# A Note on Numerical Perverse Sheaves

David B. Massey

###### Abstract.

We define and describe the properties of a class of perverse sheaves which is
very useful when the base ring is not a field.

###### 2010 Mathematics Subject Classification:

32S25, 32S15, 32S55

##  1\. Introduction

Fix a base ring $R$ that is a regular, Noetherian ring with finite Krull
dimension (e.g., ${\mathbb{Z}}$, ${\mathbb{Q}}$, or ${\mathbb{C}}$). Let $X$
be a reduced complex analytic space.

It is common when dealing with perverse sheaves of $R$-modules on $X$ to
assume that $R$ is a field. One main reason for this, perhaps the main reason,
is that, with a base field, the support and cosupport conditions for a
perverse sheaf are Verdier dual to each other, and so the Verdier dual of a
perverse sheaf is again a perverse sheaf. This property is false for general
rings $R$.

For instance, in the case where $X$ consists of a single point and
$R={\mathbb{Z}}$, a perverse sheaf is simply a complex which is quasi-
isomorphic to a finitely-generated ${\mathbb{Z}}$-module in degree 0 and
zeroes in other degrees; however if the finitely-generated
${\mathbb{Z}}$-module in degree 0 has a non-zero free part and a non-zero
torsion part, then the Verdier dual complex will have non-zero cohomology in
both degrees $0$ and $1$, and so is not perverse even up to a shift.

However, for many topological applications, we would like to use integral
cohomology and so would like to choose our base ring to be ${\mathbb{Z}}$,
which is not a field, but is a Dedekind domain. So the question is: is there a
“nice” class of perverse sheaves over a Dedekind domain (we include fields as
Dedekind domains) such that Verdier dualizing yields perverse sheaves and
which behaves well with respect to other important functors?

The answer is “yes”, and we call such complexes of sheaves
$\mathbf{F}^{\bullet}$ numerical, which in particular implies that
$\mathbf{F}^{\bullet}$ is perverse (so, technically, in the phrase “numerical
perverse sheaf” the word “perverse” is superfluous, but we included it in the
title for the sake of making our context clear).

We defined and proved properties of numerical sheaves some time ago, in [7],
but the definition and results were in the midst of much other work, and so
did not stand out as an independent topic. Hence, we are writing this note to
put numerical sheaves in the spotlight.

##  2\. Definition

We continue with a base ring $R$ that is a regular, Noetherian ring with
finite Krull dimension; we will not need to require that $R$ is a Dedekind
domain until we discuss Verdier dualizing numerical sheaves. The definitions
and results here are closely related to [8].

Let ${\mathfrak{S}}$ be a complex analytic Whitney stratification of $X$, with
connected strata. Let $\mathbf{F}^{\bullet}$ be a bounded complex of sheaves
of $R$-modules on $X$, which is constructible with respect to
${\mathfrak{S}}$. For each $S\in{\mathfrak{S}}$, we let $d_{S}:=\dim S$, and
let $({\mathbb{N}}_{S},{\mathbb{L}}_{S})$ denote complex Morse data for $S$ in
$X$, consisting of a normal slice and complex link of $S$ in $X$; see, for
instance, [1] or [6].

######  Definition 2.1.

For each $S\in{\mathfrak{S}}$ and each integer $k$, the isomorphism-type of
the finitely-generated module
$m_{S}^{k}(\mathbf{F}^{\bullet}):={\mathbb{H}}^{k-d_{S}}({\mathbb{N}}_{S},{\mathbb{L}}_{S};\mathbf{F}^{\bullet})$
is independent of the choice of $({\mathbb{N}}_{S},{\mathbb{L}}_{S})$; we
refer to $m_{S}^{k}(\mathbf{F}^{\bullet})$ as the degree $k$ Morse module of
$S$ with respect to $\mathbf{F}^{\bullet}$.

######  Remark 2.2.

The shift by $d_{S}$ above is present so that perverse sheaves can have non-
zero Morse modules in only degree $0$.

######  Definition 2.3.

A complex of sheaves $\mathbf{P}^{\bullet}$ is numerical if and only if the
only non-zero Morse modules of strata are in degree $0$ and in degree $0$ are
free $R$-modules.

We usually say just that $\mathbf{P}^{\bullet}$ is a numerical sheaf or
numerical complex.

######  Remark 2.4.

Definition 2.3 is independent of the stratification, as refining a
stratification with respect to which $\mathbf{P}^{\bullet}$ is constructible
will simply lead to having extra strata with Morse modules that are zero in
all degrees.

We use the term “numerical” since the cohomology of the complex Morse data of
a given stratum is completely determined by the single number which is the
rank in degree $0$. We cannot use the perhaps more-obvious term “free perverse
sheaf” as “free” has a categorical meaning.

The cohomology of the complex Morse data being concentrated in degree zero is
equivalent to the complex being pure with shift $0$ (see Definition 7.5.4 of
[2]). This is equivalent to the complex being perverse ([2], 9.5.2). So we
have:

######  Theorem 2.5.

(Kashiwara and Schapira) Numerical sheaves are perverse.

##  3\. Properties

As we wrote in the introduction, we proved all of our results on numerical
sheaves in [7], but they were obscured by many other results in that paper.

######  Theorem 3.1.

Let $f:X\rightarrow{\mathbb{C}}$ be complex analytic, and let
$i:X-V(f)\hookrightarrow X$ be the (open) inclusion. Suppose that
$\mathbf{P}^{\bullet}$ is a numerical sheaf on $X$. Then, the following
sheaves are also numerical:

  1. (1)

the nearby cycles, $\psi_{f}[-1]\mathbf{P}^{\bullet}$,

  2. (2)

the vanishing cycles, $\phi_{f}[-1]\mathbf{P}^{\bullet}$,

  3. (3)

$i_{*}i^{*}\mathbf{P}^{\bullet}$,

  4. (4)

$i_{!}i^{!}\mathbf{P}^{\bullet}$, and

  5. (5)

the Verdier dual ${\mathcal{D}}\mathbf{P}^{\bullet}$, provided that $R$ is a
Dedekind domain.

###### Proof.

Item 1 is Corollary 6.4 of [7]. Item 2 is Corollary 8.4 of [7]. Items 3 and 4
are Corollary 7.8 of [7].

Item 5 is immediate from Item 7 of Proposition 2.10 of [7]. ∎

##  4\. Examples

In this section, we give some examples, all of which are related to
topological data. Hence, we fix ${\mathbb{Z}}$ as the base ring.

Suppose that $X$ is a local complete intersection of dimension $d$. In the
literature on perverse sheaves, one frequently sees that Lê proved that
${\mathbb{Z}}_{X}^{\bullet}[d]$ is a perverse sheaf in [3]. In fact, what he
proved in [3] is stronger; he proved that each of the complex links of each of
the strata of any Whitney stratification of $X$ have the homotopy-type of a
bouquet of spheres in middle dimension. In cohomological terms, this means
that Lê proved:

######  Theorem 4.1.

(Lê) If $X$ is a local complete intersection of pure dimension $d$, then
${\mathbb{Z}}_{X}^{\bullet}[d]$ is numerical.

######  Example 4.2.

Suppose that $X$ is a local complete intersection of pure dimension $d$
embedded in an open subset ${\mathcal{U}}$ of ${\mathbb{C}}^{n+1}$ and suppose
that ${\mathbf{0}}\in X$. Let $f:(X,{\mathbf{0}})\rightarrow({\mathbb{C}},0)$
be complex analytic, and let $i:X-V(f)\hookrightarrow X$ be the inclusion.

Then $\psi_{f}[-1]{\mathbb{Z}}^{\bullet}_{X}[d]$,
$\phi_{f}[-1]{\mathbb{Z}}^{\bullet}_{X}[d]$, and
$i_{!}i^{!}{\mathbb{Z}}^{\bullet}_{X}[d]$ are numerical by Theorem 4.1 and
Theorem 3.1.

Now suppose that $z_{0}$ is the restriction to $V(f)$ of a generic non-zero
linear form on ${\mathbb{C}}^{n+1}$. Then
$\mathbf{P}^{\bullet}:=\phi_{z_{0}}[-1]i_{!}i^{!}{\mathbb{Z}}^{\bullet}_{X}[d]$
is numerical and either ${\mathbf{0}}$ is not in the support of
$\mathbf{P}^{\bullet}$ or ${\mathbf{0}}$ is an isolated point in the support
of $\mathbf{P}^{\bullet}$. Therefore the stalk cohomology
$H^{k}(\mathbf{P}^{\bullet})_{\mathbf{0}}$ is zero if $k\neq 0$ and
$H^{0}(\mathbf{P}^{\bullet})_{\mathbf{0}}$ is free abelian. This is of
topological interest since $H^{0}(\mathbf{P}^{\bullet})_{\mathbf{0}}\cong
H^{d-1}({\mathbb{L}}_{X,{\mathbf{0}}},{\mathbb{L}}_{V(f),{\mathbf{0}}};\,{\mathbb{Z}})$,
where ${\mathbb{L}}$ denotes the complex link of a space at a point.

######  Example 4.3.

Suppose that $X$ is embedded in an open subset ${\mathcal{U}}$ of
${\mathbb{C}}^{n+1}$ and suppose that ${\mathbf{0}}\in X$. Let
$\mathbf{P}^{\bullet}$ be a numerical sheaf on $X$. Let
${\mathbf{z}}:=(z_{0},z_{1},\dots,z_{n})$ be a generic linear choice of
coordinates for ${\mathbb{C}}^{n+1}$. We will not distinguish in our notation
between the $z_{i}$’s and their restrictions to various subsets.

In Theorem 5.4 of [4] (and also in Theorem 10.9 of [5]), we proved that there
is a chain complex of ${\mathbb{Z}}$-modules with the degree $-i$ term, for
$0\leq i\leq\dim_{0}X$, equal to

$H^{0}(\phi_{z_{i}}[-1]\psi_{z_{i-1}}[-1]\dots\psi_{z_{0}}[-1]\mathbf{P}^{\bullet})_{\mathbf{0}}$

such that the cohomology in any degree $k$ is isomorphic to
$H^{k}(\mathbf{P}^{\bullet})_{\mathbf{0}}$. We called the ranks of these
${\mathbb{Z}}$-modules the characteristic polar multiplicities, and denoted
the rank of the degree $-i$ cohomology by
$\lambda^{i}_{\mathbf{P}^{\bullet},{\mathbf{z}}}$.

Now, since $\mathbf{P}^{\bullet}$ is numerical, we now know that each
$H^{0}(\phi_{z_{i}}[-1]\psi_{z_{i-1}}[-1]\dots\psi_{z_{0}}[-1]\mathbf{P}^{\bullet})_{\mathbf{0}}$
is free abelian, and so we have a chain complex

$0\rightarrow{\mathbb{Z}}^{\lambda^{n}_{\mathbf{P}^{\bullet},{\mathbf{z}}}}\rightarrow{\mathbb{Z}}^{\lambda^{n-1}_{\mathbf{P}^{\bullet},{\mathbf{z}}}}\rightarrow\cdots\rightarrow{\mathbb{Z}}^{\lambda^{1}_{\mathbf{P}^{\bullet},{\mathbf{z}}}}\rightarrow{\mathbb{Z}}^{\lambda^{0}_{\mathbf{P}^{\bullet},{\mathbf{z}}}}\rightarrow
0$

whose homology/cohomology at the
${\mathbb{Z}}^{\lambda^{i}_{\mathbf{P}^{\bullet},{\mathbf{z}}}}$ term is
isomorphic to $H^{-i}(\mathbf{P}^{\bullet})_{\mathbf{0}}$.

In Corollary 10.10 of [5], we applied this to the case where
$\mathbf{P}^{\bullet}=\phi_{f}[-1]{\mathbb{Z}}^{\bullet}_{\mathcal{U}}[n+1]$
for $f:{\mathcal{U}}\rightarrow{\mathbb{C}}$ a complex analytic function and
concluded, in this case, that
$\lambda^{i}_{\mathbf{P}^{\bullet},{\mathbf{z}}}$ was equal to the $i$-th Lê
number of $f$ with respect to ${\mathbf{z}}$ at the origin, where the Lê
numbers $\lambda^{i}_{f,{\mathbf{z}}}$ are actually defined algebraically.
However, we had to switch to ${\mathbb{C}}$ as the base ring in order to give
our chain complex because we did not have the results on numerical sheaves.
Now we know that in the chain complex in Corollary 10.10 of [5], we can
replace the ${\mathbb{C}}$’s by ${\mathbb{Z}}$’s (and we also need to correct
the typographical error which has the indexing reversed on the Lê numbers).

## References

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  * [7] Massey, D.  Calculations with Characteristic Cycles.  https://doi.org/10.48550/arXiv.1101.4304, 2016.
  * [8] Maxim, L., Schürmann, J.  Constructible Sheaf Complexes in Complex Geometry and Applications, volume III, Chapt. 10 of Handbook of Geometry and Topology of Singularities.  Springer, 2022.