of dimension no more than $2$
or Picard number no more than $2$, Hirzebruch surfaces
[ballard2012hochschild];

  4. (4)

The product of varieties from the previous item [yang2016note];

  5. (5)

The product of two Fermat elliptic curves with the Fermat $K3$ surface
[ballard2014category, Thm. 1.6];

  6. (6)

Smooth toric Fano varieties of dimension $3$ or $4$ [ballard2019toric, Prop.
2.15, 2.16];

  7. (7)

Certain blow-ups of the projective space [pirozhkov2019rouquier, Thm. 4.1];

  8. (8)

Some weak del Pezzo surfaces [elagin-xu-zhang, Cor. 7.5];

We introduce a new method to prove Orlov’s conjecture for varieties which
admit Weinstein mirrors. More precisely, we prove:

######  Theorem 1.18.

Suppose that $Y$ is a variety over $\mathbb{C}$ of complex dimension $n$ which
admits a homological mirror given by a polarizable Weinstein pair. Then
$\operatorname{Rdim}D^{b}\operatorname{Coh}(Y)=n$ if $n\leq 3$. For $n\geq 4$,
we have $\operatorname{Rdim}D^{b}\operatorname{Coh}(Y)\leq 2n-3$.

(A Weinstein pair $(X,A)$ is said to be polarizable if $X$ admits a global
Lagrangian plane field $\xi$.) Theorem 1.18 implies the following new cases of
Orlov’s conjecture.

######  Corollary 1.19.

1.17 is true for all $3$-dimensional toric varieties over $\mathbb{C}$ and the
log Calabi–Yau surfaces in Example 5.19.

In fact, Theorem 1.18 shows that (1.10) holds for a large class of (possibly
singular) schemes and stacks over $\mathbb{C}$, see Section 5.4 for the list.
It also gives a new proof of some of the known cases of Orlov’s conjecture
listed above. Finally, Theorem 1.18 improves Rouquier’s upper bound above for
$\operatorname{Rdim}D^{b}\operatorname{Coh}(-)$ for many examples in
dimensions $n\geq 4$ (including all smooth toric varieties and DM stacks).

Our proof of Theorem 1.18 is entirely symplectic. As mentioned above, it is
already known that $\operatorname{Rdim}D^{b}\operatorname{Coh}(Y)\geq n$.
Hence, we only need to prove the claimed upper bounds. To do this, we apply
homological mirror symmetry. This transforms Theorem 1.18 into a statement
about the Rouquier dimension of wrapped Fukaya categories of Weinstein pairs;
namely, it is enough to prove that the Rouquier dimension of a (polarizable)
Weinstein pair of real dimension $2n$ is bounded above by $n$ when $n\leq 3$
and by $2n-3$ when $n\geq 4$.

The proof of this upper bound will be explained shortly (see Section 1.4.2).
The main inputs are the sectorial descent property for wrapped Fukaya
categories [gps2] and the arborealization theorem of Álvarez-
Gavela–Eliashberg–Nadler [AGEN3]. The stronger bound in Theorem 1.18 when
$n\leq 3$ ultimately comes from the fact that arboreal singularities of
dimension $\leq 3$ are automatically of Dynkin type (this fails when $n\geq
4$). However, it may be possible to improve the bounds in Theorem 1.18 in high
dimensions when one has an explicit description of the skeleton of the mirror.

###  1.4. Computational techniques

We summarize two key techniques for obtaining lower and upper bounds on the
Rouquier dimension of wrapped Fukaya categories. Both of these techniques have
a central role in this paper.

####  1.4.1. Lower bounds: commutative ring actions on wrapped Fukaya
categories

Let $\mathcal{C}$ be a pre-triangulated $A_{\infty}$ category and let
$\mathcal{T}=H^{0}(\mathcal{C})$ be its homotopy category. Let
$Z^{\bullet}(\mathcal{T})$ be the (graded-commutative) ring of graded natural
transformations of the identity on $\mathcal{T}$. Given a graded-commutative
ring $R$, a _central action_ of $R$ on the triangulated category $\mathcal{T}$
is by definition a morphism of graded-commutative rings $R\to
Z^{\bullet}(\mathcal{T})$. Concretely, this is the data of a morphism
$R\to\hom_{H(\mathcal{C})}^{\bullet}(K,K)$ for each object $K\in\mathcal{C}$
such that the induced left and right module structures on
$\hom_{H(\mathcal{C})}^{\bullet}(K,L)$ coincide up to sign.

The upshot is that the morphism spaces of objects of $\mathcal{T}$ define
modules over $R$. One can then study the triangulated category $\mathcal{T}$
using tools from (graded-)commutative algebra. This perspective leads to a
very rich theory; see e.g. [benson2008local].

The workhorse result for lower-bounding the Rouquier dimension of wrapped
Fukaya categories in this paper is the following:

######  Theorem 1.20 (= Theorem 4.2).

Let $\mathcal{T}$ be a $\mathbb{Z}/m$-graded triangulated category $(1\leq
m\leq\infty)$ and let $R$ be a finite type $k$-algebra acting centrally on
$\mathcal{T}$. Suppose that there exists a (split-)generator $G\in\mathcal{T}$
such that $\operatorname{Hom}_{\mathcal{T}}^{\bullet}(G,G)$ is a Noetherian
$R$-module. Then
$\operatorname{Rdim}\mathcal{T}\geq\operatorname{dim}_{R}\operatorname{Hom}_{\mathcal{T}}^{\bullet}(G,G).$

Theorem 4.2 can be viewed as an “affine” analog of a beautiful theorem of
Bergh–Iyengar–Krause–Oppermann [b-i-k-o, Thm. 4.2]. In fact, (as we will see)
one could directly apply [b-i-k-o, Thm. 4.2] and still obtain useful lower
bounds for certain examples. However, for our purposes, [b-i-k-o, Thm. 4.2]
suffers from two limitations. First of all, it is essentially never sharp
(there is a pesky $-1$ which comes, ultimately, from working in the
“projective” rather than “affine” setting). Secondly, it requires rather
strong hypotheses on $R$ and $\operatorname{Hom}_{\mathcal{T}}^{\bullet}(G,G)$
which do not hold in many cases of interest to us (for instance, Example 1.8
and Example 4.18 would fail).333We note that there is a typo in the definition
of the projective dimension in [b-i-k-o, Sec. 2] which propagates. As a
result, one needs to add an assumption in the statement of [b-i-k-o, Thm.
4.2], the most natural (to us) being that $R^{0}=k$. In contrast, Theorem 4.2
is sharp for many examples (such as cotangent bundles of $n$-spheres and
$n$-tori), and more widely applicable.

Our general approach to proving Theorem 4.2 via _Koszul objects_ follows [b-i-
k-o]; however, the input from graded-commutative algebra is replaced by new
arguments from (ungraded) commutative algebra. We crucially use the hypothesis
that $R$ is of finite type. This hypothesis is rather severe from the
representation theoretic perspective taken in [b-i-k-o]. However, from the
symplectic perspective which we adopt in this paper, this assumption is rather
mild.

Of course, Theorem 1.20 is useless without a good supply of central actions.
Happily, for any pre-triangulated $A_{\infty}$ category $\mathcal{C}$, there
is a canonical central action of the Hochschild cohomology ring
$HH^{\bullet}(\mathcal{C})$ on $\mathcal{T}=H^{0}(\mathcal{C})$ called the
_characteristic morphism_. Given any object $K\in\mathcal{C}$, this is just
the map $HH^{\bullet}(\mathcal{C})\to\hom^{\bullet}_{H(\mathcal{C})}(K,K)$
that projects a Hochschild cochain onto its length zero part. In general, the
characteristic morphism is neither injective nor surjective.

In the case which is relevant to us, namely when
$\mathcal{T}=H^{0}(\operatorname{Perf}\mathcal{W}(X))$ and $X$ is Weinstein,
the characteristic morphism is known to admit a concrete description in terms
of the familiar _closed-open map_. To explain this a little, recall that the
closed-open map defines a ring morphism $SH^{\bullet}(X)\to
HW^{\bullet}(L,L)$, for any object $L\in\mathcal{W}(X)$. On the other hand, if
$X$ is Weinstein, there is an isomorphism of graded-commutative rings
$HH^{\bullet}(\mathcal{W}(X))=SH^{\bullet}(X)$ [cdgg]. Now the point is simply
that the obvious diagram formed by these maps commutes. As a result, we obtain
the following corollary which was mentioned many times above.

######  Corollary 1.21 (= Corollary 4.17).

Suppose that $SH^{0}(X)$ admits a subring $R$ of finite-type over $k$, and
suppose there exists a (split-)generator $K\in\mathcal{W}(X)$ such that
$HW^{\bullet}(K,K)$ is Noetherian over $R$, where $R$ acts via the closed-open
map. Then

(1.11) $\operatorname{Rdim}\mathcal{W}(X)\geq\dim_{R}HW^{\bullet}(K,K).$

Corollary 1.21 works with any choice of gradings, but $SH^{\bullet}(X)$ will
typically be $\mathbb{Z}$-graded or $\mathbb{Z}/2$-graded, depending on the
application. The flexibility to choose different gradings is convenient in
practice, since of course $SH^{0}(X)$ will admit different subrings depending
on the choice of grading.

####  1.4.2. Upper bounds: arborealization

According to the monumental work of Ganatra–Pardon–Shende [gps2], wrapped
Fukaya categories of Liouville manifolds satisfy “Weinstein sectorial
descent”. Essentially, this means that if we (suitably) cover a Liouville
manifold $X$ by Weinstein sectors $\\{X_{\alpha}\\}_{\alpha\in I}$, then
$\mathcal{W}(X)$ can be expressed as the homotopy colimit of the wrapped
Fukaya categories of the form $\mathcal{W}(\cap_{\alpha\in J}X_{\alpha})$
(where $J$ ranges over all non-empty subsets of $I$ and the homotopy colimit
is indexed by the obvious diagram of inclusions).

The Rouquier dimension of a homotopy colimit can be bounded from above by the
Rouquier dimension of its individual pieces. This bound depends on (a) the
“depth” of the diagram (i.e. the longest chain of arrows) and (b) the Rouquier
dimension of the individual pieces. Thus, we want to control both of these
quantities.

If $X=T^{*}M$, for $M$ a closed $n$-manifold, we can construct a cover of
depth $n+1$ by the following procedure: first triangulate $M$, then set
$X_{\alpha}:=T^{*}\operatorname{star}(v_{\alpha})$ where $v_{\alpha}$ ranges
over the vertices of the triangulation. Since
$T^{*}\operatorname{star}(v_{\alpha})=T^{*}D^{n}$ after appropriate smoothing,
the Rouquier dimension of $T^{*}\operatorname{star}(v_{\alpha})$ is zero. As
we will explain, the existence of such a cover implies that
$\operatorname{Rdim}\mathcal{W}(T^{*}M)\leq n$, which is optimal in general.
(We warn the reader that we have overlooked some technical points; for
instance, stars are open so their cotangent bundles are technically not even
Liouville sectors).

We would like to mimic this argument for a general Weinstein manifold $X$.
Thus, we want to find a sectorial cover $\\{X_{\alpha}\\}$ of minimal depth,
such that the Rouquier dimension of each piece is as small as possible. This
is where arborealization enters the picture.

The arborealization program, which was initiated by Nadler
[nadler2017arboreal], aims to prove that any Weinstein manifold/pair can be
homotoped so that the singularities of its skeleton are in a distinguished
class (the so-called “arboreal singularities”). A major portion of the
arborealization program has now been implemented in a series of papers by
Álvarez-Gavela–Eliashberg–Nadler [AGEN1, AGEN2, AGEN3]. In dimension $4$, see
also the work of Starkston [starkston2018arboreal], though we will follow
[AGEN1, AGEN2, AGEN3] closely in this paper. If $X_{\alpha}$ is a “standard
arboreal sector”, then it turns out that
$\operatorname{Rdim}\mathcal{W}(X_{\alpha})=0$ if the singularity is of so-
called “Dynkin type” and $\operatorname{Rdim}\mathcal{W}(X_{\alpha})=1$
otherwise. Thus, _the Rouquier dimension of arboreal sectors is “small”._

Our strategy is therefore as follows: fix a Weinstein manifold/pair $X$ of
dimension $2n$ and apply the arborealization theorem to make its skeleton
arboreal; then triangulate the skeleton; finally, construct a good cover by
taking stars of the vertices. If all the singularities are of Dynkin type
(this is automatic if $2n\leq 6$), then one finds again that
$\operatorname{Rdim}\mathcal{W}(X)\leq n$, like for cotangent bundles. In
general, one gets a worse bound, namely $\operatorname{Rdim}\mathcal{W}(X)\leq
2n-3$.

For purely technical reasons, we have chosen to implement this argument in the
framework of microlocal sheaf theory. We then use the main result of [gps3] to
pass back to wrapped Fukaya categories. Microlocal sheaf categories satisfy
the same type of sectorial descent property as wrapped Fukaya categories, but
they are easier to work with, at least in our setting. (For example, to check
that a cover is “sectorial”, one needs to verify some technical geometric
conditions; in contrast, the notion of an open cover in microlocal sheaf
theory is purely topological). We certainly expect that the above outline can
be made to work without appealing to [gps3] and microlocal sheaves, but
perhaps at considerable technical expense.

###  1.5. Acknowledgements

We are indebted to Daniel Álvarez-Gavela for answering countless questions
about arborealization, Vivek Shende for help with microlocal sheaves and
homological mirror symmetry of toric varieties, Srikanth Iyengar for
discussions and comments related to Section 4.2, Y. Barış Kartal for telling
us the proof of Proposition 6.6, and John Pardon for many helpful
conversations about various aspects of the paper. In addition, we would like
to thank Denis Auroux, Petter Bergh, Georgios Dimitroglou Rizell, Sheel
Ganatra, Wahei Hara, Oleg Lazarev, David Nadler, Amnon Neeman and Dmitri Orlov
for answering our questions at various points in this project.

SB would like to express deep gratitude to his advisor John Pardon for
constant encouragement and support. LC learned about 1.1 from Zhengyi Zhou and
would like to thank him for many enjoyable and informative discussions. LC
would also like to thank Y. Barış Kartal for teaching him much of what he
knows about $A_{\infty}$ categories.

Part of this work was completed while LC was a Member of the Institute of
Advanced Study, and supported there by the National Science Foundation under
Grant No. DMS-1926686. LC was also partially supported by Simons Foundation
grant 385573 (Simons Collaboration on Homological Mirror Symmetry).

##  2\. Category theory

###  2.1. Notation and conventions

Unless otherwise indicated, $k$ denotes a field.

Let $\mathcal{T}$ be a category. We will routinely abuse notation by writing
$K\in\mathcal{T}$ to mean that $K$ is an object of $\mathcal{T}$. Similarly,
we write $\mathcal{I}\subseteq\mathcal{T}$ to mean that $\mathcal{I}$ is a
full subcategory of $\mathcal{T}$. Given two full subcategories
$\mathcal{I},\mathcal{J}$ of $\mathcal{T}$, we let
$\mathcal{I}\cup\mathcal{J}$ be the full subcategory with objects the union of
the objects in $\mathcal{I}$ and those in $\mathcal{J}$.

A full subcategory $\mathcal{I}\subseteq\mathcal{T}$ is said to be _strictly
full_ if is closed under isomorphisms. We will sometimes implicitly identify a
set of objects with the corresponding full subcategory (resp. identify a set
of isomorphism classes of objects with the corresponding strictly full
subcategory).

Given an object $K\in\mathcal{T}$, a _summand_ of $K$ is a triple $(Z,k,r)$
consists of an object $Z\in\mathcal{T}$ along with maps
$r\in\hom(Z,K),k\in\hom(K,Z)$ such that $k\circ r=\operatorname{id}$ and
$r\circ k$ is an idempotent. An idempotent $e\in\hom(K,K)$ is said to _split_
if $e=r\circ k$ for some summand $(Z,k,r)$. A category is said to be
_idempotent complete_ (or Karoubi complete) if all idempotents split.

###  2.2. Triangulated categories

A triangulated category $\mathcal{T}=(\mathcal{T},\Sigma,\Delta)$ is an
additive category $\mathcal{T}$ equipped with an autoequivalence
$\Sigma:\mathcal{T}\to\mathcal{T}$, called the _shift functor_ , and a class
of distinguished triangles $\Delta$. This data is required to satisfy certain
axioms. A standard reference for triangulated categories is [neeman].

We say that a triangulated category $\mathcal{T}$ is $\mathbb{Z}/m$-graded
(for $m\in\mathbb{N}_{+}\cup\\{\infty\\}$ and $\mathbb{Z}/\infty:=\mathbb{Z}$)
if $\Sigma^{m}=\operatorname{id}$. If $\mathcal{T}$ is $\mathbb{Z}/m$-graded,
we define
$\operatorname{Hom}_{\mathcal{T}}^{\bullet}(X,Y):=\oplus_{i=1}^{m}\operatorname{Hom}_{\mathcal{T}}(X,\Sigma^{i}Y)$.

Given full subcategories $\mathcal{I}_{0},\mathcal{I}_{1}$ of $\mathcal{T}$,
let $\mathcal{I}_{0}*\mathcal{I}_{1}$ be the strictly full subcategory which
is uniquely characterized by the following property: an object
$K\in\mathcal{T}$ is contained in $\mathcal{I}_{0}*\mathcal{I}_{1}$ if there
exists a distinguished triangle $K_{0}\to K\to K_{1}\to$ with
$K_{0}\in\mathcal{I}_{0}$ and $K_{1}\in\mathcal{I}_{1}$. It is a
straightforward consequence of the octahedral axiom for triangulated
categories that the operation
$(\mathcal{I}_{0},\mathcal{I}_{1})\mapsto\mathcal{I}_{0}*\mathcal{I}_{1}$ is
associative (see e.g. [b-b-d, Lem. 1.3.10]).

Given $\mathcal{I}\subseteq\mathcal{T}$ a full subcategory, let
$[\mathcal{I}]$ be the smallest strictly full subcategory of $\mathcal{T}$
containing $\mathcal{I}$ and closed under finite direct sums and shifts. Let
$\langle\mathcal{I}\rangle$ be the smallest strictly full subcategory of
$\mathcal{T}$ closed under finite direct sums, shifts and taking summands. Let
$\operatorname{ads}(\mathcal{I})$ be the smallest strictly full subcategory of
$\mathcal{T}$ containing $\mathcal{I}$ and closed under _arbitrary_ direct
sums and shifts.

We inductively define $[\mathcal{I}]_{n}=[\mathcal{I}]_{n-1}*[\mathcal{I}]$
and
$\langle\mathcal{I}\rangle_{n}=\langle\langle\mathcal{I}\rangle_{n-1}*\langle\mathcal{I}\rangle\rangle$.
We set $[\mathcal{I}]_{\infty}:=\cup_{n\in\mathbb{N}}[\mathcal{I}]_{n}$ and
$\langle\mathcal{I}\rangle_{\infty}=\cup_{n\in\mathbb{N}}\langle\mathcal{I}\rangle_{n}$.

Finally, we let $\mathcal{I}_{sd}$ be the smallest strictly full subcategory
containing all summands of $\mathcal{I}$ (this is often called the “Karoubi
completion”). A full subcategory of a triangulated category is called _thick_
if it is closed under taking summands.

######  Remark 2.1.

Some sources (e.g. [elagin-lunts]) adopt a different convention in which the
indices in $\langle-\rangle_{n}$ and $[-]_{n}$ are shifted down by $1$.

######  Lemma 2.2.

The following properties hold for all $n\in\mathbb{N}\cup\\{\infty\\}$ and any
$\mathcal{I}\subseteq\mathcal{T}$.

  1. (1)

$[\mathcal{I}]=[[\mathcal{I}]]$, $\mathcal{I}_{sd}=(\mathcal{I}_{sd})_{sd}$
and $[\mathcal{I}_{sd}]\subseteq[\mathcal{I}]_{sd}$

  2. (2)

$\langle\mathcal{I}\rangle_{n}=([\mathcal{I}]_{n})_{sd}$

  3. (3)

$[[\mathcal{I}]_{n}]=[\mathcal{I}]_{n}$ and
$\langle\langle\mathcal{I}\rangle_{n}\rangle=\langle\mathcal{I}\rangle_{n}$

  4. (4)

Given objects $G,H\in\mathcal{T}$, we have $\langle
G\rangle_{n}\subseteq\langle G\oplus H\rangle_{n}$ and $\langle
H\rangle_{n}\subseteq\langle G\oplus H\rangle_{n}$.

###### Proof.

(1) is obvious. (2) can be established by following the argument of [seidel-
book, Lem. 4.1] (and is stated without proof in [rouquier, Rmk. 3.1]). (3) is
straightforward using (2). (4) is obvious. ∎

######  Lemma 2.3.

Let $\mathcal{T}$ be a triangulated category. Let $\mathcal{I},\mathcal{J}$ be
thick full triangulated subcategories such that
$\mathcal{T}=\langle\mathcal{I}\cup\mathcal{J}\rangle_{\infty}$ and
$\hom(K,L)=0$ for all $K\in\mathcal{J}$ and $L\in\mathcal{I}$. Then
$\mathcal{T}=\langle\mathcal{J}*\mathcal{I}\rangle$.

###### Proof.

The proof is standard; see e.g. [bondal, Lem. 3.1]. Since
$\langle\mathcal{J}*\mathcal{I}\rangle$ contains $\mathcal{I}$ and
$\mathcal{J}$, it is enough to verify that
$\langle\mathcal{J}*\mathcal{I}\rangle$ is closed under cones. This is a
straightforward check using [b-b-d, Prop. 1.1.11]. ∎

######  Lemma 2.4.

Let $\mathcal{T}$ be a triangulated category. Let
$\mathcal{I}_{1},\dots,\mathcal{I}_{r}$ be full triangulated subcategories
which are mutually orthogonal, in the sense that $\hom(K,L)=0$ for all
$K\in\mathcal{I}_{k}$ and $L\in\mathcal{I}_{l}$ if $k\neq l$. Then
$\langle\mathcal{I}_{1}\cup\dots\cup\mathcal{I}_{r}\rangle_{n}=\langle\langle\mathcal{I}_{1}\rangle_{n}\cup\dots\cup\langle\mathcal{I}_{r}\rangle_{n}\rangle$
for all $n\in\mathbb{N}\cup\\{\infty\\}$. ∎

The following lemma is well-known; see e.g. [rouquier, Cor. 3.14].

######  Lemma 2.5.

Let $\mathcal{T}$ be a triangulated category and let $\mathcal{T}^{c}$ denote
the full triangulated subcategory of compact objects. Then given
$\mathcal{I}\subseteq\mathcal{T}^{c}$, we have
$\mathcal{T}^{c}\cap\langle\operatorname{ads}(\mathcal{I})\rangle_{d}=\langle\mathcal{I}\rangle_{d}$.
∎

###  2.3. $A_{\infty}$ categories

####  2.3.1. Basic definitions

Unless otherwise indicated, we follow the conventions of [seidel-book] and
[seidel-cat-dynamics] when discussing $A_{\infty}$ categories. One exception
is that all $A_{\infty}$ categories in this paper are assumed to be _strictly_
unital. We will work mostly with $\mathbb{Z}/2$-graded or $\mathbb{Z}$-graded
$A_{\infty}$ categories, which are always assumed to be defined over a field
$k$. However, the categorical reasoning in this paper applies with purely
superficial changes to the setting of $\mathbb{Z}/n$-graded $A_{\infty}$
categories for any $n\in\mathbb{N}_{+}$ (the case $n=1$ corresponds to an
ungraded $A_{\infty}$ category).

Given $A_{\infty}$ categories $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$, an
$(\mathcal{C}_{1},\mathcal{C}_{2})$-bimodule is an $A_{\infty}$ functor
$\mathcal{C}_{1}^{\operatorname{op}}\times\mathcal{C}_{2}\to\operatorname{Ch}k$,
where $\operatorname{Ch}k$ is the dg category of chain complexes of $k$-vector
spaces. Note that $(\mathcal{C}_{1},\mathcal{C}_{2})$-bimodules form a dg
category, which is denoted by
$(\mathcal{C}_{1},\mathcal{C}_{2})-\operatorname{mod}$. As usual, two such
bimodules are said to be quasi-isomorphic if they are isomorphic in the
homotopy category
$H^{0}((\mathcal{C}_{1},\mathcal{C}_{2})-\operatorname{mod})$. Let
$\Delta_{\mathcal{C}}$ be the diagonal bimodule over an $A_{\infty}$ category
$\mathcal{C}$. A right (resp. left) module over $\mathcal{C}$ is a
$(k\times\mathcal{C})$-bimodule (resp. a $(\mathcal{C}\times k)$-bimodule).
The right (resp. left) modules over $\mathcal{C}$ form a dg category
$\operatorname{Mod}\mathcal{C}$ (resp.
$\operatorname{Mod}\mathcal{C}^{\operatorname{op}}$).

Let $\\{\mathcal{C}_{i}\\}_{i=1,2,3}$ be $A_{\infty}$ categories. Given a
$(\mathcal{C}_{1},\mathcal{C}_{2})$-bimodule $\mathcal{P}$ and a
$(\mathcal{C}_{2},\mathcal{C}_{3})$-bimodule $\mathcal{Q}$, we can form their
convolution (bimodule tensor product)
$\mathcal{P}\otimes_{\mathcal{C}_{2}}\mathcal{Q}$. Using the same notations,
the convolution has the following properties:

  * •

it is strictly associative;

  * •

we have quasi-isomorphisms between bimodules
$\Delta_{\mathcal{C}_{1}}\otimes_{\mathcal{C}_{1}}\mathcal{P}\to\mathcal{P}$
and
$\mathcal{P}\otimes_{\mathcal{C}_{2}}\Delta_{\mathcal{C}_{2}}\to\mathcal{P}$.

  * •

the convolution $-\otimes_{\mathcal{C}_{2}}\mathcal{Q}$ induces a dg functor
$(\mathcal{C}_{1},\mathcal{C}_{2})-\operatorname{mod}$ to
$(\mathcal{C}_{1},\mathcal{C}_{3})-\operatorname{mod}$. Similarly for
$\mathcal{Q}\otimes_{\mathcal{C}_{3}}-$.

There is a canonical Yoneda embedding
$\mathcal{C}\to\operatorname{Mod}\mathcal{C}$. Let
$\operatorname{Tw}\mathcal{C}\subseteq\operatorname{Mod}\mathcal{C}$ be the
closure of the image of the Yoneda embedding under taking mapping cones. Let
$\operatorname{Perf}\mathcal{C}\subseteq\operatorname{Mod}\mathcal{C}$ be the
closure of the image of the Yoneda embedding under taking mapping cones and
summands. Note that we have

(2.1) $\operatorname{Perf}\mathcal{C}=(\operatorname{Mod}\mathcal{C})^{c},$

where $(-)^{c}$ means taking compact objects; see [gps3, (A.2) and Prop. A.3].

An $A_{\infty}$ category $\mathcal{C}$ is said to be _pre-triangulated_ if the
canonical embedding $\mathcal{C}\hookrightarrow\operatorname{Tw}\mathcal{C}$
is a quasi-equivalence. The homotopy category $H^{0}(\mathcal{C})$ of a pre-
triangulated $A_{\infty}$ category is a triangulated category.

Given an $A_{\infty}$ category $\mathcal{C}$ and a set of objects
$\mathcal{A}$, we can form the _quotient_ $A_{\infty}$ category
$\mathcal{C}/\mathcal{A}$ (see [lyubashenko-ovsienko, drinfeld]), which comes
equipped with a canonical map $q:\mathcal{C}\to\mathcal{C}/\mathcal{A}$.
Suppose $\mathcal{B}$ is another $A_{\infty}$ category. Given a functor
$\mathcal{C}\to\mathcal{B}$ which sends $\mathcal{A}$ to acyclic objects,
there is an induced functor $\mathcal{C}/\mathcal{A}\to\mathcal{B}$ (see
[gps3, Sec. A.7]). We note that the canonical map
$(\operatorname{Tw}\mathcal{C})/\mathcal{A}\to\operatorname{Tw}(\mathcal{C}/\mathcal{A})$
is a quasi-equivalence, and the canonical map
$\operatorname{Perf}\mathcal{C}/\mathcal{A}\to\operatorname{Perf}(\mathcal{C}/\mathcal{A})$
is a Morita equivalence (it is not in general a quasi-equivalence).

####  2.3.2. Equivalent constructions of the derived category of a (dg)
$k$-algebra

In the literature, one encounters various definitions of the derived category
of a (dg) $k$-algebra. We therefore collect some standard facts which will be
implicitly assumed in the sequel.

######  Lemma 2.6.

Let $\mathcal{A}$ be a dg $k$-algebra. The following constructions of the
derived category $D(\mathcal{A})$ produce equivalent triangulated categories.

  1. (1)

Consider the quotient (in the sense of Drinfeld [drinfeld]) of the dg category
of dg $\mathcal{A}$-modules by the dg subcategory of acyclic dg
$\mathcal{A}$-modules. Then pass to $H^{0}(-)$.

  2. (2)

View $\mathcal{A}$ as an $A_{\infty}$ algebra and consider
$\operatorname{Mod}\mathcal{A}$, the $A_{\infty}$ category of $A_{\infty}$
$\mathcal{A}$-modules. Now pass to $H^{0}(-)$.

∎

######  Lemma 2.7.

Let $A$ be a $k$-algebra. The following constructions of the derived category
$D(A)$ produce equivalent triangulated categories.

  1. (1)

Consider the abelian category of modules over $A$ and let $K(A)$ be the
homotopy category. Then take the quotient (in the sense of Verdier) of $K(A)$
by the subcategory of acyclic complexes.

  2. (2)

View $A$ as a dg algebra concentrated in degree zero with the trivial
differential. Now apply either one of the equivalent constructions of Lemma
2.6.

∎

In contrast to Lemma 2.7, if $A$ is a graded $k$ algebra, it is in general
_not_ the case that the derived category of the abelian category of graded
$A$-modules coincides with the derived category of dg $A$-modules.

######  Remark 2.8.

Let $A$ be a $k$-algebra. Viewing $A$ as a dg $k$-algebra, we can consider the
$A_{\infty}$ category $\operatorname{Perf}A$. Then
$H^{0}(\operatorname{Perf}A)\subseteq H^{0}(\operatorname{Mod}A)=D(A)$ is
equivalent to the smallest idempotent complete triangulated subcategory of
$D(A)$ containing $A$ (where $A$ is viewed as a module over itself). It is
shown in [stacks-project, Prop. 07LT] that the objects of this latter category
are precisely the perfect complexes, i.e. the complexes of $A$-modules quasi-
isomorphic to a bounded complex of finite projective $A$-modules.

###  2.4. The Rouquier dimension

In his remarkable paper [rouquier], Rouquier introduced the following notion
of dimension for triangulated categories.

######  Definition 2.9.

The _Rouquier dimension_ of a triangulated category $\mathcal{T}$ is the
smallest $n\in\mathbb{N}\cup\\{\infty\\}$ such that there exists an object
$G\in\mathcal{T}$ with $\mathcal{T}=\langle G\rangle_{n+1}$. We denote the
Rouquier dimension of $\mathcal{T}$ by $\operatorname{Rdim}\mathcal{T}$.

The Rouquier dimension of an $A_{\infty}$ category $\mathcal{C}$ is defined to
be the Rouquier dimension of $H^{0}(\operatorname{Perf}\mathcal{C})$ and is
denoted by $\operatorname{Rdim}\mathcal{C}$.

We now collect some standard properties of the Rouquier dimension which will
be needed in the sequel. The following lemma is a useful inequality for the
Rouquier dimension of semi-orthogonal decompositions.

######  Lemma 2.10.

Let $\mathcal{T}$ be a triangulated category. Let
$\mathcal{I}_{1},\dots,\mathcal{I}_{m}$ be thick full triangulated
subcategories satisfying the following conditions:

  * (i)

for any $j>i$, we have $\hom(K,L)=0$ for all $K\in\mathcal{I}_{j}$ and
$L\in\mathcal{I}_{i}$;

  * (ii)

$\langle\mathcal{I}_{1}\cup\dots\cup\mathcal{I}_{m}\rangle_{\infty}=\mathcal{T}$.

Then

(2.2)
$\operatorname{Rdim}\mathcal{T}\leq\sum_{l=1}^{m}(\operatorname{Rdim}\mathcal{I}_{l}+1)-1.$

(2.2) is in general far from sharp, as can be seen by taking for instance
$\mathcal{I}_{2},\dots,\mathcal{I}_{m}=0$.

###### Proof.

Note that
$\mathcal{T}=\langle\mathcal{I}_{1}\cup\dots\cup\mathcal{I}_{m}\rangle_{\infty}=\langle\langle\mathcal{I}_{1}\cup\dots\cup\mathcal{I}_{m-1}\rangle_{\infty}\cup\mathcal{I}_{m}\rangle_{\infty}$.
Hence it is sufficient to treat the case where $m=2$.

Suppose $\operatorname{Rdim}\mathcal{I}_{1}=n_{1}$ and
$\operatorname{Rdim}\mathcal{I}_{2}=n_{2}$. This means that there exist
objects $G\in\mathcal{I}_{1}$ and $H\in\mathcal{I}_{2}$ such that
$\mathcal{I}_{1}=\langle G\rangle_{n_{1}+1}$ and $\mathcal{I}_{2}=\langle
H\rangle_{n_{2}+1}$.

We now have, by Lemma 2.3, $\mathcal{T}=\langle\langle
H\rangle_{n_{2}+1}*\langle G\rangle_{n_{1}+1}\rangle\subseteq\langle\langle
G\oplus H\rangle_{n_{2}+1}*\langle G\oplus
H\rangle_{n_{1}+1}\rangle=\langle([G\oplus H]_{n_{2}+1})_{sd}*([G\oplus
H]_{n_{1}+1})_{sd}\rangle\subseteq\langle([G\oplus H]_{n_{2}+1}*[G\oplus
H]_{n_{1}+1})_{sd}\rangle=\langle([G\oplus
H]_{n_{1}+n_{2}+2})_{sd}\rangle=\langle\langle G\oplus
H\rangle_{n_{1}+n_{2}+2}\rangle$. ∎

The following statements show that the Rouquier dimension is non-increasing
under quotients of categories.

######  Lemma 2.11 (Lem. 3.4 in [rouquier]).

Let $\mathcal{T}$ and $\mathcal{T}^{\prime}$ be triangulated categories. If
there exists an exact functor $\mathcal{T}\to\mathcal{T}^{\prime}$ with dense
image (i.e. every object in $\mathcal{T}^{\prime}$ is isomorphic to a summand
of an object in the image), then
$\operatorname{Rdim}\mathcal{T}^{\prime}\leq\operatorname{Rdim}\mathcal{T}$. ∎

######  Corollary 2.12.

Let $\mathcal{C}$ be an $A_{\infty}$ category and let $\mathcal{A}$ denote a
set of objects. Then
$\operatorname{Rdim}\mathcal{C}/\mathcal{A}\leq\operatorname{Rdim}\mathcal{C}$.
∎

When a triangulated category comes from an $A_{\infty}$ category, the
following lemma shows that the Rouquier dimension is bounded above by the
length of the shortest resolution of the diagonal bimodule.

######  Lemma 2.13.

Let $\mathcal{C}$ be an $A_{\infty}$ category and let $\Delta_{\mathcal{C}}$
denote the diagonal bimodule. Suppose there exists a resolution of length
$\ell\in\mathbb{N}_{+}$

(2.3) $0=\mathcal{M}_{0}\to\mathcal{M}_{1}\to\dots\to\mathcal{M}_{\ell}$

of $(\mathcal{C},\mathcal{C}$)-bimodules where

  * •

$\Delta_{\mathcal{C}}$ is a summand of $\mathcal{M}_{\ell}$;

  * •

there exists perfect modules
$\mathcal{P}^{r}_{i}\in\operatorname{Perf}\mathcal{C}$ and
$\mathcal{P}^{l}_{i}\in\operatorname{Perf}\mathcal{C}^{\operatorname{op}}$
such that
$\operatorname{cone}(\mathcal{M}_{i-1}\to\mathcal{M}_{i})\in[\mathcal{P}^{l}_{i}\otimes\mathcal{P}^{r}_{i}]$
in $H^{0}((\mathcal{C},\mathcal{C})-\operatorname{mod})$.

Then $\operatorname{Rdim}\mathcal{C}\leq\ell-1$.

###### Proof.

Given $Q\in\operatorname{Perf}\mathcal{C}$, we convolve $\mathcal{Q}$ with the
above resolution (i.e. we apply the functor
$\mathcal{Q}\otimes_{\mathcal{C}}-$). Note that
$\mathcal{Q}\otimes_{\mathcal{C}}\mathcal{P}^{l}_{i}\otimes\mathcal{P}^{r}_{i}\simeq
H^{\bullet}(\mathcal{Q}\otimes_{\mathcal{C}}\mathcal{P}^{l}_{i})\otimes\mathcal{P}^{r}_{i}$.
Writing
$C_{i}:=H^{\bullet}(\mathcal{Q}\otimes_{\mathcal{C}}\mathcal{P}^{l}_{i})$, it
follows that
$\mathcal{Q}\in\langle\oplus_{i=1}^{\ell}(C_{i}\otimes\mathcal{P}^{r}_{i})\rangle_{\ell}\subseteq\langle\operatorname{ads}(\mathcal{G})\rangle_{\ell}$,
where $\mathcal{G}=\oplus_{i=1}^{\ell}\mathcal{P}^{r}_{i}$. Since
$\mathcal{Q}$ and $\mathcal{G}$ are compact objects in
$\operatorname{Mod}\mathcal{C}$ (see (2.1)), this implies that
$\mathcal{Q}\in\langle\mathcal{G}\rangle_{\ell}$ by Lemma 2.5. ∎

###  2.5. Homotopy colimits

Given a diagram of $A_{\infty}$ categories
$\\{\mathcal{C}_{\sigma}\\}_{\sigma\in\Sigma}$ indexed by a poset $\Sigma$,
one can form the _Grothendieck construction_ (or semiorthogonal gluing)

(2.4) $\operatorname{Groth}_{\sigma\in\Sigma}\mathcal{C}_{\sigma}.$

This is an $A_{\infty}$ category with
$\operatorname{Ob}(\operatorname{Groth}_{\sigma\in\Sigma}\mathcal{C}_{\sigma})=\bigsqcup_{\sigma\in\Sigma}\operatorname{Ob}\mathcal{C}_{\sigma}$.
It satisfies the following important property: if $K\in\mathcal{C}_{\sigma}$
and $L\in\mathcal{C}_{\sigma^{\prime}}$ are objects, then the space of
morphism from $K$ to $L$ in
$\operatorname{Groth}_{\sigma\in\Sigma}\mathcal{C}_{\sigma}$ is zero unless
$\sigma\leq\sigma^{\prime}$.

The category $\operatorname{Groth}_{\sigma\in\Sigma}\mathcal{C}_{\sigma}$
admits a distinguished collection of morphism $A_{\Sigma}$ called _adjacency
morphisms_. The _homotopy colimit_ of the diagram
$\\{\mathcal{C}_{\sigma}\\}_{\sigma\in\Sigma}$ is defined to be the
localization of the Grothendieck construction at the adjacency morphisms:

(2.5)
$\operatorname{hocolim}_{\sigma\in\Sigma}:=\left(\operatorname{Groth}_{\sigma\in\Sigma}\mathcal{C}_{\sigma}\right)[A_{\Sigma}]^{-1}.$

We refer the reader to [gps2, A.4] for details on the Grothendieck
construction and homotopy colimit. We will not need any properties of these
constructions beyond those mentioned above.

We now collect some lemmas about the Rouquier dimension of homotopy colimits.
We begin with a small piece of notation: given a finite poset $\Sigma$ and an
element $\sigma\in\Sigma$, we let $h(\sigma)\in\mathbb{N}_{+}$ be the length
of the longest chain $\sigma_{1}<\dots<\sigma_{h(\sigma)}=\sigma$ terminating
at $\sigma$. Note that $h(\sigma)=h(\sigma^{\prime})$ implies that $\sigma$
and $\sigma^{\prime}$ are either incomparable or equal. Let $D$ be the maximum
of $h(\sigma)$ for all $\sigma\in\Sigma$.

Let us now consider the triangulated categories
$\mathcal{T}=H^{0}(\operatorname{Perf}\operatorname{Groth}_{\sigma\in\Sigma}\mathcal{C}_{\sigma})$
and
$\mathcal{T}^{\prime}=H^{0}(\operatorname{Perf}\operatorname{hocolim}_{\sigma\in\Sigma}\mathcal{C}_{\sigma})$.
We write
$\mathcal{I}_{\ell}=\langle\bigcup_{h(\sigma)=\ell}\mathcal{C}_{\sigma}\rangle\subseteq\mathcal{T}$.

######  Lemma 2.14.

$\operatorname{Rdim}\mathcal{I}_{\ell}=\operatorname{max}_{h(\sigma)=\ell}\operatorname{Rdim}\mathcal{C}_{\sigma}$.

###### Proof.

Follows from Lemma 2.4 and the fact that
$\mathcal{C}_{\sigma},\mathcal{C}_{\sigma^{\prime}}$ are orthogonal whenever
$\sigma\neq\sigma^{\prime}$. ∎

######  Lemma 2.15.

We have
$\operatorname{Rdim}\mathcal{T}^{\prime}\leq\operatorname{Rdim}\mathcal{T}\leq\sum_{\ell=1}^{D}(\operatorname{Rdim}\mathcal{I}_{\ell}+1)-1$.

###### Proof.

The first inequality follows from Corollary 2.12. For the second one, observe
that
$\langle\bigcup_{\ell=1}^{L}\mathcal{I}_{\ell}\rangle_{\infty}=\mathcal{T}$.
As noted above, we have $\hom_{\mathcal{T}}(K,L)=0$ if $K\in\mathcal{I}_{j}$
and $L\in\mathcal{I}_{i}$ for $j>i$. The claim thus follows from Lemma 2.10. ∎

###  2.6. The Rouquier dimension of some triangulated categories

We collect some computations from the literature to be used later. For many
other interesting computations, we refer to [elagin2020calculating].

####  2.6.1. Coherent sheaves

We let $D^{b}\operatorname{Coh}(Y)$ denote the bounded derived category of
coherent sheaves over a scheme $Y$, which is always assumed to be separated
and of finite type over a field.

The following facts are due to Rouquier [rouquier].

  * •

If $Y$ is a separated scheme of finite type over a perfect field, then
$\operatorname{Rdim}D^{b}\operatorname{Coh}(Y)$ is finite; [rouquier, Thm.
7.38].

  * •

If $Y$ is a smooth quasi-projective scheme over a field, then
$\operatorname{Rdim}D^{b}\operatorname{Coh}(Y)\leq 2\operatorname{dim}(Y)$;
[rouquier, Prop. 7.9].

  * •

If $Y$ is a reduced separated scheme of finite type over a field, then
$\operatorname{Rdim}D^{b}\operatorname{Coh}(Y)\geq\operatorname{dim}(Y)$;
[rouquier, Prop 7.16].

  * •

If $Y$ is a smooth affine scheme of finite type over a field, then
$\operatorname{Rdim}D^{b}\operatorname{Coh}(Y)=\operatorname{dim}(Y)$;
[rouquier, Thm. 7.17].

It follows from the above facts that
$\operatorname{Rdim}D^{b}\operatorname{Coh}(Y)=\dim Y$ if $Y$ is affine. Orlov
conjectured in [orlov2009remarks, Conj. 10] that this equality holds for any
smooth quasi-projective variety. We will return to this conjecture in Section
5.4.

If $Y$ is a variety which is not assumed to be smooth, then the Rouquier
dimension $D^{b}\operatorname{Coh}(Y)$ can be bigger than its Krull dimension.
The following example, which was communicated to us by W. Hara, shows that
there exist projective curves whose which have arbitrarily large Rouquier
dimension.

######  Example 2.16.

Fix a positive integer $m$. Wantanabe [watanabe, Thm. 1] constructed a
$k$-algebra $(A,\mathfrak{m})$ which is a complete intersection Noetherian
local domain of Krull dimension $1$ and with
$\operatorname{dim}_{R/\mathfrak{m}}\mathfrak{m}/\mathfrak{m}^{2}=m$.
Moreover, by construction, $A$ is the localization of a finite type
$k$-algebra $B$ at a maximal ideal.

Let $Y_{0}=\operatorname{Spec}B$ and let $Y\subseteq\mathbb{P}^{n}$ be the
projective compactification of $Y_{0}$. We then have
$\operatorname{Rdim}D^{b}\operatorname{Coh}(Y)\geq\operatorname{Rdim}D^{b}\operatorname{Coh}(Y_{0})\geq\operatorname{Rdim}D^{b}Coh(R)$,
where the second inequality is by [aihara-takahashi, Lem. 4.2]. We now appeal
to [b-i-k-o, Cor. 5.10], which implies that
$\operatorname{Rdim}D^{b}Coh(R)\geq m-\operatorname{dim}Y-1=m-2$.

Ballard–Favero gave examples of singular varieties whose Rouquier dimension
equals their Krull dimension (see [ballard2012hochschild, Cor. 3.3]). We will
prove in Section 5.4 that such an equality holds for all toric boundary
divisors of smooth quasiprojective DM stacks over the complex numbers of
dimension at most $3$ and some other singular varieties.

####  2.6.2. Representation categories of quivers

In this paper, a _quiver_ $\Gamma$ is a directed graph which is connected,
finite, and acyclic (i.e. having no loops and no cycles).

Given a quiver $\Gamma$ and a field $k$, we can consider the _path algebra_
$k[\Gamma]$. As a vector space, $k[\Gamma]$ is generated by words of
composable arrows; in particular, it is finite dimensional as a $k$-vector
space since $\Gamma$ is acyclic. The product of two words is given by
concatenation when this makes sense and is defined to be zero otherwise. The
path algebra of a quiver is in general non-commutative. Unless otherwise
indicated, we always view $k[\Gamma]$ as an _ungraded_ algebra (although it
admits a natural grading by path length).

For a quiver $\Gamma$, we will be interested in the $A_{\infty}$ category
$\operatorname{Perf}k[\Gamma]$, as well as in the associated triangulated
category $H^{0}(\operatorname{Perf}k[\Gamma])$. As noted in Remark 2.8, the
later category is equivalent to the smallest idempotent complete triangulated
subcategory of $D(k[\Gamma])$ containing $k[\Gamma]$.

A quiver is said to be of Dynkin ADE type if its underlying graph is a Dynkin
diagram of type $A_{n},D_{n},E_{6},E_{7},E_{8}$.

The following proposition is a special case of [elagin2020calculating, Prop.
4.3, 4.4] and is ultimately an application of Gabriel’s theorem [quivers-book,
Chap. 4]. (We warn the reader that the notation $\operatorname{Perf}(A)$ in
[elagin2020calculating] denotes the triangulated category which we have been
denoting by $H^{0}(\operatorname{Perf}A)$.)

######  Proposition 2.17.

Let $\Gamma$ be a quiver.

  * •

If $\Gamma$ is of Dynkin ADE type, then
$\operatorname{Rdim}(\operatorname{Perf}k[\Gamma])=0$;

  * •

if $\Gamma$ is not of Dynkin ADE type, then
$\operatorname{Rdim}(\operatorname{Perf}k[\Gamma])=1$.

##  3\. Wrapped Fukaya categories

###  3.1. Liouville manifolds and related objects

####  3.1.1. Basic definitions

A Liouville vector field on a symplectic manifold $(X,\omega)$ is a vector
field $V$ which satisfies $\mathcal{L}_{V}\omega=\omega$. Dually, a Liouville
$1$-form $\lambda$ on a symplectic manifold $(X,\omega)$ is a primitive for
$\omega$.

A _Liouville cobordism_ is an exact symplectic manifold $(X_{0},\lambda_{0})$
with boundary $\partial X_{0}=\partial_{+}X_{0}\sqcup\partial_{-}X_{0}$, such
that the Liouville vector field is outward pointing along $\partial_{+}X_{0}$
and inward pointing along $\partial_{-}X_{0}$. If
$\partial_{-}X_{0}=\emptyset$, such a cobordism is called a _Liouville
domain_.

A _Liouville manifold_ is an exact symplectic manifold $(X,\lambda)$ which is
modeled near infinity on the symplectization of a closed contact manifold
$(Y,\xi=\operatorname{ker}\lambda_{Y})$. More formally, this means that there
exists a proper embedding

(3.1) $e:(SY,\lambda_{Y})\hookrightarrow(X,\lambda)$

which covers the complement of a compact set and satisfies
$e^{*}\lambda=\lambda_{Y}$. Here $(SY,\lambda_{Y})=\\{\sigma\in
T^{*}Y\mid\sigma|_{\xi}=0\\}$ denotes the symplectization of $(Y,\xi)$, where
$\lambda_{Y}$ is the (restriction of) the tautological $1$-form on $T^{*}Y$. A
Liouville manifold can equivalently be viewed as the result of completing a
Liouville domain by gluing on the positive symplectization of its boundary.
The contact manifold $Y$ is written as $\partial_{\infty}X$ and is called the
_ideal contact boundary_ of $X.$

A _Liouville sector_ is a Liouville manifold with boundary which is modeled
near infinity on the positive symplectization of a contact manifold with
convex boundary (see [gps1, Def. 2.4] for the precise definition). A Liouville
manifold is in particular a Liouville sector. One example of a Liouville
sector which is not a Liouville manifold is the cotangent bundle of a compact
manifold with boundary. Given a Liouville sector $(X,\lambda)$, a _stop_ is an
arbitrary closed subset $\mathfrak{f}\subseteq(\partial_{\infty}X)^{\circ}$.
Such a pair $(X,\mathfrak{f})$ is often called a stopped Liouville sector.

A Liouville manifold $(X,\lambda)$ is said to be _Weinstein_ if the Liouville
vector field $V$ is gradient-like with respect to a proper Morse function
$\phi$. Note that according to this definition, being Weinstein is a
_property_ of Liouville manifolds rather than an extra structure. A Liouville
cobordism is said to be Weinstein if its completion is Weinstein; a Liouville
sector is said to be Weinstein if its convex completion [gps1, Sec. 2.7] is
Weinstein. A _Liouville pair_ consists of a pair $(X,A)$ such that $X$ is a
Liouville manifold and $A$ (possibly empty) is a Liouville hypersurface
embedded inside the ideal contact boundary $\partial_{\infty}X$, i.e. the
induced contact form $\lambda_{\partial_{\infty}X}$ from the Liouville
$1$–form $\lambda$ restricts to the Liouville $1$–form on $A$. The pair
$(X,A)$ is called a _Weinstein pair_ of both $X$ and $A$ are Weinstein. A
_deformation_ of Liouville manifolds/sectors/cobordisms is a one parameter
family of such objects.

####  3.1.2. Skeleta

The _skeleton_ $\mathfrak{c}_{X}\subset X$ of a Liouville manifold or sector
$(X,\lambda)$ is the set of points which do not escape to infinity under the
positive Liouville flow. (More formally: $p\in\mathfrak{c}_{X}\subset X$ iff
there exists a compact subset $K\subset X$ containing $p$ and such that $p$
does leave $K$ under the positive Liouville flow). The skeleton is sometimes
called the _core_ in the literature.

The (relative) skeleton $\mathfrak{c}_{X,\mathfrak{f}}$ of a stopped Liouville
manifold $(X,\mathfrak{f})$ is defined to be the set of points which do not
escape to $\partial_{\infty}X-\mathfrak{f}$ under the positive Liouville flow.
If $(X,A)$ is a Liouville pair, we similarly define the (relative) skeleton
$\mathfrak{c}_{X,A}$ to be the set of points which do not escape to
$\partial_{\infty}X-\mathfrak{c}_{A}$ under the positive Liouville flow.

The skeleton of an arbitrary Liouville manifold always has measure zero. If
$(X,\lambda)$ is Weinstein, then a choice of a gradient-like proper Morse
function $\phi:X\to\mathbb{R}$ defines a handlebody decomposition of $X$. The
skeleton is then precisely the union of the stable manifolds associated to
this decomposition. In particular, the skeleton is stratified by isotropic
submanifolds. In contrast, the skeleton of an arbitrary Liouville manifold can
be much larger (see e.g. McDuff’s examples in [mcduff]).

###  3.2. Invariants of Liouville manifolds

The _wrapped Fukaya category_ $\mathcal{W}(X)$ of a Liouville manifold
$(X,\lambda)$ is an $A_{\infty}$ category. The objects of this category are
exact, cylindrical Lagrangian submanifolds. The $A_{\infty}$ operations are
constructed by counting pseudoholomorphic curves. The wrapped Fukaya category
was first constructed by Abouzaid–Seidel [abouzaid-seidel] and has become a
fundamental invariant in the study of Liouville manifolds.

More generally, one can also consider the wrapped Fukaya category
$\mathcal{W}(X,\mathfrak{f})$ of a stopped Liouville sector. The relevant
theory was developed by Ganatra–Pardon–Shende (see [gps2, Sec. 2] for an
overview), following earlier work of Sylvan [sylvan-stops].

The wrapped Fukaya category is invariant up to quasi-equivalence under
deformations of the underlying Liouville manifold/sector. More generally, the
wrapped Fukaya category of a stopped Liouville manifold is invariant under
deformations which preserve the contactomorphism type of the complement of the
stop [gps2, Thm. 1.4].

The _symplectic cohomology_ $SH^{\bullet}(X)$ of a Liouville manifold
$(X,\lambda)$ is a graded-commutative ring (meaning that
$ab=(-1)^{|a||b|}ba$). The generators can be interpreted as orbits for a
Hamiltonian on $X$ which grows suitably fast at infinity, and the differential
counts pseudoholomorphic curves. Our grading conventions for symplectic
cohomology are uniquely determined by stipulating that the differential
increases grading and that the unit lives in degree zero. They match those of
e.g. [seidel-biased-view] and [ganatra].

Symplectic cohomology was originally introduced by Floer and Hofer [floer-
hofer] and subsequently recast by Viterbo [viterbo]. We refer to [seidel-
biased-view] for a beautiful survey which also matches our sign and grading
conventions.

######  Remark 3.1 (Orientations/gradings).

Note that one must fix additional orientations/grading data in order for the
above invariants to be $\mathbb{Z}$-graded and defined over a field of
characteristic zero. We generally avoid discussing signs/gradings in this
paper (see instead [gps3, Sec. 5.3]).

######  Remark 3.2 (Technical setups).

There are multiple technical setups for Floer theoretic invariants on
Liouville manifolds in the literature. In particular, there are (at least)
three distinct versions of the wrapped Fukaya category of a Liouville
manifold. The first is the so-called “linear” version of [abouzaid-seidel].
The second is the so-called “quadratic” version, which first appeared in
[abouzaid-generation]. The third is the so-called “localization” version which
was first implemented in the literature by Ganatra–Pardon–Shende in [gps1].
Some standard references (such as [abbondandolo-schwarz] and [ganatra]) also
construct Floer theoretic invariants using almost-complex structures of
“rescaled contact type”, which is in contrast to the previously cited
references which only use almost-complex structures of contact type.

The fact that all of these setups yield isomorphic theories is a folklore
result which is routinely assumed in the literature. However, to the best of
our knowledge, detailed proofs for most of these equivalences have yet to
appear (see however [sylvan, Prop. 2.6] concerning the comparison between the
quadratic and localization versions). In this paper, the reader is free to
assume throughout that the wrapped Fukaya category is defined according to the
localization setup of [gps1] (this is the only setup available for sectors).
However, we will quote facts from [abbondandolo-schwarz] and [ganatra] in
Section 4.4, and we will sometimes also appeal to certain mirror symmetry
results from the literature which are proved using other setups.

###  3.3. Structural facts about wrapped Fukaya categories

We collect some structural properties of wrapped Fukaya categories which will
be used in the sequel. Unless otherwise indicated, the theory summarized here
is due to Ganatra–Pardon–Shende [gps1, gps2].

######  Definition 3.3 (Def. 1.6 in [gps2]).

A closed subset $\mathfrak{c}$ of a symplectic manifold $W$ is said to be
_mostly Lagrangian_ if it admits a decomposition
$\mathfrak{c}=\mathfrak{c}^{\operatorname{crit}}\cup\mathfrak{c}^{subcrit}$,
where $\mathfrak{c}^{crit}$ is a Lagrangian submanifold (in general non-
compact) and $\mathfrak{c}^{\operatorname{subcrit}}$ is closed and covered by
the smooth image of a second countable manifold of dimension at most
$\frac{1}{2}\operatorname{dim}W-1$. We similarly define the notion of a
_mostly Legendrian_ subset of a contact manifold.

If $\mathfrak{c}_{X}\subset X$ is mostly Lagrangian, a _generalized cocore_
associated to a connected component of
$\mathfrak{c}_{X}^{\operatorname{crit}}$ is an exact (cylindrical at infinity)
Lagrangian which intersects this component transversally in a single point.
There is a closely related notion for stops: suppose
$\mathfrak{c}\subseteq\partial_{\infty}X$ is a stop and let
$p\in\mathfrak{c}^{\operatorname{crit}}$. Then there is an associated
Lagrangian submanifold $L_{p}\subseteq X$ called the _linking disk_ , which is
well-defined up to isotopy and depends only on the connected component of
$\mathfrak{c}^{\operatorname{crit}}$ containing $p$.

######  Example 3.4.

If the stable manifolds of $(X,\lambda)$ are properly embedded (this condition
holds after a generic perturbation of $\lambda$), then the skeleton is mostly
Lagrangian. The cocores of the critical handles are in particular generalized
cocores. Similarly, the skeleton of a Weinstein hypersurface
$V\subseteq\partial_{\infty}X$ is mostly Legendrian after possibly perturbing
$V$.

If $X$ is a Liouville sector and $\mathfrak{c}\subseteq\partial_{\infty}X$ is
a stop, then there is a natural functor (often called “pushforward” or “stop-
removal”)

(3.2) $\mathcal{W}(X,\mathfrak{c})\to\mathcal{W}(X).$

This functor is characterized as follows when $\mathfrak{c}$ is mostly
Legendrian.

######  Theorem 3.5 (see Thm. 1.16 in [gps2]).

If $\mathfrak{c}\subseteq\partial_{\infty}X$ is mostly Legendrian, then (3.2)
induces a quasi-equivalence

(3.3)
$\mathcal{W}(X,\mathfrak{c})/\mathcal{D}\xrightarrow{\sim}\mathcal{W}(X),$

where $\mathcal{D}$ is the collection of linking disks associated to
$\mathfrak{c}^{\operatorname{crit}}$.

It is a fundamental fact due independently to Chaintraine–Dimitroglou
Rizell–Gigghini–Golovko [cdgg] and Ganatra–Pardon–Shende [gps2, Thm. 1.10]
that the wrapped Fukaya category of a Weinstein manifold is generated by
cocores. We will need a slightly more refined statement, which follows from
essentially the same proof. This is the content of Theorem 3.6 below.

To set the stage, let $(X,\lambda)$ be a Liouville manifold. Suppose that the
skeleton
$\mathfrak{c}_{X}=\mathfrak{c}^{\operatorname{crit}}\cup\mathfrak{c}^{\operatorname{subcrit}}$
is mostly Lagrangian, and that every connected component of
$\mathfrak{c}^{\operatorname{crit}}$ admits a generalized cocore. Let $L$ be
an object of $\mathcal{W}(X)$ such that the underlying Lagrangian satisfies
the following conditions:

  * •

$L\cap\mathfrak{c}_{V}\subseteq\mathfrak{c}_{V}^{\operatorname{crit}}$;

  * •

$L$ intersects $\mathfrak{c}_{V}^{\operatorname{crit}}$ transversally.

######  Theorem 3.6.

With the above assumptions, suppose that
$|L\cap\mathfrak{c}_{V}|=r\in\mathbb{N}$. Then there exists a resolution

(3.4) $0=K_{0}\to K_{1}\to\dots\to K_{r-1}\to K_{r}=L$

where each $\operatorname{cone}(K_{i-1}\to K_{i})$ is isomorphic in
$H^{0}(\mathcal{W}(X))$ to a generalized cocore.

###### Proof.

The proof is essentially the same as that of [gps2, Thm. 1.11]. After possibly
modifying $L$ outside a compact set, we may assume that $\lambda|_{L}$ has a
compactly-supported primitive [gps2, Lem. 6.1]. Let $f:X\to\mathbb{R}$ be an
extension of this primitive to $X$.

Consider the functors

(3.5)
$\mathcal{W}(X)\hookrightarrow\mathcal{W}(X\times\mathbb{C},\mathfrak{c}_{X}\times\\{\pm\infty\\})\xrightarrow{\sim}\mathcal{W}(X\times\mathbb{C},\mathfrak{c}_{X}\times\\{+\infty\\}\cup\mathfrak{c}_{X}\times\\{e^{4\pi
i/3}\cdot\infty\\}),$

where the first arrow is the Künneth embedding [gps2, Thm. 1.5] and the second
arrow is induced by the isotopy of stops
$\mathfrak{c}_{X}\times\\{-\infty\\}\rightsquigarrow\mathfrak{c}_{X}\times\\{e^{\pi
i\cdot(1+s)}\infty\\})$ for $s\in[0,1/3]$, which is a quasi-equivalence by
[gps2, Thm. 1.4].

On objects, (3.5) sends $L\mapsto L\tilde{\times}i\mathbb{R}$, which is the
_cylindrization_ of the product Lagrangian $L\times i\mathbb{R}$ as defined in
[gps2, Sec. 6.2]. The cylindrization is well-defined up to a contractible
choice of Lagrangian isotopies. It will be convenient to construct a nice
cylindrization. To this end, let $C\gg 1$ and $0<\epsilon\ll 1$ be constants
which will be fixed later. Let
$\phi(r,\theta)=\phi(r):\mathbb{C}\to\mathbb{R}$ be a function with the
following properties:

  * •

the derivative of $\phi$ in the radial direction is non-negative;

  * •

$\phi(r)=0$ for $r\in[0,C-\epsilon]$ and $\phi(r)=1$ if $r\geq C$.

To define the cylindrization, we consider the Liouville $1$-form
$\lambda_{X}+\lambda_{\mathbb{C}}-d(\phi f)$, where
$\lambda_{\mathbb{C}}=r^{2}d\theta$. The associated Liouville vector field is

(3.6) $(Z_{X}+\phi X_{f})+(\frac{1}{2}r\partial_{r}+fX_{\phi}).$

Assuming $C$ is large enough, it was shown in [gps2, Sec. 6.2] that there is a
unique Hamiltonian isotopy $\Phi$ which is the identity on a Liouville
subdomain of $X\times\mathbb{C}$ and sends $Z_{X}+\frac{1}{2}r\partial_{r}$
(i.e. the Liouville vector field of $\lambda_{X}+\lambda_{\mathbb{C}}$) to
(3.6).

For $t\in[0,2/3]$, let $L_{t}:=L\times e^{\pi i\cdot(t+1/2)}\mathbb{R}$. We
define $\tilde{L}_{t}:=\Phi(L_{t})$ to be the cylindrization.

Consider now the contact manifold $\mathcal{O}=X\times\\{r=C\\}$, with contact
form $\lambda_{X}+C^{2}d\theta$. This defines coordinates on (a region of)
$(\partial_{\infty}(X\times\mathbb{C}),\lambda_{X}+\lambda_{\mathbb{C}})$. We
set $\Lambda_{t}=\mathcal{O}\cap L_{t}$ and
$\tilde{\Lambda}_{t}=\mathcal{O}\cap\tilde{L}_{t}$. By construction,
$\tilde{\Lambda}_{t}\subseteq\mathcal{O}$ is Legendrian. Moreover, by taking
$\epsilon$ small enough, we can ensure that $\tilde{L}_{t}$ is arbitrarily
close to the Legendrian lift of $L_{t}$ in any $C^{k}$ norm. In particular,
the projection of $\tilde{L}_{t}$ onto
$X\times\\{r=C,\theta=\pi\cdot(t+1/2)\\}$ is arbitrarily close to
$\Lambda_{t}$.

We now consider the isotopy $\tilde{L}_{t}$ for $t\in[0,2/3]$. The
intersection points of $\tilde{L}_{t}$ with the stop
$\mathfrak{c}_{X}\times\\{+\infty\\}$ are in correspondence with the
intersection points of $\Lambda_{t}$ with
$\mathfrak{c}_{X}\times\\{\infty\\}$, and hence with the intersection points
$L\cap\mathfrak{c}_{X}$. By assumption, these intersections are all transverse
and contained in $\mathfrak{c}_{X}^{\operatorname{crit}}$.

The rest of the argument is identical to the proof of [gps2, Thm. 1.11] and is
therefore only sketched. We may assume (e.g. by a small perturbation of the
stop) that the intersections happens at discrete times
$0<t_{1}<\dots<t_{r}<2/3$. At each intersection point $t_{i}$, the wrapping
exact triangle [gps2, Thm. 1.9] gives $L_{t_{i}+\delta}\to L_{t_{i}-\delta}\to
D_{i}\xrightarrow{[1]}$, where $D_{i}$ is a linking disk. One can verify that
$L_{t_{r}+\delta}$ is the zero object (for example, because it is contained in
a Liouville sector deformation equivalent to
$X\times\mathbb{C}_{\operatorname{Im}\geq 0}$ whose wrapped Fukaya category
thus vanishes). Finally, it can be shown that the linking disks correspond to
generalized cocores under (3.5). ∎

######  Remark 3.7.

Theorem 3.6 also follows from [cdgg, Prop. 9.3] (in a slightly less general
form). Note that in the statement of [cdgg, Prop. 9.3], the authors implicitly
assume that the Liouville structure has already been put in a “nice” form (see
the discussion in [cdgg, Sec. 2.2, 9.1]). However, these deformations can
always be assumed to preserve the skeleton and the cocores set-wise so do not
cause any problems.

###  3.4. The Rouquier dimension of wrapped Fukaya categories

We now arrive at the main topic of this paper.

######  Definition 3.8.

Let $(X,\lambda)$ be a Liouville manifold. We let

(3.7) $\operatorname{Rdim}\mathcal{W}(X)\in\mathbb{N}\cup\\{\infty\\}$

denote the Rouquier dimension of the wrapped Fukaya category $\mathcal{W}(X)$,
as defined in Definition 2.9.

More generally, if $(X,\mathfrak{f})$ is a stopped Liouville sector, we let
$\operatorname{Rdim}\mathcal{W}(X,\mathfrak{f})$ denote the Rouquier dimension
of its wrapped Fukaya category.

######  Remark 3.9 (Orientations/gradings).

Of course, the category $\mathcal{W}(X)$ depends on auxiliary
orientations/grading data, so $\operatorname{Rdim}\mathcal{W}(X)$ also depends
on this data. Thus $\operatorname{Rdim}\mathcal{W}(-)$ should be viewed as an
invariant of Liouville manifolds _equipped with orientation/grading data_.

To obtain an honest invariant of Liouville manifolds defined in maximum
generality, one could simply choose to work exclusively with
$\mathbb{Z}/2$-coefficients and $\mathbb{Z}/2$-gradings. On the other hand,
for the purpose of this paper, we are mostly interested in the Rouquier
dimension as a tool for symplectic and algebro-geometric applications. Thus we
will typically restrict our discussion to whichever choice of
coefficients/gradings is most convenient for computations or applications.

In the following examples, we collect some preliminary facts about the
Rouquier dimension of wrapped Fukaya categories.

######  Example 3.10.

If $X$ is Weinstein, then it can be shown using either [cdgg, Thm. 1.1] and
[gps2, Thm. 1.10] that the diagonal
$\Delta\subseteq\mathcal{W}(\overline{X}\times X,-\lambda\oplus\lambda)$ is
generated by product-cocores. It then follows by combining Theorem 7.1 in the
appendix with Lemma 2.13 that

(3.8) $\operatorname{Rdim}\mathcal{W}(X)<\infty.$

We do not know if there exists a Liouville manifold whose wrapped Fukaya
category has infinite Rouquier dimension.

######  Example 3.11.

If $\mathcal{W}(X)=0$, then $\operatorname{Rdim}\mathcal{W}(X)=0$. This holds
of course for subcritical Weinstein manifolds. More generally, the same holds
for _flexible_ Weinstein manifold, namely those Weinstein manifolds which
admit a handlebody presentation where all critical handles are attached along
loose Legendrians.

######  Example 3.12 (Point-like objects).

Suppose that $\mathcal{W}(X)$ is equipped with $\mathbb{Z}$-grading and admits
a point-like object (e.g. an exact Lagrangian torus). Then it follows from
[elagin-lunts-koszul, Thm. 5.2] that
$\operatorname{Rdim}\mathcal{W}(X)\geq\frac{1}{2}\operatorname{dim}_{\mathbb{R}}X$.
(For an $A_{\infty}$ category $\mathcal{C}$, we say that $L\in\mathcal{C}$ is
_point-like_ if (i) $\mathcal{C}$-module $\mathcal{Y}^{r}_{L}$ is proper and;
(ii) $\hom(L,L)=\Lambda^{\bullet}V$ as $A_{\infty}$ algebras, where the right
hand side is the exterior algebra (viewed as an $A_{\infty}$ algebra) of an
$r$-dimensional graded vector space $V$ which is supported in degree $1$.
Strictly speaking, [elagin-lunts-koszul, Thm. 5.2] applies only to proper
categories. However, one can verify that the proof only uses the weaker
assumption (i).)

######  Example 3.13 (Homological mirror symmetry).

If $X$ happens to be mirror to an algebraic variety $X^{\vee}$, then we can
immediately transfer the computations of
$\operatorname{Rdim}D^{b}\operatorname{Coh}(X^{\vee})$ discussed in Section
2.6 into computations of $\operatorname{Rdim}\mathcal{W}(X)$. A partial list
of homological mirror symmetry results for Weinstein manifolds and sectors is
[hacking2020homological, lekili-polishchuk, lekili-ueda, gammage-bh,
gammage2017mirror, pomerleano2021intrinsic]. We leave it to the reader to
verify which results from Section 2.6 apply in each of these instances.

##  4\. Lower bounds: central actions on the wrapped Fukaya category

The main goal of this section is to prove Corollary 4.17, which gives a lower
bound on the Rouquier dimension of certain wrapped Fukaya categories. We do
this by considering a central action of $HH^{\bullet}(\mathcal{W}(X))$ on
$\mathcal{W}(X)$ and adapting methods of Bergh–Iyengar–Krause–Opperman [b-i-
k-o] to deduce lower bounds on the Rouquier dimension. The crucial point is
that this central action can be understood geometrically in terms of the
closed-open string map.

###  4.1. Central actions on triangulated categories

Let $\mathcal{T}=(\mathcal{T},\Sigma,\Delta)$ be a triangulated category. Let
$Z^{d}(\mathcal{T})$ be the set of natural transformations
$\eta:\operatorname{id}\to\Sigma^{d}$ such that
$\eta\circ\Sigma=(-1)^{d}\Sigma\circ\eta$. Observe that
$Z^{\bullet}(\mathcal{T})=\oplus_{d\in\mathbb{Z}}Z^{d}(\mathcal{T})$ inherits
the structure of a graded-commutative ring by composition. We call
$Z^{\bullet}(\mathcal{T})$ the _graded center_ of $\mathcal{T}$.

We will be mainly interested in the case where
$\mathcal{T}=H^{0}(\mathcal{C})$ for $\mathcal{C}$ being a pre-triangulated
$A_{\infty}$ category. In that case, the graded center of $H^{0}(\mathcal{C})$
is precisely the set of graded natural transformations of the identity in
$H^{\bullet}(\mathcal{C})$; cf. [lowen, Sec. 4.2]. Concretely, the elements of
the graded center of $H^{0}(\mathcal{C})$ are products

(4.1)
$\zeta=(\zeta_{K})_{K\in\mathcal{C}}\in\prod_{K\in\mathcal{C}}H^{\bullet}(K,K)$

subject to the naturality condition that the following diagram commutes

(4.2) ${H^{\bullet}(K,L)}$${H^{\bullet}(L,L)\otimes
H^{\bullet}(K,L)}$${H^{\bullet}(K,L)\otimes
H^{\bullet}(K,K)}$${H^{\bullet}(K,L).}$$\scriptstyle{1\otimes\zeta_{K}}$$\scriptstyle{\zeta_{L}\otimes
1}$

Given a graded-commutative ring $R$, a _central action_ of $R$ on the
triangulated category $\mathcal{T}$ is a morphism of graded-commutative rings
$R\to Z^{\bullet}(\mathcal{T})$.

Any triangulated category $\mathcal{T}$ which comes from a pre-triangulated
$A_{\infty}$ category $\mathcal{C}$ (i.e. which is of the form
$\mathcal{T}=H^{0}(\mathcal{C})$) admits a canonical central action

(4.3) $\chi_{\mathcal{C}}:HH^{\bullet}(\mathcal{C},\mathcal{C})\to
Z^{\bullet}(\mathcal{T}),$

where $HH^{\bullet}(\mathcal{C},\mathcal{C})$ is the Hoschschild cohomology of
$\mathcal{C}$. In general, this map is neither injective nor surjective.
(Recall that the Hochschild cohomology of an $A_{\infty}$ category is
naturally a graded-commutative ring. We refer to [ritter-smith, Sec. 2.13] for
a very detailed overview of Hochschild cohomology of $A_{\infty}$ categories.)

The morphism (4.3) which realizes this canonical action is called the
_characteristic morphism_. To describe the characteristic morphism concretely,
let us recall the bar model for the Hochschild cochain complex.

(4.4) $CC^{k}(\mathcal{C},\mathcal{C})=\prod_{\begin{smallmatrix}l\geq 0\\\
K_{0},\ldots,K_{l}\in\mathcal{C}\end{smallmatrix}}\operatorname{Hom}_{k-l}(\hom_{\mathcal{C}}(X_{l-1},X_{l})\otimes\dots\otimes\hom_{\mathcal{C}}(X_{0},X_{1}),\hom_{\mathcal{C}}(X_{0},X_{l}))$

where $\operatorname{Hom}_{s}(-,-)$ is the space of degree $s$ morphisms of
graded vector spaces.

The differential applied to an element $\phi$ of degree $d$ is

(4.5) $\displaystyle\delta(\varphi)(a_{l},\dots,a_{1})$
$\displaystyle=\sum_{r,s}(-1)^{\clubsuit_{1}}\mu^{l-r+s}_{\mathcal{C}}(a_{l},\dots,a_{r+1},\varphi^{r-s+1}(a_{r},\dots,a_{s}),a_{s-1},\dots,a_{1})$
(4.6)
$\displaystyle-(-1)^{d}\sum_{r,s}(-1)^{\clubsuit_{2}}\varphi^{l-r+s}(a_{l},\dots,a_{r+1},\mu^{r-s+1}_{\mathcal{C}}(a_{r},\dots,a_{s}),a_{s-1},\dots,a_{1}),$

where $\clubsuit_{1}:=d(|a_{s-1}|+\dots+|a_{1}|-(s-1))$ and
$\clubsuit_{2}=(|a_{s-1}|+\dots+|a_{1}|-(s-1))$.

Given any object $K\in\mathcal{C}$, there is a natural projection map

(4.7) $\displaystyle\pi_{K}:CC^{k}(\mathcal{C},\mathcal{C})$
$\displaystyle\to\operatorname{Hom}_{k}(k,\hom_{\mathcal{C}}(K,K))\equiv\hom_{\mathcal{C}}(K,K)$
(4.8) $\displaystyle\varphi\mapsto\varphi^{0}_{K}\equiv\varphi^{0}_{K}(1).$

It can be verified (cf. [ritter-smith, Sec. 2.14]) that the projection map is
a morphism of rings. The characteristic morphism is then precisely

(4.9) $\chi_{\mathcal{C}}(\varphi)=(\varphi^{0}_{K}(1))_{K\in\mathcal{C}}.$

The fact that this is a central action follows from the definition of the
Hochschild differential. Indeed, given a closed element $x\in\hom(K,L)$, we
have
$0=\delta(\varphi)(x)=\mu^{2}_{\mathcal{C}}(\varphi_{L}^{0}(1),x)-\mu^{2}_{\mathcal{C}}(x,\varphi_{K}^{0}(1))$.
This matches (4.2).

######  Remark 4.1.

The Hochschild cohomology of an $A_{\infty}$ category is canonically
equivalent to the space of natural transformations of the identity functor,
i.e. the “derived center” of the category. The characteristic morphism thus
realizes a map from the derived center of an $A_{\infty}$ category to the
center of its derived category. For more on this perspective, see [neumann-
szymik] and the references therein.

###  4.2. Central actions and the Rouquier dimension

The goal of this section is to prove the following theorem. We refer to
Section 1.4.1 for motivation and a discussion of the proof.

######  Theorem 4.2.

Let $\mathcal{T}$ be a $\mathbb{Z}/m$-graded triangulated category $(1\leq
m\leq\infty)$ and let $R$ be a finite type $k$-algebra acting centrally on
$\mathcal{T}$. Suppose that there exists a (split-)generator $G\in\mathcal{T}$
such that $\operatorname{Hom}_{\mathcal{T}}^{\bullet}(G,G)$ is a Noetherian
$R$-module. Then
$\operatorname{Rdim}\mathcal{T}\geq\operatorname{dim}_{R}\operatorname{Hom}_{\mathcal{T}}^{\bullet}(G,G).$

####  4.2.1. Recollections from commutative algebra

Given a module $M$ over a commutative ring $R$, the _annihilator_ is the ideal
$\operatorname{Ann}_{R}(M):=\\{r\in R\mid r\cdot m=0\text{ for all }m\in
M\\}\subseteq R$.

The _Krull dimension_ of a commutative ring $R$ is defined as the length of
the longest strictly increasing chain of prime ideals
$\mathfrak{p}_{0}\subsetneq\dots\subsetneq\mathfrak{p}_{l}$ and is denoted by
$\operatorname{dim}(R)$. Given a finitely generated module $M$ over $R$, one
defines the Krull dimension of $M$ to be
$\operatorname{dim}_{R}(M):=\operatorname{dim}(R/\operatorname{Ann}_{R}(M))$.
Finally, the Krull dimension of a (Zariski) closed subset of
$\operatorname{Spec}R$ is the length of the longest strictly increasing chain
of closed subsets. When the ring $R$ is clear from the context, we usually
write the Krull dimension as
$\operatorname{dim}(M):=\operatorname{dim}_{R}(M)$.

The _support_ of a module $M$ over $R$ is the set
$\operatorname{Supp}_{R}(M):=\\{\mathfrak{p}\in\operatorname{Spec}R\mid
M_{\mathfrak{p}}\neq 0\\}$. The support enjoys the following standard
properties:

######  Fact 4.3 (Properties of the support).

Let $R$ be a commutative ring.

  1. (1)

If $M$ is a finitely generated $R$ module, then
$\operatorname{Supp}_{R}(M)=V(\operatorname{Ann}_{R}(M)):=\\{\mathfrak{p}\in\operatorname{Spec}R\mid\operatorname{Ann}_{R}(M)\subseteq\mathfrak{p}\\}$.
In particular,
$\operatorname{dim}_{R}(M)=\operatorname{dim}(\operatorname{Supp}_{R}(M))$.

  2. (2)

If $0\to M_{1}\to M\to M_{2}\to 0$ is an exact sequence of $R$-modules, then
$\operatorname{Supp}_{R}(M)=\operatorname{Supp}_{R}(M_{1})\cup\operatorname{Supp}_{R}(M_{2})$.

  3. (3)

If $M,N$ are finitely generated $R$-modules, then
$\operatorname{Supp}_{R}(M\otimes_{R}N)=\operatorname{Supp}_{R}(M)\cap\operatorname{Supp}_{R}(N)$.

  4. (4)

If $M$ is a finitely generated $R$-module and $I\subseteq R$ is an ideal, then
$\operatorname{Supp}_{R/I}(M/I)=V(I)\cap\operatorname{Supp}_{R}(M)$.

We also need to review the notion of regular sequences.

######  Definition 4.4.

Let $R$ be a commutative ring and let $M$ be an $R$-module. A sequence
$x_{1},\dots,x_{n}$ is _$M$ -regular_ (or just _regular_) if for each $k\leq
n$, we have that $x_{k}$ is not a zero-divisor in $M/(x_{1},\dots,x_{k-1})M$,
and that $(x_{1},\dots,x_{k})M\neq M$. Given an ideal $I\subseteq R$, we let
$\operatorname{depth}_{I}M$ denote the length of the longest $M$-regular
sequence contained in $I$. (If $R$ is local and $I\subseteq R$ is the maximal
ideal, we usually just write $\operatorname{depth}_{I}M$ as
$\operatorname{depth}M$).

######  Fact 4.5 (Regular sequences).

Let $R$ be a commutative ring and let $M$ be a module.

  1. (1)

Suppose $R$ is Noetherian and $M$ is finitely generated. If
$\mathfrak{m}\in\operatorname{Spec}R$ is maximal, then

(4.10)
$\operatorname{depth}_{\mathfrak{m}}M=\operatorname{depth}_{\mathfrak{m}R_{\mathfrak{m}}}M_{\mathfrak{m}}.$

  2. (2)

Let $x_{1},\dots,x_{n}$ be elements of $R$ and let and $a_{1},\dots,a_{n}>0$
be integers. Then $x_{1},…,x_{n}$ is an $M$-regular sequence if and only if
$x_{1}^{a_{1}},\dots,x_{n}^{a_{n}}$ is an $M$-regular sequence.

  3. (3)

Suppose that $x_{1},\dots,x_{n}$ is an $M$-regular sequence. Suppose that
there exists $\mathfrak{p}\in\operatorname{Supp}_{R}(M)$ such that
$x_{i}\in\mathfrak{p}$. Then $x_{1},\dots,x_{n}$, viewed as a sequence of
elements of $R_{\mathfrak{p}}$, is an $M_{\mathfrak{p}}$ regular sequence.

###### Proof.

(1) follows from [mathew-notes, Cor. 1.27]. (2) is [stacks-project, Tag 0AUH].
For (3), note that since multiplication by $x_{i}$ is injective on
$M/(x_{1},\dots,x_{i-1})M$, the same is true after localizing (recall that
localization is exact). Finally, it follows from Nakayama’s lemma that
$M_{\mathfrak{p}}/(x_{1},\dots,x_{n})M_{\mathfrak{p}}=0$ iff
$M_{\mathfrak{p}}=0$. ∎

######  Lemma 4.6.

Let $R$ be a finite type algebra over a field $k$. Suppose that $R$ is
equidimensional (meaning that all its minimal primes have the same dimension).
Given a minimal prime $\mathfrak{p}\subset R$, if $f\in
R\setminus\mathfrak{p}$, then
$\operatorname{dim}(R)=\operatorname{dim}(R_{f})$.

###### Proof.

Observe that
$\operatorname{dim}(R_{f})=\operatorname{dim}(R_{f}/\mathfrak{p})$. We have a
canonical isomorphism $R_{f}/\mathfrak{p}=(R/\mathfrak{p})_{f}$. Finally,
since $R/\mathfrak{p}$ and $(R/\mathfrak{p})_{f}$ are integral domains and
finite type algebras over $k$, it follows from [vakil, Thm. 11.2.1] that
$\operatorname{dim}(R)=\operatorname{dim}(R/\mathfrak{p})=\operatorname{tr.deg}K(R/\mathfrak{p})=\operatorname{tr.deg}K((R/\mathfrak{p})_{f})=\operatorname{dim}((R/\mathfrak{p})_{f})=\operatorname{dim}(R_{f})$.
∎

Note that the hypothesis that $R$ is of finite type is essential. (For
example, $R=k[x]_{(x)}$ has Krull dimension $1$. However, $x$ is not contained
in the unique minimal prime $(0)$ and $R_{x}=k(x)$ has dimension $0$.)

######  Corollary 4.7.

Let $R$ be a finite type algebra over a field $k$ and let $M$ be a finitely
generated $R$-module. Let $\mathfrak{p}\in\operatorname{Ass}_{R}(M)$ be an
associated prime with the property that
$\operatorname{dim}(M/\mathfrak{p})=\operatorname{dim}(M)$. If $f\in
R\setminus\mathfrak{p}$ and $f$ is contained in every other associated prime
of $M$, then:

  * •

$\operatorname{Ass}_{R_{f}}(M_{f})$ has exactly one associated prime (hence
$\operatorname{Supp}_{R_{f}}(M_{f})$ has exactly one minimal prime).

  * •

$\operatorname{dim}(M)=\operatorname{dim}(M_{f})$.

###### Proof.

It follows from [stacks-project, Tag 05BZ] that $\mathfrak{p}$ is the unique
associated prime of $M_{f}$. Hence [stacks-project, Tag 02CE] implies that it
is also the unique minimal prime. This is the first claim.

We have that
$\operatorname{dim}(M_{f})=\operatorname{dim}(M_{f}/\mathfrak{p})=\operatorname{dim}((M/\mathfrak{p})_{f})$.
It is enough to show that
$\operatorname{dim}(M/\mathfrak{p})=\operatorname{dim}((M/\mathfrak{p})_{f})$.
It follows from 4.3 that
$\operatorname{Supp}(M/\mathfrak{p})=V(\mathfrak{p})=\operatorname{Spec}(R/\mathfrak{p})$,
while the support of $(M/\mathfrak{p})_{f}$ is $V(\mathfrak{p})\cap
D(f)=\operatorname{Spec}((R/\mathfrak{p})_{f})$ (where
$D(f):=\\{\mathfrak{q}\in\operatorname{Spec}R\mid f\notin\mathfrak{q}\\}$).
Hence the second claim follows from Lemma 4.6 applied to $R/\mathfrak{p}$. ∎

######  Proposition 4.8.

Let $R$ be a finite type algebra over a field $k$ and let $M$ be a finite
$R$-module of Krull dimension $c\geq 0$. Then there exists an $M$-regular
sequence of length $c$.

###### Proof.

Choose a minimal associated prime $\mathfrak{p}\in\operatorname{Ass}(M)$ such
that $M/\mathfrak{p}$ has the same dimension as $M$ (see [stacks-project, Tag
02CE]). Since $M$ is finite, $\operatorname{Ass}_{R}(M)$ is finite [stacks-
project, Tag 00LC]. Choose an element $f_{1}\in R$ such that
$f_{1}\notin\mathfrak{p}$ but $f_{1}\in\mathfrak{q}$ for all
$\mathfrak{q}\in\operatorname{Ass}(M)\setminus\mathfrak{p}$. According to
Corollary 4.7, $\operatorname{dim}(M_{f_{1}})=\operatorname{dim}(M)=c$.

Now choose a tower of primes
$\mathfrak{p}R_{f_{1}}=\mathfrak{p}_{0}\subseteq...\subseteq\mathfrak{p}_{c}$
in $R_{f_{1}}$ of maximal length. Pick
$r_{1}\in\mathfrak{p}_{1}\setminus\mathfrak{p}_{0}$. Then $r_{1}$ is not a
zero-divisor of $M_{f_{1}}$ (indeed, by [stacks-project, Tag 00LD]
$\mathfrak{p}_{0}$ contains all zero divisors). According to 4.3(4), the
dimension of the quotient module is at least $c-1$.

We now replace $M$ with $M_{f_{1}}/r_{1}$ and $R$ with $R_{f_{1}}/r_{1}$, and
repeat the same argument. After $c$ steps, we end up with
$(\dots(M_{f_{1}}/r_{1})_{f_{2}}\dots/r_{c-1})_{f_{c}}/r_{c}$. This is the
same as $M_{f_{1}\dots f_{c}}/(r_{1},\dots,r_{c})$. (Note that this module has
dimension zero, so it is not the zero module!) So $r_{1},\dots,r_{c}$ is a
regular sequence for $M_{f_{1}\dots f_{c}}$.

Since $M_{f_{1}\dots f_{c}}$ is not zero, it has a maximal ideal
$\mathfrak{m}$ in its support. We now localize at $\mathfrak{m}$. Observe
first that $(M_{f_{1}\dots f_{c}})_{\mathfrak{m}}=M_{\mathfrak{m}}$ (indeed,
it follows from 4.3(3) that $\mathfrak{m}\in D(f_{1}\dots f_{m})$, so
$f_{1}\dots f_{m}$ is a unit in $R_{\mathfrak{m}}$). Moreover, it follows from
4.5(3) that $r_{1},\dots,r_{c}$ is still a regular sequence for the
$R_{\mathfrak{m}}$ module $M_{\mathfrak{m}}$. The conclusion now follows from
4.5(1). ∎

######  Remark 4.9.

In fact, more is true: with the notation of the proof of Proposition 4.8, one
can apply [kaplansky, Sec. 3.1, Ex. 23] to show that there exists an
$M$-regular sequence generating $\mathfrak{m}$ which is _stable under
permutations_. We thank Srikanth Iyengar for this reference.

####  4.2.2. Support varieties

We begin with some standard facts about Noetherian modules. First, recall that
a module over a Noetherian ring is Noetherian iff it is finitely generated.
Also, Noetherian modules are well behaved under exact sequences, meaning that
if $0\to M_{1}\to M\to M_{2}\to 0$ is an exact sequence of $R$-modules, then
$M$ is Noetherian iff $M_{1},M_{2}$ are Noetherian.

######  Lemma 4.10.

Assume that $R$ is Noetherian and acts centrally on $\mathcal{T}$. Given any
object $X\in\mathcal{T}$, the subcategory
$\\{Y\in\mathcal{T}\mid\operatorname{Hom}_{\mathcal{T}}^{\bullet}(X,Y)\;\text{is
Noetherian over }R\\}\subseteq\mathcal{T}$ is a thick triangulated
subcategory. Similarly for
$\\{Y\in\mathcal{T}\mid\operatorname{Hom}_{\mathcal{T}}^{\bullet}(Y,X)\;\text{is
Noetherian over }R\\}\subseteq\mathcal{T}$.

###### Proof.

We only treat the first case as the second one is similar. Consider an exact
triangle $Y_{1}\to Q\to Y_{2}\to$ where
$\operatorname{Hom}_{\mathcal{T}}^{\bullet}(X,Y_{i})$ is Noetherian over $R$.
Then we have an exact sequence $0\to
M\to\operatorname{Hom}_{\mathcal{T}}^{\bullet}(X,Q)\to N\to 0$ where
$M=\operatorname{Hom}_{\mathcal{T}}^{\bullet}(X,Y_{1})/\operatorname{ker}(\operatorname{Hom}_{\mathcal{T}}^{\bullet}(X,Y_{1})\to\operatorname{Hom}_{\mathcal{T}}^{\bullet}(X,Q))$
and
$N=\operatorname{im}(\operatorname{Hom}_{\mathcal{T}}^{\bullet}(X,Q)\to\operatorname{Hom}_{\mathcal{T}}^{\bullet}(X,Y_{2}))$.
Observe that $M,N$ are Noetherian over $R$. Hence so is the middle term.

Suppose that
$\operatorname{Hom}_{\mathcal{T}}^{\bullet}(X,Y_{1})\hookrightarrow\operatorname{Hom}_{\mathcal{T}}^{\bullet}(X,Y_{2})$,
e.g. if $Y_{1}$ is a direct summand of $Y_{2}$. If
$\operatorname{Hom}_{\mathcal{T}}^{\bullet}(X,Y_{2})$ is Noetherian over $R$,
it follows that $\operatorname{Hom}_{\mathcal{T}}^{\bullet}(X,Y_{1})$ is too.
∎

We also need the following lemma, which can be deduced from 4.3.

######  Lemma 4.11.

Let $R$ be a commutative Noetherian ring which acts centrally on
$\mathcal{T}$. Fix an object $X\in\mathcal{T}$. Then for any subset
$\mathcal{U}\subseteq\operatorname{Spec}R$, the subcategories
$\\{Y\in\mathcal{T}\mid\operatorname{Supp}_{R}\operatorname{Hom}_{\mathcal{T}}^{\bullet}(X,Y)\in\mathcal{U}\\}$
and
$\\{Y\in\mathcal{T}\mid\operatorname{Supp}_{R}\operatorname{Hom}_{\mathcal{T}}^{\bullet}(Y,X)\in\mathcal{U}\\}$
are thick triangulated subcategories. ∎

As a corollary of Lemma 4.10 and Lemma 4.11, we get:

######  Corollary 4.12.

Let $R$ be a commutative Noetherian ring which acts centrally on
$\mathcal{T}$. Suppose that $G\in\mathcal{T}$ is a generator and let
$X,Y\in\mathcal{T}$ be arbitrary objects.

  * •

If $\operatorname{Hom}_{\mathcal{T}}^{\bullet}(G,G)$ is a Noetherian
$R$-module, then $\operatorname{Hom}_{\mathcal{T}}^{\bullet}(X,Y)$ is also
Noetherian.

  * •

We have
$\operatorname{Supp}_{R}(\operatorname{Hom}_{\mathcal{T}}^{\bullet}(X,Y))\subseteq\operatorname{Supp}_{R}(\operatorname{Hom}_{\mathcal{T}}^{\bullet}(G,G))$.
In particular, all generators have the same support.

∎

####  4.2.3. Proof of Theorem 4.2

A standard (and possibly the only) tool for obtaining lower bounds on the
Rouquier dimension of a triangulated category is the so-called “ghost lemma”.

######  Lemma 4.13 (Ghost lemma; see Lem. 4.2 in [elagin-lunts-koszul]).

Let $\mathcal{T}$ be a triangulated category and let $F,G$ be objects. Suppose
there exist morphisms

(4.11)
$K_{c}\xrightarrow{\theta_{c}}K_{c-1}\xrightarrow{\theta_{c-1}}\dots\xrightarrow{\theta_{1}}K_{0}$

such that the following conditions hold:

  1. (1)

$\operatorname{Hom}^{\bullet}_{\mathcal{T}}(G,\theta_{i})=0$ for each
$i=1,\dots,c$;

  2. (2)

$\operatorname{Hom}^{\bullet}_{\mathcal{T}}(F,\theta_{1}\dots\theta_{c})\neq
0$.

Then one has $F\notin\langle G\rangle_{c}$.

Following [b-i-k-o], we recall the notion of a Koszul object. Suppose that $R$
is a commutative ring acting centrally on a triangulated category
$\mathcal{T}$. Given $X\in\mathcal{T}$ and $r\in R$, we say that
${X/\\!\\!/r}$ is a _Koszul object_ for $r$ if it fits into the exact triangle

(4.12) $X\xrightarrow{r}X\to{X/\\!\\!/r}\to\Sigma X.$

Note that ${X/\\!\\!/r}$ is well-defined up to isomorphism.

By splicing together exact triangles, we get

(4.13) $X\xrightarrow{r}X\to{X/\\!\\!/r}\to\Sigma X\xrightarrow{-r}\Sigma X.$

More generally, given a sequence of elements $r_{1},\dots,r_{n}\in R$, we
inductively define ${X/\\!\\!/\bm{r}}:={X/\\!\\!/(r_{1},\dots,r_{n})}$ as the
unique object fitting into the triangle

(4.14)
${X/\\!\\!/(r_{1},\dots,r_{n-1})}\xrightarrow{r_{n}}{X/\\!\\!/(r_{1},\dots,r_{n-1})}\to{X/\\!\\!/(r_{1},\dots,r_{n})}\to\Sigma({X/\\!\\!/(r_{1},\dots,r_{n-1})}).$

######  Lemma 4.14.

Suppose that $a_{1},\dots,a_{n}\in R$ is an $M$-regular sequence, where
$M=\operatorname{Hom}_{\mathcal{T}}^{\bullet}(X,X)$. Then there are natural
isomorphisms of $R$-modules:

  1. (1)

$\operatorname{Hom}_{\mathcal{T}}^{\bullet}({X/\\!\\!/(a_{1},\dots,a_{n})},X)\simeq
M[-n]/(r_{1},\dots,r_{n});$

  2. (2)

$\operatorname{Hom}_{\mathcal{T}}^{\bullet}(X,{X/\\!\\!/(a_{1},\dots,a_{n})})\simeq
M/(r_{1},\dots,r_{n})$

###### Proof.

We only treat (i) because (ii) is similar. Suppose first $n=1$. Apply the
functor $\operatorname{Hom}_{\mathcal{T}}^{\bullet}(-,X)$ to (4.13) (with
$r=a_{1}$) gives an exact sequence of $R$-modules

(4.15)
$M[-1]\xrightarrow{-a_{1}}M[-1]\to\operatorname{Hom}_{\mathcal{T}}^{\bullet}({X/\\!\\!/a_{1}},X)\to
M\xrightarrow{a_{1}}M,$

from which the desired claim follows. In general, suppose that the claim holds
for all regular sequences of length at most $n-1$. Consider the spliced
triangle

(4.16)
$\displaystyle{X/\\!\\!/(a_{1},\dots,a_{n-1})}\xrightarrow{a_{n}}{X/\\!\\!/(a_{1},\dots,a_{n-1})}\to{X/\\!\\!/(a_{1},\dots,a_{n})}$
$\displaystyle\to\Sigma({X/\\!\\!/(a_{1},\dots,a_{n-1})})\to\Sigma({X/\\!\\!/(a_{1},\dots,a_{n-1})}).$

and apply the same argument. ∎

We will also need the following lemma.

######  Lemma 4.15 (cf. Lem. 5.11(1) in [benson2008local] and Lem. 3.1 in
[b-i-k-o]).

Let $n\geq 1$ be an integer and set $s=2^{n}$. For any sequence of elements
$r_{1},\dots,r_{n}\in R$, we have
$r_{i}^{s}\cdot\operatorname{Hom}_{\mathcal{T}}^{\bullet}({X/\\!\\!/\bm{r}},-)=0=\operatorname{Hom}_{\mathcal{T}}^{\bullet}(-,{X/\\!\\!/\bm{r}})\cdot
r_{i}^{s}$. ∎

###### Proof of Theorem 4.2.

By assumption, $N:=\operatorname{Hom}_{\mathcal{T}}^{\bullet}(G,G)$ is a
Noetherian $R$-module. Let $n=\operatorname{dim}_{R}(N)$. If $G^{\prime}$ is
another generator, then it follows from Corollary 4.12 that
$\operatorname{Hom}_{\mathcal{T}}^{\bullet}(G^{\prime},G^{\prime})$ also a
Noetherian $R$-module of Krull dimension $n$. Hence, we can restrict our
attention to $G$. Our goal is to construct an object which is not contained in
$\langle G\rangle_{n}$.

By Proposition 4.8, there exists an $N$-regular sequence $r_{1},\dots,r_{n}$.
Set $s=2^{n}$ and note that $r_{1}^{s},\dots,r_{n}^{s}$ is still a regular
sequence (4.5). For $l\leq n$, consider the exact triangles (obtained by
rotating (4.14) and setting $X=G$)

(4.17)
$\Sigma^{-1}({G/\\!\\!/(r_{1}^{s},\dots,r_{l}^{s})})\xrightarrow{-w[-1]}{G/\\!\\!/(r_{1}^{s},\dots,r_{l-1}^{s})}\xrightarrow{r_{l}^{s}}{G/\\!\\!/(r_{1}^{s},\dots,r_{l-1}^{s})}\to{G/\\!\\!/(r_{1}^{s},\dots,r_{l}^{s})},$

where $w$ is the connecting map. Suspending $l-1$ more times, we get

(4.18)
$\displaystyle\Sigma^{-l}({G/\\!\\!/(r_{1}^{s},\dots,r_{l}^{s})})\xrightarrow{\theta_{l}}\Sigma^{-(l-1)}({G/\\!\\!/(r_{1}^{s},\dots,r_{l-1}^{s})})$
$\displaystyle\xrightarrow{\pm
r_{l}^{s}}\Sigma^{-(l-1)}({G/\\!\\!/(r_{1}^{s},\dots,r_{l-1}^{s})})\to\Sigma^{-(l-1)}({G/\\!\\!/(r_{1}^{s},\dots,r_{l}^{s})}).$

(note that each application of $\Sigma^{-1}$ has the effect of changing the
sign of $\cdot r_{l}^{s}$; here $\theta_{l}$ is defined as the morphism
obtained by applying $\Sigma^{-(l-1)}(-)$ to $-w[-1]$).

Applying the functor $\operatorname{Hom}_{\mathcal{T}}^{\bullet}(G,-)$, it
follows from Lemma 4.14(2) that we have a natural isomorphism
$\operatorname{Hom}_{\mathcal{T}}^{\bullet}(G,\Sigma^{-(l-1)}({G/\\!\\!/r_{1}^{s},\dots,r_{l-1}^{s}}))\simeq
N/(r_{1}^{s},\dots,r_{l-1}^{s})[-(l-1)]$. By regularity, $\pm r_{l}^{s}$ acts
injectively, and it follows that
$\operatorname{Hom}_{\mathcal{T}}^{\bullet}(G,\theta_{l})=0$ by (4.18).

Next, it follows from Lemma 4.15 that $\pm r_{l}^{s}$ acts by zero on
$\operatorname{Hom}({G/\\!\\!/\bm{r}},\Sigma^{-(l-1)}({G/\\!\\!/(r_{1}^{s},\dots,r_{l-1}^{s})})$,
which implies that
$\operatorname{Hom}_{\mathcal{T}}^{\bullet}({G/\\!\\!/\bm{r}},\theta_{l})$ is
surjective by (4.18). But Lemma 4.14(1) implies that
$\operatorname{Hom}_{\mathcal{T}}^{\bullet}({G/\\!\\!/\bm{r}},G)\simeq
N[-n]/(r_{1},\dots,r_{n})$. By regularity, $N[-n]/(r_{1},\dots,r_{n})\neq 0$.
Since this module is the target of the surjective map
$\operatorname{Hom}_{\mathcal{T}}^{\bullet}(G,\theta_{1}\dots\theta_{n})$,
this map is nonzero. We conclude by Lemma 4.13 (with ${G/\\!\\!/\bm{r}}$ in
place of $F$) that ${G/\\!\\!/\bm{r}}\notin\langle G\rangle_{n}$. ∎

###  4.3. Central actions on Fukaya categories

We now specialize the discussion of the previous sections to wrapped Fukaya
categories. For simplicity, we assume that the wrapped Fukaya category is
defined with field coefficients and $\mathbb{Z}/2$-gradings.

Given a Liouville manifold $(X,\lambda)$, we wish to consider the canonical
central action of $HH^{\bullet}(\mathcal{W}(X),\mathcal{W}(X))$ on the
triangulated category $H^{0}(\operatorname{Perf}\mathcal{W}(X))$ furnished by
the characteristic morphism. The following lemma states that this
characteristic morphism admits a purely symplectic topological interpretation,
in terms of the familiar closed-open string map which is constructed for
instance in [ganatra, Sec. 5.4].

######  Lemma 4.16.

For any object $K\in\mathcal{W}(X)$, the following diagram of unital rings
commutes

(4.19)
${SH^{\bullet}(X)}$${HH^{\bullet}(\mathcal{W}(X),\mathcal{W}(X))}$${HH^{\bullet}(\operatorname{Perf}\mathcal{W}(X),\operatorname{Perf}\mathcal{W}(X))}$${HW^{\bullet}(K,K)}$$\scriptstyle{\mathcal{CO}}$$\scriptstyle{\mathcal{CO}(-)^{0}_{K}(1)}$$\scriptstyle{\simeq}$

###### Proof.

The fact that $\mathcal{CO}$ is a ring morphism is proved in [ganatra, Prop.
5.3], and the left triangle is then tautologically commutative. The top right
arrow is induced by the canonical (Yoneda) embedding
$\mathcal{W}(X)\to\operatorname{Perf}\mathcal{W}(X)$, and the left triangle is
also tautologically commutative; cf. [ritter-smith, Lem. 2.11]. The fact that
the top right arrow is an isomorphism of graded commutative rings is a
manifestation of the familiar Morita invariance of Hochschild cohomology (see
e.g. [seidel-cat-dynamics, Rmk. 9.5]). ∎

######  Corollary 4.17.

Let $R\subseteq SH^{0}(X)$ be a subring which is of finite type as an algebra
over $k$. Suppose that there exists a (split-)generator $K\in\mathcal{W}(X)$
such that $\operatorname{Hom}_{\mathcal{T}}^{\bullet}(K,K)$ is a Noetherian
$R$-module, where $R$ acts via the closed-open map. Then

(4.20)
$\operatorname{Rdim}\mathcal{W}(X)\geq\dim_{R}\operatorname{Hom}_{\mathcal{T}}^{\bullet}(K,K).$

###### Proof.

Combine Theorem 4.2 and Lemma 4.16. ∎

We will see in the next section how to perform concrete computations via
string topology. For now, let us record the following rather trivial example.

######  Example 4.18.

If $X=T^{*}\mathbb{T}^{n}$ and $F$ is a cotangent fiber, then we have
$\mathcal{CO}:SH^{\bullet}(X)=\Lambda^{\bullet}[v_{1},\dots,v_{n}]\otimes
k[x_{1}^{\pm},\dots,x_{n}^{\pm}]\to
k[x_{1}^{\pm},\dots,x_{n}^{\pm}]=HW^{\bullet}(F,F)$, where
$|v_{i}|=1,|x_{i}|=0$ and $v$ are in degree $1$ and $\mathcal{CO}(1\otimes
x_{i})=x_{i}$. Hence (taking $R=k[x_{1}^{\pm},\dots,x_{n}^{\pm}]$, we find
that $\mathcal{W}(T^{*}\mathbb{T}^{n})\geq n$, where $\mathcal{W}(-)$ is
defined with $k$-coefficients).

For a related (but significantly more complicated) example, see Example 1.8.

###  4.4. Computations via string topology

We now explain how to obtain concrete lower bounds for the Rouquier dimension
of certain cotangent bundles via string topology. We will always assume in
this section that the wrapped Fukaya category is defined with $\mathbb{Q}$
coefficients and $\mathbb{Z}/2$-gradings (so in particular, all Liouville
manifolds carry the appropriate orientations/grading data). Similarly, all
singular homology groups considered in this subsection are assumed to have
coefficients in $\mathbb{Q}$ and all graded objects (such as
homology/cohomology grousp) are assumed to be $\mathbb{Z}/2$-graded (although
the gradings all come from a $\mathbb{Z}$-grading, so the reader is welcome to
work with $\mathbb{Z}$-gradings if they prefer).

Given a closed, oriented manifold $M$ of dimension $n$, the (shifted) homology
of its free loop space $H_{\bullet+n}(\mathcal{L}M)$ carries the structure of
a graded-commutative algebra (in fact a BV algebra). The product is called the
_Chas–Sullivan product_ and was introduced by Chas and Sullivan in [chas-
sullivan]. At an intuitive level, this product is constructed as follows:
choose cycles $\sigma_{p}\in C_{p}(\mathcal{L}M)$ and $\sigma_{q}\in
C_{q}(\mathcal{L}M)$, and consider the intersection
$\operatorname{ev}(\sigma_{p})\cap\operatorname{ev}(\sigma_{q})$, where
$\operatorname{ev}:\mathcal{L}M\to M$ is the evaluation map sending a loop
$\gamma:S^{1}=\mathbb{R}/\mathbb{Z}\to M$ to $\gamma(1)\in M$. Along this
intersection, we can concatenate loops coming from $\sigma_{p}$ with loops
from $\sigma_{q}$. This defines a $p+q-n$ chain in $\mathcal{L}M$, which is
precisely the desired product of $\sigma_{p}$ and $\sigma_{q}$.

Next, it will be useful to recall the construction of the _umkehr_ (or “wrong-
way”) map in differential topology. Let $\mathcal{M}$ be a smooth manifold and
let $\mathcal{N}\subseteq\mathcal{M}$ be a closed, co-oriented submanifold of
codimension $d$. Let $\tau_{\mathcal{M}/\mathcal{N}}\to\mathcal{N}$ be the
normal bundle of $\mathcal{N}\subseteq\mathcal{M}$ and let
$\mathcal{U}\subseteq\mathcal{M}$ be a tubular neighborhood. We have a
sequence of maps

(4.21) $H_{\bullet}(\mathcal{M})\to
H_{\bullet}(\mathcal{M},\mathcal{M}-\mathcal{N})\to
H_{\bullet}(\mathcal{U},\mathcal{U}-\mathcal{N})\to
H_{\bullet}(\tau_{\mathcal{M}/\mathcal{N}},\tau_{\mathcal{M}/\mathcal{N}}-0_{\mathcal{N}})\to
H_{\bullet-d}(\mathcal{N}),$

where we are considering singular homology. (The first map is the natural
projection, the second one is obtained by excision and the fourth one is the
Thom isomorphism.) The composition (4.21) is called the _umkehr_ map
associated to the embedding $\mathcal{N}\subseteq\mathcal{M}$. Note that it is
not necessary for $\mathcal{M},\mathcal{N}$ to be finite dimensional in this
construction. It works as long as $\mathcal{N}\subseteq\mathcal{M}$ has finite
codimension and admits a tubular neighborhood.

We will in fact mostly be interested in the umkehr map associated to the
embedding $\Omega_{p}M\hookrightarrow\mathcal{L}M$, where $M$ is a closed
manifold of dimension $n<\infty$ with a basepoint $p\in M$.

The associated umkehr map

(4.22) $\mathcal{I}:H_{\bullet+n}(\mathcal{L}M)\to H_{\bullet}(\Omega_{p}M)$

is called the _intersection morphism_. It was also first introduced (under a
slightly different guise) by Chas and Sullivan [chas-sullivan] and plays an
important role in string topology. The construction of the intersection
morphism via the umkehr map is found in [q-string-jems, Sec. 9].

The connection to symplectic topology arises via the work of
Abbondandolo–Schwarz [abbondandolo-schwarz].

######  Theorem 4.19 (Abbondandolo–Schwarz, [abbondandolo-schwarz]).

Let $M$ be a closed manifold of dimension $n$. Let $F_{p}\subseteq T^{*}M$ be
a cotangent fiber. Then the following diagram commutes

(4.23)
${H_{-\bullet+n}(\mathcal{L}M)}$${SH^{\bullet}(T^{*}M)}$${H_{-\bullet}(\Omega_{p}M)}$${HW^{\bullet}(F_{p},F_{p})}$$\scriptstyle{\simeq}$$\scriptstyle{\mathcal{I}}$$\scriptstyle{\mathcal{CO}(-)^{0}_{F_{p}}(1)}$$\scriptstyle{\simeq}$

###### Proof.

All of this is contained in [abbondandolo-schwarz]. However, since the
notation and terminology in loc. cit. differs from ours, we give a brief
overview for the reader.

Given a manifold $M$, the wrapped Floer cohomology of a cotangent fiber is
denoted by $HF^{\Omega}(T^{*}M)$ in [abbondandolo-schwarz] and called “Floer
homology for Hamiltonian orbits with Dirichlet boundary conditions”. The
intersection morphism that we denote by $\mathcal{I}$ is denoted by $i_{!}$.
The authors describe how to realize this map in Morse theory in [abbondandolo-
schwarz, Sec. 2.2], which also depends on material from the appendix. The
closed-open map is denoted by $I_{!}$ and constructed in [abbondandolo-
schwarz, Sec. 3.5]. The top horizontal arrow in (4.23) corresponds to
[abbondandolo-schwarz, (3)], while the bottom horizontal arrow is
[abbondandolo-schwarz, Thm. B]. The fact that (4.23) commutes amounts to the
assertion that the diagram [abbondandolo-schwarz, (7)] has a Floer theoretic
counterpart; as is often the case in Floer theory, the proof consists in
showing that the two ways of traveling around the diagram correspond to the
two ends of a cobordism of moduli spaces (see [abbondandolo-schwarz, Thm.
4.11]). ∎

We now discuss some concrete computations. We begin by recording the following
lemma.

######  Lemma 4.20.

Let $M$ and $N$ be closed oriented manifolds of dimension $m$ and $n$
respectively. Then the following diagram of vector spaces commutes, where the
vertical arrow is the usual Eilenberg–Zilber map:

(4.24) ${H_{\bullet+m}(\mathcal{L}M)\otimes
H_{\bullet+n}(\mathcal{L}N)}$${H_{\bullet}(\Omega_{*}M)\otimes
H_{\bullet}(\Omega_{*}N)}$${H_{\bullet+m+n}(\mathcal{L}(M\times
N))}$${H_{\bullet}(\Omega_{*}(M\times
N))}$$\scriptstyle{\mathcal{I}\otimes\mathcal{I}}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\mathcal{I}}$

The vertical arrows are in fact isomorphisms of algebras with respect to the
Chas–Sullivan and Pontryagin products.

###### Proof.

There are canonical splittings of the based loop and free loop spaces which
are compatible with the inclusions (of the former into the latter). It follows
from the Künneth formula and the definition of the umkehr map that the umkehr
map takes products of spaces to tensor products. Finally, it is apparent from
the above description of the Chas–Sullivan product that it is compatible with
the Künneth isomorphism, which means that the vertical left arrow is indeed an
isomorphism of algebras. Similar considerations apply for the Pontryagin
product. ∎

The following theorem is extremely useful, as it essentially reduces the
computation of the intersection morphism $\mathcal{I}$ to rational homotopy
theory.

######  Theorem 4.21 (Félix–Thomas–Vigué-Poirrier; see Thm. F. in [q-string-
jems]).

Let $M$ be a simply-connected closed oriented manifold of dimension $n$ with
basepoint $p$. Then there exists a commutative diagram of morphisms of graded-
commutative algebras

(4.25)
${HH^{\bullet}(C^{\bullet}(M),C^{\bullet}(M))}$${H_{\bullet+n}(\mathcal{L}M)}$${HH^{\bullet}(C^{\bullet}(M),\mathbb{Q})}$${H_{\bullet}(\Omega_{p}M),}$$\scriptstyle{HH^{\bullet}(C^{\bullet}(M),\epsilon)}$$\scriptstyle{\simeq}$$\scriptstyle{\mathcal{I}}$$\scriptstyle{\simeq}$

where $\epsilon:C^{\bullet}(M)\to\mathbb{Q}$ is the augmentation induced by
the inclusion $p\hookrightarrow M$.

Given a graded vector space $V=\\{V_{i}\\}_{i\in\mathbb{Z}}$, let
$T(V):=\oplus_{i\geq 0}V^{\otimes i}$ be the _tensor algebra_ on $V$. Let
$I\subseteq T(V)$ be the ideal generated by elements of the form $x\otimes
y-(-1)^{|x||y|}y\otimes x$ with $x,y\in V$. Then $\Lambda V:=T(V)/I$ be the
_free commutative graded algebra_ on $V$. See [q-homotopy-book, Sec. 3(a)]. If
$V$ is generated by a single element $v$, we write $T(V),\Lambda(V)$ as
$T(v),\Lambda v$ respectively.

Let us now record some useful facts drawn from [f-t-v, Sec. 6.1]. First of
all, $HH^{\bullet}(C^{\bullet}(S^{n}),\mathbb{Q})=T(v)$ with $|v|=n-1$. If
$n>1$ is odd, then
$HH^{\bullet}(C^{\bullet}(S^{n}),C^{\bullet}(S^{n}))\simeq\Lambda u\otimes
T(v)$, where $|u|=-n$ and $|v|=n-1$. We have $\mathcal{I}=\epsilon\otimes
1:\Lambda u\otimes T(v)\to T(v)$, where we implicitly identify both columns of
(4.25) under the horizontal isomorphisms.

If $n>1$ is even, then

(4.26) $HH^{\bullet}(C^{\bullet}(S^{n}),C^{\bullet}(S^{n}))\simeq\Lambda
b\otimes k[a,c]/(2ac,a^{2},ab)$

with $|a|=-n,|b|=-1$ and $|c|=2n-2$. We have $\mathcal{I}(a)=\mathcal{I}(b)=0$
and $\mathcal{I}(c)=v^{2}$.

######  Example 4.22 (Products of odd dimensional spheres).

Let $M=S^{n_{1}}\times\dots\times S^{n_{k}}$ be a product of odd dimensional
spheres with $n_{i}\geq 3$. Fix a basepoint $p\in M$. If $k=1$, then it
follows from the computations above that there is an inclusion
$\mathbb{Q}[x]\hookrightarrow H_{\bullet+n_{1}}(\mathcal{L}S^{n_{1}})$ sending
$x\mapsto 1\otimes v$, such that the composition $\mathcal{Q}[x]\to
H_{\bullet+n_{1}}(\mathcal{L}S^{n_{1}})\to H_{\bullet}(\Omega S^{n_{1}})$ is
an isomorphism.

For $k>1$, similar considerations along with Lemma 4.20 furnish maps

(4.27) $\mathbb{Q}[x_{1},\dots,x_{k}]\hookrightarrow
H_{\bullet+n_{1}}(\mathcal{L}M)\to H_{\bullet}(\Omega M)$

where $|x_{i}|=n_{i}-1$ and the composition is an isomorphism.

By combining Theorem 4.19 and Corollary 4.17, it follows that

(4.28) $\operatorname{Rdim}\mathcal{W}(T^{*}M)\geq k.$

######  Example 4.23 (Compact Lie groups).

Let $G$ be a simply-connected compact Lie group. Then $G$ has a Sullivan
minimal model of the form $(\Lambda V,d=0)$, where
$V=\\{V^{i}\\}_{i\in\mathbb{N}}$ is a graded $\mathbb{Q}$-vector space
supported in odd degrees and satisfying $V^{1}=0$. Thus $G$ has the rational
homotopy type of a product of odd dimensional spheres.

The dimension of $V$ is precisely the rank of the group $G$ [f-o-t, Thm.
3.33]. It follows from 4.22 that we have

(4.29) $\operatorname{Rdim}\mathcal{W}(T^{*}G)\geq\operatorname{rank}G.$

For the reader’s convenience, we list the simply-connected simple compact Lie
groups in Figure 1. It follows from the classification theorem that all other
simply-connected compact Lie groups are obtained from these by taking
products. For $i=6,7,8$, we write $\tilde{E}_{i}$ to denote the universal
cover of the exceptional Lie group $E_{i}$.

Simply-connected compact Lie group | Real dimension | Rank | $\operatorname{Rdim}\mathcal{W}(T^{*}G)$
---|---|---|---
$\operatorname{Sp}(n),n\geq 1$ | $n(2n+1)$ | $n$ | $\geq n$
$\operatorname{SU}(n),n\geq 3$ | $n^{2}-1$ | $n-1$ | $\geq n$
$\operatorname{Spin}(n),n\geq 7$ | $n(n-1)/2$ | $\lfloor n/2\rfloor$ | $\geq\lfloor n/2\rfloor$
$G_{2}$ | $14$ | 2 | $\geq 2$
$F_{4}$ | $52$ | $4$ | $\geq 4$
$\tilde{E}_{6}$ | $156$ | $6$ | $\geq 6$
$\tilde{E}_{7}$ | $266$ | $7$ | $\geq 7$
$\tilde{E}_{8}$ | $496$ | $8$ | $\geq 8$
Figure 1. The simply-connected compact Lie groups

######  Example 4.24 (Products of arbitrary spheres).

We again assume $M=S^{n_{1}}\times\dots\times S^{n_{k}}$. Suppose first that
$k=1$ and $n_{1}>1$ is even. Then there is a natural inclusion
$\mathbb{Q}[x]\hookrightarrow\Lambda b\otimes\mathbb{Q}[a,c]/(2ac,a^{2},ab)$
with $x\mapsto c$. Composing with the intersection morphism gives the map
$\mathbb{Q}[c]\to\mathbb{Q}[v]$ sending $c\mapsto v^{2}$. More generally, we
can apply Lemma 4.20, to get a map

(4.30) $\mathbb{Q}[x_{1},\dots,x_{k}]\to
H_{\bullet+n_{1}}(\mathcal{L}M)\to\mathbb{Q}[v_{1},\dots,v_{k}],$

where

  * •

$|x_{i}|=n_{i}-1$ and $x_{i}\mapsto v_{i}$ if $n_{i}$ is odd;

  * •

$|x_{i}|=2n_{i}-2$ and $x_{i}\mapsto v_{i}^{2}$ if $n_{i}$ is even.

Observe that $\mathbb{Q}[v_{1},\dots,v_{k}]$ is a Noetherian module over
$\mathbb{Q}[x_{1},\dots,x_{k}]$ (indeed, it is finitely generated as a module
over a Noetherian ring). By combining Theorem 4.19 and Corollary 4.17, it
follows that

(4.31) $\operatorname{Rdim}\mathcal{W}(T^{*}M)\geq k.$

######  Example 4.25 (Complex projective spaces).

Let us also briefly mention the case of complex projective spaces. The
cohomology of $\Omega\mathbb{CP}^{n}$ is computed in [seeliger, Thm. 1.3]. It
contains a free subalgebra $\mathbb{Q}[y]$ where $|y|=2n$. On the other hand,
as explained in e.g. [seeliger, Sec. 3], we have
$\Omega\mathbb{CP}^{n}=S^{1}\times\Omega S^{2n+1}$ and hence
$H_{\bullet}(\Omega\mathbb{CP}^{n})=H_{\bullet}(S^{1})\otimes
H_{\bullet}(\Omega S^{2n+1})$. It is shown in [briggs-gelinas, Ex. 5.7] that
the characteristic morphism hits the class $1\otimes v$ of degree $2n$ in
$H_{\bullet}(\Omega\mathbb{CP}^{n})$. For degree reasons, the composition of
$\mathbb{Q}[y]\to H_{\bullet+2n}(\mathcal{L}\mathbb{CP}^{n})\to
H_{\bullet}(\Omega\mathbb{CP}^{n})$ must send $y$ to $1\otimes v$.

By a similar argument to Example 4.22, we find that if
$M=M_{1}\times\dots\times M_{k}$ where each $M_{i}$ is a sphere of dimension
at least $2$ or a complex projective space of arbitrary dimension, then

(4.32) $\operatorname{Rdim}\mathcal{W}(M)\geq k.$

##  5\. Upper bounds: constructing arboreal covers

In this section, we take advantage of recent advances on symplectic
flexibility, namely the arborealization program [AGEN1, AGEN2, AGEN3], to
obtain upper bounds on the Rouquier dimensions of (partially) wrapped Fukaya
categories of _polarized_ Weinstein manifolds. Here a polarization of a
Weinstein manifold $(X,\lambda)$ means a choice of a global Lagrangian plane
field $\xi\subset TX$. Combining these upper bounds with known results in
homological mirror symmetry, we settle several new cases of a conjecture of
Orlov (see Section 5.4). We will use the notations $\operatorname{skel}(X)$
and $\mathfrak{c}_{X}$ interchangeably throughout this section to denote the
skeleton of $(X,\lambda)$.

Before delving into the technical details, let us illustrate the ideas of this
section by considering the example $(X,\lambda)=(T^{*}M,pdq)$, where $M$ is an
$n$-dimensional closed manifold. Choose a smooth triangulation (see the
discussion in Section 5.1 for a precise definition) $\mathcal{S}$ of $M$ and
denote by $\\{v_{\alpha}\\}_{\alpha\in I}$ the set of $0$-simplices of
$\mathcal{S}$. Denote by $\operatorname{star}(v_{\alpha})$ the star of the
vertex $v_{\alpha}$ and let $\overline{\operatorname{star}(v_{\alpha})}$ be
its closure. Then $\overline{\operatorname{star}(v_{\alpha})}$ is an embedded
polyhedron. We choose an inward smoothing of each
$\overline{\operatorname{star}(v_{\alpha})}$ to obtain a smooth manifold with
boundary $M_{\alpha}$ such that the boundaries $\\{\partial
M_{\alpha}\\}_{\alpha\in I}$ intersect transversely. By [gps2, Ex. 1.25], the
collection of cotangent bundles $\\{T^{*}M_{\alpha}\\}_{\alpha\in I}$ forms a
_sectorial cover_ (see [gps2, Def. 1.24]) of the Weinstein manifold
$(T^{*}M,pdq)$. Then [gps2, Thm. 1.27] asserts a pre-triangulated equivalence

(5.1) $\operatorname{hocolim}_{\emptyset\neq J\subset
I}\mathcal{W}(\bigcap_{\alpha\in
J}T^{*}M_{\alpha})\xrightarrow{\sim}\mathcal{W}(T^{*}M).$

The sectorial cover $\\{T^{*}M_{\alpha}\\}_{\alpha\in I}$ admits the following
special properties:

  1. (1)

it has depth at most $n+1$ if the inward smoothing is small enough; i.e. if
$\bigcap_{\alpha\in J}T^{*}M_{\alpha}\neq\emptyset$, then $|J|\leq n+1$;

  2. (2)

each local piece $\bigcap_{\alpha\in J}T^{*}M_{\alpha}$ is Weinstein homotopic
to the cotangent bundle of a closed $n$-dimensional ball $\overline{D}^{n}$.

Note that $\mathcal{W}(T^{*}\overline{D}^{n})$ is equivalent to
$\operatorname{Perf}(k[A_{1}])$ where $A_{1}$ is the quiver with only one
vertex so $\operatorname{Rdim}\mathcal{W}(T^{*}\overline{D}^{n})=0$ by
Proposition 2.17. Then Lemma 2.15 and 2.14 and the fact that this cover has
depth at most $n+1$ implies $\operatorname{Rdim}\mathcal{W}(T^{*}M)\leq n$
because of (5.1).

We wish to apply the above ideas to general Weinstein manifolds or sectors
with arboreal skeleta (see Section 5.2). For purely technical reasons, it is
more convenient to implement this argument using microlocal sheaf theory. We
therefore study the _Nadler–Shende microlocal sheaf_ (see Section 5.3) as an
alternative model of the wrapped Fukaya category, and appeal to [gps3, Thm.
1.4] to translate our results back into the framework of wrapped Fukaya
categories. The Nadler–Shende sheaf just depends on the topology of the
skeleta and satisfies a local-to-global descent formula similar to (5.1). Then
we can cover the skeleta by the stars of topological triangulations and
conclude our result Theorem 5.16 as above.

###  5.1. Stratifications and Goresky’s triangulation theorem

A _stratification_ $\mathcal{S}$ of a topological space $X$ is a locally
finite decomposition $\\{X_{\sigma}\\}_{\sigma\in\mathcal{S}}$ of $X$ into
disjoint, locally closed subsets (called strata444The strata are required to
be connected under some conventions. We find it more convenient to also allow
disconnected strata – however, this is just a matter of terminology which does
not affect the substances of the arguments.) which satisfies the frontier
condition: if $\overline{X_{\alpha}}\cap X_{\beta}\neq\emptyset$, then
$X_{\beta}\subseteq X_{\alpha}$. The pair $(X,\mathcal{S})$ is said to be a
_stratified space_. Note that $\mathcal{S}$ is naturally a partially ordered
set, where we set $X_{\alpha}\leq X_{\beta}$ if and only if
$X_{\alpha}\subseteq\overline{X_{\beta}}$.

Given a stratum $X_{\alpha}$, the (open) star of $X_{\alpha}$ is defined to be
the union of strata from $\mathcal{S}$ whose closures contain $X_{\alpha}$ and
is denoted by $\operatorname{star}(\alpha)$. The _link_ of $X_{\alpha}$ is
denoted by $\operatorname{link}(\alpha)$ and defined as the closure of
$\operatorname{star}(\alpha)$ minus its interior.

Given a fixed topological space $X$, a stratification $\mathcal{S}^{\prime}$
of $X$ is said to be a _refinement_ of some coarser stratification
$\mathcal{S}$ of $X$ if every stratum in $\mathcal{S}^{\prime}$ is contained
in a stratum in $\mathcal{S}$ (equivalently, every stratum in $\mathcal{S}$ is
a union of strata in $\mathcal{S}^{\prime}$).

If $X$ is a topological subspace of a manifold (without loss of generality
$X\subseteq\mathbb{R}^{N}$), then a stratification $\mathcal{S}$ of $X$ is
said to be $C^{\mu}$ (for $\mu\in\mathbb{Z}_{\geq 0}\cup\\{\infty\\}$) if each
stratum is a locally closed $C^{\mu}$ submanifold.

######  Definition 5.1.

A $C^{1}$ stratification $\mathcal{S}$ of $X\subset\mathbb{R}^{N}$ is called
_Whitney_ if the following condition is satisfied.

_Whitney (b)_ : given any pair of strata $X_{\alpha}$ and $X_{\beta}$ for
$\alpha,\beta\in\mathcal{S}$ with $X_{\alpha}<X_{\beta}$, suppose that
$x_{i}\in X_{\beta}$ and $y_{i}\in X_{\alpha}$ are a sequences of points both
converging to $y\in X_{\alpha}$. Suppose also that $T_{x_{i}}X_{\beta}$
converges to a subspace $T\subseteq T_{y}\mathbb{R}^{N}$, and the secant lines
$\overline{x_{i}y_{i}}$ also converge to a line $\ell\subseteq
T_{y}\mathbb{R}^{N}$. Then $\ell\subseteq T$.

Next we review the notion of a triangulation. A $k$-dimensional _simplex_ is a
subset of $\mathbb{R}^{d}$ (for some $d>0$) which is the convex hull of some
finite set of independent points $p_{0},\dots,p_{k}$. The _faces_ of a simplex
are defined in the obvious way. A _simplicial complex_ (see
[munkres2016elementary, Def. 7.1]) is a set $\mathcal{K}$ of simplices in
$\mathbb{R}^{d}$ (for some fixed $d>0$) which satisfies the following
conditions:

  1. (1)

every face of a simplex $\sigma\in\mathcal{K}$ is also in $\mathcal{K}$;

  2. (2)

the non-empty intersection of any two simplices
$\sigma_{1},\sigma_{2}\in\mathcal{K}$ is a face of both $\sigma_{1}$ and
$\sigma_{2}$;

  3. (3)

local finiteness (i.e. each point of $|\mathcal{K}|$ is contained in finitely
many simplices, where $|\mathcal{K}|$ is the topological space formed by the
union of the simplices from $\mathcal{K}$).

Note that the space $|\mathcal{K}|$ admits a stratification whose strata are
given by interiors of simplices from $\mathcal{K}$ and we denote this
stratification by $\mathcal{S}_{\mathcal{K}}$. Each stratum of
$\mathcal{S}_{\mathcal{K}}$ is naturally a smooth submanifold of
$\mathbb{R}^{d}$ and it is easy to see that $\mathcal{S}_{\mathcal{K}}$
defines a Whitney stratification.

The notion of a simplicial subcomplex is defined in the obvious way.

######  Definition 5.2.

Given a smooth manifold $M$, a _smooth triangulation_ of $M$ consists of the
following data:

  1. (1)

a simplicial complex $\mathcal{K}$ and a simplicial subcomplex
$\mathcal{L}\subset\mathcal{K}$;

  2. (2)

a homeomorphism $f:|\mathcal{K}|\setminus|\mathcal{L}|\to X$ such that for
each simplex $\sigma\in\mathcal{K}$ and $x\in\sigma\setminus|\mathcal{L}|$,
there is a neighborhood $U$ of $x$ in the plane containing $\sigma$ and a
smooth embedding $\tilde{f}:U\to X$ extending $f|_{U\cap\sigma}$.

######  Definition 5.3.

For a Whitney stratified space $(X,\mathcal{S})$ with smooth strata, a
triangulation of $(X,\mathcal{S})$ is given by a simplicial complex
$\mathcal{K}$ and a homeomorphism $f:|\mathcal{K}|\to X$ such that for each
stratum $X_{\alpha}$, the set $f^{-1}(X_{\alpha})$ is the underlying