# Recollements for graded gentle algebras from spherical band objects

Pierre Bodin

###### Abstract.

In this paper we study the localization of a derived category of a graded
gentle algebra by a subcategory generated by a spherical band object. This
object corresponds to a simple closed curve under the equivalence between the
perfect derived category of the graded gentle algebra and the partially
wrapped Fukaya category of the associated graded marked surface, as
established by Haiden, Katzarkov and Kontsevich.

We describe this localization as a recollement that involves the derived
category of a new graded algebra given by quiver and relations. This leads us
to the introduction of the class of graded pinched gentle algebras, a
generalization of graded gentle algebras. We then show that these algebras are
in bijection with graded marked surfaces with conical singularities. Moreover,
under this correspondence the localization process amounts to the contraction
of the closed curve.

###### Contents

  1. 1 Introduction
    1. 1.1 Context
    2. 1.2 Definitions and main results
    3. 1.3 Graded pinched gentle algebras
  2. 2 Recollections on DG categories
  3. 3 Spectral sequences for Drinfeld quotients
    1. 3.1 Recollections on spectral sequences
    2. 3.2 Application to Drinfeld’s quotients
  4. 4 Admissible dissections and graded pinched gentle algebras
    1. 4.1 Adapted admissible dissections
    2. 4.2 Pinched graded marked surfaces
  5. 5 Formality of the quotient
    1. 5.1 The quasi-equivalence
    2. 5.2 Computation of $\mathrm{H^{*}}(\mathcal{A}/\mathcal{B})_{\circ}$
  6. 6 Proof of the main results

##  1\. Introduction

###  1.1. Context

Gentle algebras, since their introduction in the context of iterated tilted
algebras of type $A$ [AH81, AH82] and $\tilde{A}$ [AS87], were found to be
linked to many other areas of mathematics. Originating in the theory of
cluster algebras from triangulated surfaces [FG09, FST08], geometric models
for gentle algebras have now been developed in numerous directions. For
instance, relying on the description of their representation theory, developed
in particular in [WW85, GP68, BR87, CB89, Kra91], geometric models from
triangulations and partial triangulations of surfaces for modules over gentle
algebras were given in [LF09, ABCJP10, BCS18].

Gentle algebras also arose in the context of Homological Miror Symmetry as
endomorphism algebras of formal generators in partially wrapped Fukaya
categories of graded marked surfaces [HKK17]. Conversely, in works such as
[LP20, OPS18, PPP19, BCS18], any graded gentle algebra is associated to a
graded marked surface with graded admissible dissection. In the homologically
smooth case, [LP20] showed that the bounded derived category of the algebra in
equivalent to the partially wrapped Fukaya category of the associated graded
marked surface. Moreover, in the ungraded case, based on the description of
the bounded derived category of gentle algebra given in [BM03, BD05, ALP16,
CPS19, CPS21], it is shown in [OPS18] that the marked surface with admissible
dissection serves as a model for the indecomposable objects, the irreducible
morphisms and their cones, and the Auslander-Reiten triangles.

These geometric models have since been fruitful for the study of the bounded
derived categories of gentle algebras. For example, this led to a complete
derived invariant for graded gentle algebras [LP20, APS23, Opp19, JSW23],
generalizing a numerical derived invariant of [AAG08](see also [BH07]).

More generally, many algebraic operations on the derived category find an
interpretation in terms of transformations at the level of the surface model.
For example, [CS23, CJS23, JSW23] illustrated the usefulness of this
correspondence in the study of the silting theory of the derived category. In
particular, silting mutation corresponds to the changing of graded arcs and in
some cases to the flipping of diagonals in a quadrilateral. Another example is
the correspondence established in [Opp19] between spherical twists of the
derived category and Dehn twists on the surface.

In this paper we are interested in the role played in this context by the
localization of the derived category by triangulated subcategories generated
by a collection of orthogonal spherical objects.

On one hand, the study of the localization at a boundary arc was done in
[HKK17], relying on the localization of strictly unital
$A_{\infty}$-categories introduced in [LO06]. A more general study of such
localization at an admissible collection of arcs was done in [CS23, CJS23]. It
is shown that the surface obtained by cutting along the arcs, and what the
authors call the surface obtained by supporting along the arcs, give rise to a
recollement at the level of the derived categories (see for instance
[CJS23](Definition 1.5) for a definition).

On the other hand, spherical objects were introduced in [ST01] in order to
construct categorical actions of the braid group. Since then, they have been
shown to play an important role in the study of triangulated categories, and
further similar braid group actions were investigated (see for instance [ST01,
KS02, GTW17, AL17, Opp19] for more details).

Our first result (Theorem 1.3) is a recollement induced by a spherical band
object in the derived category of a graded gentle algebra $\Lambda$. We have
the following recollement:

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where the left hand side gives an explicit description of the localization as
the derived category of an algebra $\Lambda_{(\alpha,\beta)}$ enjoying a
simple definition in terms of quiver with relations, where the ideal is not
necessarily admissible. This leads us to the introduction of the class of
graded pinched gentle algebras (see Definition 1.6), which in particular
contains the class of graded gentle algebras.

We show in Theorem 1.10 and Proposition 1.11 that Theorem 1.3 generalizes to
the class of graded pinched gentle algebras. In particular, this gives the
possibility to iterate the localization process.

On the geometric side, we show in section 4.2 that the isomorphism classes of
graded pinched gentle algebras are in bijection with pinched marked surfaces
with admissible graded dissection, an analogy of graded marked surfaces for
surfaces with conical singularities (see Definition 4.8 and Proposition 4.10).
This is analogous to the correspondence established for gentle algebras in
[OPS18, BCS18, PPP19]. Moreover, the pinched surface associated to
$\Lambda_{(\alpha,\beta)}$ is obtained by contracting the surface associated
to $\Lambda$ along the simple closed curve corresponding to the spherical band
object (Proposition 4.13). Figure 1.1 gives an example of such contraction.

Figure 1.1.  A marked surface with a simple closed curve (blue), and its
pinched
marked surfaces obtained by contraction.

Finally, results on derived equivalence of graded gentle algebras given in
[HKK17, LP20] allow us to give a class of derived equivalences between graded
pinched gentle algebras (Proposition 4.15). A more in depth analysis of the
role of the pinched graded marked surfaces as a model for the bounded derived
category of graded pinched algebras will be made in a future work.

The paper is structured as follows. In the rest of this section we introduce
some definitions and state the main theorem, first for gentle algebras
(subsection 1.2), then for pinched gentle algebras (subsection 1.3). In
section 2 we recall some constructions in DG categories, such as Drinfeld’s DG
quotient [Dri04] which will be used in the proof of the main theorem, and we
introduce some notations. Some basic definitions and results on spectral
sequences are recalled in section 3, and applied to describe the first page of
a spectral sequence on morphism spaces in DG quotients. In section 4 we first
show how one can use the surface model of a gentle algebra in order to choose
a set of generators of the derived category which simplify the calculations
concerning a spherical band object. Then in subsection 4.2 we introduced the
notion of pinched graded marked surfaces and establish the correspondence with
graded pinched gentle algebras. The section 5 is devoted to the proof of a
technical lemma, namely the formality of the quotient algebra. Finally the
proof of the main theorem is made in section 6.

### Acknowledgements

I am grateful to Giulio Salvatori for sharing his notes on surfacehedra, where
the notion of pinched surface is also studied. This paper is part of my Ph.D.
thesis. I would like to thank my advisors Thomas Brüstle and Pierre-Guy
Plamondon.

###  1.2. Definitions and main results

We first introduce some definitions and state our main result in the more
restricted context of graded gentle algebras.

Conventions: A quiver $Q$ is a quadruple $Q=(Q_{0},Q_{1},\sigma,\tau)$ with
finite sets of vertices $Q_{0}$ and arrows $Q_{1}$, and with
$\sigma,\tau:Q_{1}\rightarrow Q_{0}$ the source and target functions. We
consider right modules with the convention of composition (a path
$\beta\alpha$ in a path algebra has source $\sigma(\alpha)$ and target
$\tau(\beta$)), and we use the cohomological convention for complexes. For
each $a\in Q_{0}$, let $e_{a}$ be the path of length zero at $a$ and
$P_{a}=e_{a}\Lambda$ be the associated indecomposable projective
$\Lambda$-module. In the rest of the paper $K$ will denote a field.

######  Definition 1.1.

Let $\Lambda=(KQ/\langle I\rangle,|\ .\ |)$ be a $\mathbb{Z}$-graded algebra.

  1. $\bullet$

$\Lambda$ is said to be gentle if:

     * -

Every vertex of $Q$ has at most two incoming and two outgoing arrows,

     * -

$I$ is a set of paths of length two satisfying: for all $\alpha\in Q_{1}$,
there is at most one arrow $\beta$ such that $0\neq\alpha\beta\in I$; at most
one arrow $\gamma$ such that $0\neq\gamma\alpha\in I$; at most one arrow
$\beta^{\prime}$ such that $0\neq\alpha\beta^{\prime}\notin I$; at most one
arrow $\gamma^{\prime}$ such that $0\neq\gamma^{\prime}\alpha\notin I$.

Suppose now that $\Lambda$ is gentle.

  1. $\bullet$

A graded Kronecker of $\Lambda$ is a pair of arrows $(\alpha,\beta)$ such that
$\alpha$ and $\beta$ have the same source and the same target, the same
degree, and such that they are not loops. It is said to be acyclic if $\alpha$
and $\beta$ do not belong to an oriented cycle of $(Q,I)$.

  1. $\bullet$

let $\omega=\alpha+\mu\beta$ for some graded Kronecker $(\alpha,\beta)$ of
$\Lambda$ and some $\mu\in K^{*}$. The localization of $\Lambda$ at $\omega$
is the graded algebra
$\Lambda[\omega^{-1}]=(K\widetilde{Q}/\langle\widetilde{I}\rangle,\widetilde{|\
.\ |})$ defined by:

    1. -

adding to $Q$ an arrow $\delta$ from 2 to 1 of degree $-|\alpha|$:

${1}$${2}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\scriptstyle{\delta}$

  1.     1. -

adding the relations $\delta\omega-e_{1},\omega\delta-e_{2}$.

  1. $\bullet$

Let $(\alpha,\beta)$ be a graded Kronecker of $\Lambda$. Let
$1:=\sigma(\alpha)=\sigma(\beta)$, $2:=\tau(\alpha)=\tau(\beta)$, and
$\alpha^{+},\alpha^{-},\beta^{+},\beta^{-}$ be either zero or if they exist,
be the (possibly equal) arrows of $Q$ satisfying
$\alpha^{+}\beta,\beta\alpha^{-},\beta^{+}\alpha,\alpha\beta^{-}\in I$ (see
Example 1.2 left).

The pinching of $\Lambda$ at $(\alpha,\beta)$ is the graded algebra
$\Lambda_{(\alpha,\beta)}:=(KQ^{\prime}/\langle I^{\prime}\rangle,|\ .\
|^{\prime})$ defined by:

    1. -

removing $\alpha$ and $\beta$;

    2. -

merging 1 and 2 into a new vertex called 1;

    3. -

adding a loop $\gamma$ of degree 0 at vertex 1;

    4. -

setting $|\alpha^{+}|^{\prime}=|\alpha^{+}|+|\alpha|$ and
$|\beta^{+}|^{\prime}=|\beta^{+}|+|\beta|$, and leaving unchanged the degree
of all the other arrows;

    5. -

keeping all relations in $I$ that don’t involve $\alpha$ and $\beta$, and
adding the relations

$\\{\alpha^{+}\beta^{-},\beta^{+}\alpha^{-},\beta^{+}(\gamma+e_{1}),(\gamma+e_{1})\beta^{-},\alpha^{+}(\gamma-
e_{1}),(\gamma-e_{1})\alpha^{-}\\}$.

  2. $\bullet$

A band object supported by $(\alpha,\beta)$ is an element of the perfect
derived category $per(\Lambda)$ isomorphic to a shift of a twisted complex
(see Definition 2.3) of the form $(P_{2}[|\alpha|]\bigoplus
P_{1}[1],\partial=\begin{pmatrix}0&\alpha+\mu\beta\\\ 0&0\end{pmatrix})$, for
some $\mu\in K^{*}$. We call $\mu$ the parameter of the band object.

######  Example 1.2.

The quiver and relations of a gentle algebra $\Lambda_{1}$ with a graded
Kronecker $(\alpha,\beta)$, and of the associated pinching
${\Lambda_{1}}_{(\alpha,\beta)}$:

${{\tilde{0}}}$${{\tilde{3}}}$${{\tilde{3}}}$${{\tilde{0}}}$${1}$${2}$${1}$${0}$${3}$${0}$${3}$$\scriptstyle{\beta^{-}}$$\scriptstyle{\alpha^{-}}$$\scriptstyle{\beta^{+}}$$\scriptstyle{\alpha^{+}}$$\scriptstyle{\alpha^{-}}$$\scriptstyle{\beta^{-}}$$\scriptstyle{\beta^{+}}$$\scriptstyle{\alpha^{+}}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\scriptstyle{\Lambda_{1}\
\mathrm{with\ graded\ Kronecker}\
(\alpha,\beta)}$$\scriptstyle{{\Lambda_{1}}_{(\alpha,\beta)}}$$\scriptstyle{\langle\alpha^{+}\beta^{-},\
\beta^{+}\alpha^{-},}$$\scriptstyle{\ \beta^{+}(\gamma+e_{1}),\
(\gamma+e_{1})\beta^{-},}$$\scriptstyle{\ \alpha^{+}(\gamma-e_{1}),\ (\gamma-
e_{1})\alpha^{-}\rangle}$$\scriptstyle{\gamma}$

Recall that graded gentle algebras are in one-to-one correspondence with
graded marked surfaces with admissible dissection, and that under this
correspondence, graded curves are associated to objects of the bounded derived
category (see subsection 4.1 and [BCS18, OPS18, PPP19] for more details). Our
first main result is the following.

######  Theorem 1.3.

Let $\Lambda^{\prime}$ be a graded gentle algebra associated to the graded
marked surface with admissible dissection $(S,M,\Delta,G)$. Let $\gamma$ be a
simple closed curve on $S$ with winding number zero that does not enclose a
subsurface containing only punctures, and let $B_{\gamma}$ be an associated
band object at the base of a tube.

  1. (1)

There exists a graded gentle algebra $\Lambda$ and an equivalence
$\Psi:per(\Lambda^{\prime})\rightarrow per(\Lambda)$ such that
$\Psi(B_{\gamma})$ is a band object supported by an acyclic graded Kronecker
$(\alpha,\beta)$ of $\Lambda$, with parameter $\mu$.

  2. (2)

Let $\omega=\alpha+\mu\beta$. There is a recollement:

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}\hbox{${\mathcal{D}(\Lambda[\omega^{-1}])}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
23.70949pt\hfil&\hfil\hskip 39.48608pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{${\mathcal{D}(\Lambda)}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
15.48611pt\hfil&\hfil\hskip 59.12216pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
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{}{}}{}{}{{}{}}}}}{}{{{{{}}{
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{}{}{}}{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{{{}{}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{{}}{}{{}}{}{{{{}}}{{{}}}}{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{{}}{}{{}}{}{{}} {}{}\pgfsys@beginscope\pgfsys@invoke{
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{}{}{}}{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{{{}{}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{{}}{}{{}}{}{{{{}}}{{{}}}}{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{{}}{}{{}}{}{{}} {}{}\pgfsys@beginscope\pgfsys@invoke{
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{}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
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{}{}{}}{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{{{}{}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{{}}{}{{}}{}{{{{}}}{{{}}}}{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{}} {}{}\pgfsys@beginscope\pgfsys@invoke{
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}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{27.87338pt}{-9.38887pt}\pgfsys@lineto{4.67337pt}{-9.38887pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}}{}{ {}{}{}}{}{ {}{}{}} {{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
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{{}}{}{{}}{}{{{{}}}{{{}}}}{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{{}}{}{{}}{}{{}} {}{}\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
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where $x$ is of degree 1.

  3. (3)

There is an equivalence:

$per(\Lambda^{\prime})/thick(B_{\gamma})\simeq per(\Lambda[\omega^{-1}]),$

where the left hand side is the Verdier quotient.

Moreover, if the characteristic of $K$ is different from 2, there is:

  1. (2’)

A recollement:

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}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
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{{}}{}{{}}{}{{{{}}}{{{}}}}{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{{}}{{{}}{{}}}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{}}{}{}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}{{}} {}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{25.98048pt}{-3.43056pt}\pgfsys@lineto{2.78047pt}{-3.43056pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{2.58049pt}{-3.43056pt}\pgfsys@invoke{
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}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$

where $x$ is of degree 1.

  2. (3’)

An equivalence:

$per(\Lambda^{\prime})/thick(B_{\gamma})\simeq per(\Lambda_{(\alpha,\beta)}).$

######  Remark 1.4.

Theorem 1.3 (1) is a reformulation of Corollary 4.6, and (2) and (2’) (resp.
(3) and (3’)) is a direct consequence of Theorem 1.10 (resp. Proposition
1.11).

######  Example 1.5.

The algebras $\Lambda_{1}$ and ${\Lambda_{1}}_{(\alpha,\beta)}$ of Example 1.2
give an example of algebras occurring in the recollement of Theorem 1.3 (2).
An illustration of Theorem 1.3 (1) is given by examples 4.4 and 4.7 by letting
$\Lambda^{\prime}=\Lambda_{0}$ and $\Lambda=\Lambda_{1}$.

###  1.3. Graded pinched gentle algebras

The pinching of a graded gentle algebra at a graded Kronecker leads us to the
introduction of the class of graded pinched gentle algebras.

######  Definition 1.6.

A graded $K$-algebra $\Lambda=(KQ/\langle I\rangle,|\ .\ |)$ is said to be a
graded pinched gentle algebra if there is a decomposition
$Q_{1}=Q_{1}^{g}\bigsqcup Q_{1}^{p}$ and $I=I^{g}\bigsqcup I^{p}$ such that

  1. $\bullet$

$(Q^{g}:=(Q_{0},Q_{1}^{g},\sigma,\tau),\langle I^{g}\rangle)$ is a gentle
bound quiver. The associated gentle algebra is denoted
$\Lambda^{g}:=KQ^{g}/\langle I^{g}\rangle$,

  2. $\bullet$

$Q_{1}^{p}$ is a set of degree zero loops supported on different vertices.

  3. $\bullet$

For $v\in Q_{0}$, the arrows
$\alpha_{v}^{-},\alpha_{v}^{+},\beta_{v}^{-},\beta_{v}^{+}\in Q_{1}^{g}$
satisfying
$\tau(\alpha_{v}^{-})=\sigma(\alpha_{v}^{+})=v=\sigma(\beta_{v}^{+})=\tau(\beta_{v}^{-})$
and $\beta_{v}^{+}\alpha_{v}^{-},\alpha_{v}^{+}\beta_{v}^{-}\in I^{g}$ (with
the possibility of being zero, and with possible compatible identifications
between $\\{\alpha_{v}^{-},\beta_{v}^{-}\\}$ and
$\\{\alpha_{v}^{+},\beta_{v}^{+}\\}$) can be chosen such that:

$I^{p}=\\{\beta_{v}^{+}(\gamma_{v}+e_{v}),(\gamma_{v}+e_{v})\beta_{v}^{-},\alpha_{v}^{+}(\gamma_{v}-e_{v}),(\gamma_{v}-e_{v})\alpha_{v}^{-}\
|\ v\in Q_{0},\ \gamma_{v}\in Q_{1}^{p}\ \mathrm{such\ that}\
\sigma(\gamma_{v})=v\\}$.

######  Remark 1.7.

The “pinched” or “vanishing” relations of $I^{p}$ are such that $\forall i\neq
j\in Q_{0}$, $e_{j}\Lambda e_{i}\simeq e_{j}\Lambda^{g}e_{i}$.

In the special case where $\alpha_{v}^{-}$ and $\alpha_{v}^{+}$ (or
equivalently $\beta_{v}^{-}$ and $\beta_{v}^{+}$) are both zero, the pinched
relations at $v$ can be seen as gentle by letting
$\gamma_{v}^{\prime}=\gamma_{v}+e_{v}$.

######  Example 1.8.

The algebra ${\Lambda_{1}}_{(\alpha,\beta)}$ of Example 1.2 is a pinched
gentle algebra whose associated gentle algebra
$({\Lambda_{1}}_{(\alpha,\beta)})^{g}$ is:

${{\tilde{3}}}$${{\tilde{0}}}$${1}$${0}$${3}$$\scriptstyle{\alpha^{-}}$$\scriptstyle{\beta^{-}}$$\scriptstyle{\beta^{+}}$$\scriptstyle{\alpha^{+}}$$\scriptstyle{({\Lambda_{1}}_{(\alpha,\beta)})^{g}}$

The notion of graded Kronecker and pinching extend naturally to the context of
graded pinched gentle algebras.

######  Definition 1.9.

Let $\Lambda=(KQ/\langle I\rangle,|\ .\ |)$ be a graded pinched gentle
algebra.

  1. $\bullet$

A graded Kronecker $(\alpha,\beta)$ of $\Lambda$ is a graded Kronecker of the
associated gentle algebra $\Lambda^{g}$ such that there is no loop $\gamma\in
Q_{1}^{p}$ based at the source or target of $\alpha$. It is said to be acyclic
if $\alpha$ and $\beta$ do not belong to an oriented cycle of $(Q^{g},I^{g})$.

  2. $\bullet$

For $(\alpha,\beta)$ a graded Kronecker of $\Lambda$, the pinching of
$\Lambda$ at $(\alpha,\beta)$ is the graded pinched gentle algebra
$\Lambda_{(\alpha,\beta)}$ obtained by performing the same local
transformation of $(Q,I)$ as in Definition 1.1.

  3. $\bullet$

The notion of a band object supported by a graded Kronecker is as in
Definition 1.1.

We now state the main results of this paper. Since every graded gentle algebra
can be seen as a graded pinched gentle algebra, Theorem 1.3 (2) is a
particular instance of Theorem 1.10, and Theorem 1.3 (3) a particular instance
of Proposition 1.11. Stating the results in this generality also allows us to
localize at a collection of disjoint simple closed curves by iterating the
process.

######  Theorem 1.10.

Let $\Lambda$ be a graded pinched gentle algebra with an acyclic graded
Kronecker $(\alpha,\beta)$, and let $\omega=\alpha+\mu\beta$ for some $\mu\in
K^{*}$. There exists a recollement:

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to20.65pt{\pgfpicture\makeatletter\hbox{\hskip
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\mathcal{D}(\Lambda[\omega^{-1}])}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
23.70949pt\hfil&\hfil\hskip 39.48608pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\mathcal{D}(\Lambda)}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
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where $x$ is of degree 1.

If the characteristic of $K$ is different from 2, there is an equivalence
$\mathcal{D}(\Lambda[\omega^{-1}])\simeq\mathcal{D}(\Lambda_{(\alpha,\beta)})$.

######  Proposition 1.11.

Let $\Lambda$ be a graded pinched gentle algebra and $B$ a band object of
$per(\Lambda)$ supported by an acyclic graded Kronecker $(\alpha,\beta)$, with
parameter $\mu$. There is an equivalence:

$per(\Lambda)/thick(B)\simeq per(\Lambda[\omega^{-1}]),$

where the left hand side is the Verdier quotient.

If the characteristic of $K$ is different from 2, there is an equivalence
$per(\Lambda[\omega^{-1}])\simeq per(\Lambda_{(\alpha,\beta)})$.

The proof of Theorem 1.10 and Proposition 1.11 is made in section 6.

##  2\. Recollections on DG categories

We recall some definition and properties about DG categories, mainly using
[CC19], [BK91] and [Dri04].

######  Definition 2.1.

  1. $\bullet$

Let $K$ be a field. A DG category $\mathcal{A}$ is a $K$-category where each
Hom set $\mathcal{A}(x,y)$ is a complex of $K$-vector spaces:

$\mathcal{A}(x,y)=(\bigoplus\limits_{p\in\mathbb{Z}}\mathcal{A}(x,y)^{p},d_{\mathcal{A}}),$

and such that the composition
$\mathcal{A}(y,z)\otimes\mathcal{A}(x,y)\rightarrow\mathcal{A}(x,z)$ is a
chain map:

$|g\circ f|=|g|+|f|$ and $d_{\mathcal{A}}(g\circ f)=d_{\mathcal{A}}(g)\circ
f+(-1)^{|g|}g\circ d_{\mathcal{A}}(f)$,

for $f$ and $g$ homogeneous, and where $|g|=p$ for $g\in\mathcal{A}(y,z)^{p}$.

  2. $\bullet$

A DG functor $F:\mathcal{A}\rightarrow\mathcal{B}$ between to DG categories
$\mathcal{A}$ and $\mathcal{B}$ is a $K$-linear functor such that

$F_{x,y}:\mathcal{A}(x,y)\rightarrow\mathcal{B}(F(x),F(y))$

is a chain map, that is $|F(f)|=|f|$ and
$F(d_{\mathcal{A}}(f))=d_{\mathcal{B}}(F(f))$.

  3. $\bullet$

The graded homotopy category $H^{*}(\mathcal{A})$ has the same objects as
$\mathcal{A}$ and the morphism spaces are given by the direct sums of the
cohomology groups of the Hom-complexes of $\mathcal{A}$. The homotopy category
$H^{0}(\mathcal{A})$ is defined by keeping only the cohomology in degree zero.
One can check that these are indeed categories.

  4. $\bullet$

The category $\mathcal{A}^{pre-tr}$ of one-sided twisted complexes over
$\mathcal{A}$ is defined in [BK91] as follow:

    1. -

Objects in $\mathcal{A}^{pre-tr}$ are formal expressions
$(\bigoplus\limits_{i=1}^{n}C_{i}[r_{i}],\partial)$ where $n\geq 0$,
$C_{i}\in\mathcal{A}$, $r_{i}\in\mathbb{Z}$, $\partial=(\partial_{ij})$,
$\partial_{ij}\in\mathcal{A}(C_{j},C_{i})[r_{i}-r_{j}]$ is homogeneous of
degree 1, $\partial_{ij}=0$ for $i\geq j$ and
$d_{\textrm{naive}}\partial+\partial^{2}=0$, where
$d_{\textrm{naive}}\partial:=(d_{\mathcal{A}}(\partial_{ij}))$.

    2. -

$f=(f_{ij})\in\mathcal{A}^{pre-
tr}((\bigoplus\limits_{j=1}^{n}C_{j}[r_{j}],\partial),(\bigoplus\limits_{i=1}^{m}C^{\prime}_{i}[r^{\prime}_{i}],\partial^{\prime}))$
verify $f_{ij}\in\mathcal{A}(C_{j},C^{\prime}_{i})[r^{\prime}_{i}-r_{j}]$ and
the composition is the matrix multiplication.

    3. -

The differential is defined by
$df:=d_{\textrm{naive}}f+\partial^{\prime}f-(-1)^{l}f\partial$ if
deg$f_{ij}=l$.

The DG category $\mathcal{A}$ can be seen as a full DG subcategory of
$\mathcal{A}^{pre-tr}$ by sending its objects to one-sided twisted complexes
concentrated in degree zero.

For $f:X\rightarrow Y$ a closed morphism of degree 0 of $\mathcal{A}$, let
$Cone(f)$ be the object $(Y\bigoplus X[1],\begin{pmatrix}0&f\\\
0&0\end{pmatrix})\in\mathcal{A}^{pre-tr}$. The category $\mathcal{A}^{pre-tr}$
can be thought as adding formally to $\mathcal{A}$ all cones of morphisms, all
cones of morphisms between cones, etc.

######  Notations 2.2.

Let $\mathcal{A}$ be a DG category and
$C=(\bigoplus\limits_{j=1}^{n}C_{j}[r_{j}],\partial),C^{\prime}=(\bigoplus\limits_{i=1}^{m}C^{\prime}_{i}[r^{\prime}_{i}],\partial^{\prime})\in\mathcal{A}^{pre-
tr}$.

An homogeneous basis $\\{b_{h}^{kl}\\}_{1\leq h\leq n^{kl}}$ of
$\mathcal{A}(C_{l},C^{\prime}_{k})$ for each $l\in\\{1,...,n\\}$ and
$k\in\\{1,...,m\\}$ induces a basis $\\{[b_{h}^{kl}]\ |\ 1\leq k\leq m,\
\mathrm{and}\ 1\leq l\leq n,\ \mathrm{and}\ 1\leq h\leq n^{kl}\\}$ of
$\mathcal{A}^{pre-tr}(C,C^{\prime})$, where $[b_{h}^{kl}]_{ij}=b_{h}^{kl}$ if
$i=k$ and $j=l$, and zero otherwise. The degrees in $\mathcal{A}^{pre-tr}$ are
then $|[b_{h}^{kl}]|^{pre-tr}=|b_{h}^{kl}|+r_{l}-r^{\prime}_{k}$.

We will extend this bracket notation to linear combinations and sometimes drop
them when there is no ambiguity and the context is clear.

######  Definition 2.3.

[Dri04](§2.4) A (non-empty) DG category $\mathcal{A}$ is said to be
pretriangulated if:

\- For every object $X\in\mathcal{A}$ and every $k\in\mathbb{Z}$, the object
$X[k]\in\mathcal{A}^{pre-tr}$ is homotopy equivalent (that is, becomes
equivalent in the homotopy category) to an object of $\mathcal{A}$.

-For every closed morphism $f$ in $\mathcal{A}$ of degree 0, the object $Cone(f)\in\mathcal{A}^{pre-tr}$ is homotopy equivalent to an object of $\mathcal{A}$.

$\mathcal{A}$ is said to be strongly pretriangulated if in what precedes the
homotopy equivalences are replaced by invertible closed morphism of degree 0.

######  Proposition 2.4.

[BK91](§4 - Prop 2)

Let $\mathcal{A}$ be a DG category and $\mathcal{A}^{pre-tr}$ the associated
category of one-sided twisted complexes. The homotopy category
$H^{0}(\mathcal{A}^{pre-tr})$ has a natural structure of triangulated
category. It is thus called the triangulated category associated to
$\mathcal{A}$ and denoted by $\mathcal{A}^{tr}$.

######  Remark 2.5.

This comes from the fact, explained in [BK91], that for a DG category
$\mathcal{A}$, the category $\mathcal{A}^{pre-tr}$ is always strongly
pretriangulated. The distinguished triangles are the ones isomorphic to those
of the form $X\rightarrow Y\rightarrow Cone(f)\rightarrow X[1]$, where
$f:X\rightarrow Y$ is a closed morphism in $\mathcal{A}$ of degree 0.

######  Definition 2.6.

[Dri04](§2.3)

A DG functor $F:\mathcal{A}\rightarrow\mathcal{B}$ is said to be a quasi-
equivalence if the induced functor $H^{*}(F):H^{*}(\mathcal{A})\rightarrow
H^{*}(\mathcal{B})$ is fully faithful and $H^{0}(F)$ is essentially
surjective.

It is said to be a DG equivalence if it is fully faithful and for every object
$X\in\mathcal{B}$ there exists an object $X^{\prime}\in F(\mathcal{A})$ and a
closed isomorphism $f:X\rightarrow X^{\prime}$ of degree 0.

######  Proposition 2.7.

[Dri04](Prop 2.5)

If $F:\mathcal{A}\rightarrow\mathcal{B}$ is a quasi-equivalence then the
associated functor $F^{pre-tr}:\mathcal{A}^{pre-
tr}\rightarrow\mathcal{B}^{pre-tr}$ is also a quasi-equivalence.

######  Remark 2.8.

We introduce the notation $full_{\mathcal{C}}(\Omega)$ to designate the full
subcategory of a category $\mathcal{C}$ on a collection of objects $\Omega$ of
$\mathcal{C}$.

Let $\mathcal{A}$ be a DG category. By construction of the category of one-
sided twisted complexes, $(\mathcal{A}^{pre-tr})^{pre-tr}$ is DG equivalent to
$\mathcal{A}^{pre-tr}$. In particular, for any collection $\Omega$ of objects
of $\mathcal{A}^{pre-tr}$, the following inclusions are DG equivalences:
$\mathcal{A}^{pre-tr}\rightarrow(full_{\mathcal{A}^{pre-
tr}}(Ob(\mathcal{A})\bigcup\Omega))^{pre-tr}\rightarrow(\mathcal{A}^{pre-
tr})^{pre-tr}$,

and this induces a triangle equivalence
$\mathcal{A}^{tr}\simeq(full_{\mathcal{A}^{pre-
tr}}(Ob(\mathcal{A})\bigcup\Omega))^{tr}$.

Drinfeld introduced and studied in [Dri04] a notion of DG quotient.

######  Definition 2.9.

[Dri04](§3.1) Let $\mathcal{A}$ be a DG category and $\mathcal{B}$ a full
subcategory.

The quotient DG category $\mathcal{A}/\mathcal{B}$ has the same objects as
$\mathcal{A}$. It has a new morphism $\epsilon_{U}:U\rightarrow U$ of degree
$-1$ for all object $U$ of $\mathcal{B}$, and all of its compositions with
existing morphisms. Formally, one has an isomorphism of vector spaces (but not
of complexes),

$\mathrm{Hom}_{\mathcal{A}/\mathcal{B}}(X,Y)\leftarrow\bigoplus\limits_{n\in\mathbb{N}}\mathrm{Hom}_{\mathcal{A}/\mathcal{B}}^{\langle
n\rangle}(X,Y)$

where $\mathrm{Hom}_{\mathcal{A}/\mathcal{B}}^{\langle n\rangle}(X,Y)$ is the
direct sum, over all families $(U_{i})_{1\leq i\leq n}$ of objects in
$\mathcal{B}$, of tensor products
$\mathrm{Hom}_{\mathcal{A}}(U_{n},U_{n+1})\otimes
K[1]\otimes\mathrm{Hom}_{\mathcal{A}}(U_{n-1},U_{n})\otimes...\otimes
K[1]\otimes\mathrm{Hom}_{\mathcal{A}}(U_{0},U_{1})$ where $U_{n+1}=Y,U_{0}=X$
and $K[1]$ is the complex with only $K$ in degree -1.

If $\epsilon$ is the canonical generator of $K[1]$, the application sends the
product $f_{n}\otimes\epsilon\otimes...\otimes\epsilon\otimes f_{0}$ to the
composition $f_{n}\epsilon_{U_{n}}f_{n-1}...\epsilon_{U_{1}}f_{0}$. The
differential is given by the Leibniz rule and $d(\epsilon_{U})=id_{U}$ for all
$U$ in $\mathcal{B}$. In particular
$\mathrm{Hom}_{\mathcal{A}/\mathcal{B}}^{\langle
0\rangle}(X,Y)=\mathrm{Hom}_{\mathcal{A}}(X,Y)$.

This DG quotient is an enhancement of the triangulated quotient:

######  Theorem 2.10.

[Dri04](Th 3.4) There is a triangulated equivalence

$(\mathcal{A}/\mathcal{B})^{tr}\simeq\mathcal{A}^{tr}/\mathcal{B}^{tr}.$

######  Definition 2.11.

Let $\Lambda:=(KQ/\langle I\rangle,|\ .\ |,d)$ be a DG algebra given by a DG
quiver with relations. We view $\Lambda$ as a DG category
$\mathcal{P}(\Lambda)$ with objects $\mathcal{P}(\Lambda)_{0}=Q_{0}$ and
morphism spaces $\mathcal{P}(\Lambda)(i,j)=e_{j}\Lambda e_{i}$, with the
induced grading and differential.

######  Proposition 2.12.

There is an equivalence of triangulated categories
$\mathcal{P}(\Lambda)^{tr}\simeq per(\Lambda)$.

Proof: As noted in Remark 2.5, $\mathcal{P}(\Lambda)^{pre-tr}$ is strongly
pretriangulated, and thus according to [CC19](Rem 6.2.5 (2)), the inclusion
$\mathcal{P}(\Lambda)^{tr}\simeq(\mathcal{P}(\Lambda)^{pre-tr})^{tr}\subset
per(\mathcal{P}(\Lambda)^{op})$ is a triangulated equivalence up to direct
summands.

$\square$

######  Setting 2.13.

Let $\Lambda$ be a graded pinched gentle algebra (see Definition 1.6) and $B$
in $\mathcal{P}(\Lambda)^{pre-tr}$ be a band object supported by an acyclic
graded Kronecker of $\Lambda$ as in Definition 1.9. Let
$\mathcal{A}:=full_{\mathcal{P}(\Lambda)^{pre-
tr}}(\mathcal{P}(\Lambda)_{0}\bigcup\\{B\\})$ and
$\mathcal{B}=full_{\mathcal{A}}(B)$.

######  Theorem 2.14.

[BK91](§4 - Th 1)

The category $\mathcal{B}^{tr}$ is the smallest strictly full triangulated
subcategory of $\mathcal{A}^{tr}\simeq per(\Lambda)$ containing $B$.

We will denote $\mathcal{B}^{tr}$ by $\langle B\rangle_{per(\Lambda)}$ or
$\langle B\rangle$ when the context is clear.

##  3\. Spectral sequences for Drinfeld quotients

###  3.1. Recollections on spectral sequences

Given a full subcategory $\mathcal{B}$ of a DG category $\mathcal{A}$, the
$\mathrm{Hom}$ spaces of $\mathcal{A}/\mathcal{B}$ naturally inherit a
filtration which gives rise to a spectral sequence that can be used to compute
the $\mathrm{Hom}$ spaces of $H^{*}\mathcal{A}/\mathcal{B}$. The main
reference in this section is McCleary’s book A user’s guide to spectral
sequences [McC].

######  Definition 3.1.

[McC](Def 2.1)

A differential bigraded module over a ring $R$ is a collection of $R$-modules
$\\{E^{p,q}\\}$ where $p$ and $q$ are integers, together with an $R$-linear
mapping, $d:E^{*,*}\rightarrow E^{*,*}$, the differential, of bidegree
$(s,1-s)$ for some integer $s$, and satisfying $d\circ d=0$.

With the differential, we can take the homology of a differential bigraded
module:

$H^{p,q}(E^{*,*},d)=\mathrm{Ker}\ d:E^{p,q}\rightarrow E^{p+s,q-s+1}\ /\
\mathrm{Im}\ d:E^{p-s,q+s-1}\rightarrow E^{p,q}$

######  Definition 3.2.

[McC](Def 2.2)

A spectral sequence of cohomological type starting on page $a$ is a collection
of differential bigraded $R$-modules $\\{E_{r}^{*,*},d_{r}\\}$, where $r\geq
a$ and such that for all $p,q,r$, $E_{r+1}^{p,q}$ is isomorphic to
$H^{p,q}(E_{r}^{*,*},d_{r})$.

Suppose that $a=2$. For the sake of clarity we omit the bigrading. Let

$Z_{2}=\mathrm{Ker}\ d_{2}\ \mathrm{and}\ B_{2}=\mathrm{Im}\ d_{2}.$

The condition $d_{2}\circ d_{2}=0$ implies $B_{2}\subset Z_{2}\subset E_{2}$
and by definition $E_{3}\simeq Z_{2}/B_{2}$. Write $\overline{Z_{3}}$ for
$\mathrm{Ker}\ d_{3}:E_{3}\rightarrow E_{3}$. Since $\overline{Z_{3}}$ is a
submodule of $E_{3}$, it can be written as $Z_{3}/B_{2}$ where $Z_{3}$ is a
submodule of $Z_{2}$. Similarly $\overline{B_{3}}:=\mathrm{Im}\ d_{3}$ is
isomorphic to $B_{3}/B_{2}$ and so

$E_{4}\simeq\overline{Z}_{3}/\overline{B}_{3}\simeq(Z_{3}/B_{2})/(B_{3}/B_{2})\simeq
Z_{3}/B_{3}.$

These data can be presented as a tower of inclusions: $B_{2}\subset
B_{3}\subset Z_{3}\subset Z_{2}\subset E_{2}$. Iterating this process, we
present the spectral sequence as an infinite tower of submodules of $E_{2}$:

$B_{2}\subset B_{3}\subset...\subset B_{n}\subset...\ ...\subset
Z_{n}\subset...\subset Z_{3}\subset Z_{2}\subset E_{2}$

with the property that $E_{n+1}\simeq Z_{n}/B_{n}$ and the differential
$d_{n+1}$ can be taken as a mapping $Z_{n}/B_{n}\rightarrow Z_{n}/B_{n}$,
which has kernel $Z_{n+1}/B_{n}$ and image $B_{n+1}/B_{n}$.

Let $Z_{\infty}=\bigcap_{n}Z_{n}$ and $B_{\infty}=\bigcup B_{n}$. From the
tower of inclusions it is clear that $B_{\infty}\subset Z_{\infty}$ and we put
$E_{\infty}=Z_{\infty}/B_{\infty}$.

######  Definition 3.3.

An increasing filtration

$...\subset F^{p-1}C\subset F^{p}C\subset F^{p+1}C\subset...\subset C$

of a cochain complex $(C,d)$ of $R$-modules is said to be:

  1. (1)

compatible with the differential if $d$ restrict to a map $d:F^{p}C\rightarrow
F^{p}C$ for each $p\in\mathbb{Z}$,

  2. (2)

exhaustive if $C=\bigcup_{s}F^{s}C$,

  3. (3)

bounded below if for each $n$, there is a value $s(n)$ with
$F^{s(n)}C^{n}=\\{0\\}$.

If $F$ is compatible with the differential, we refer to $(C,d)$ as a filtered
cochain complex and designate it by $(C,d,F)$.

If $H^{*}$ is a graded $R$-module and $F$ is a filtration on $H^{*}$, we
define the associated bigraded module by

$E^{p,q}(H^{*},F)=F^{p}H^{p+q}/F^{p-1}H^{p+q}$.

######  Definition 3.4.

[McC](Def 2.4) A spectral sequence $\\{E_{r}^{*,*},d_{r}\\}$ is said to
converge to a graded $R$-module $H^{*}$ if there is a filtration $F$ on
$H^{*}$ such that

$E_{\infty}^{p,q}\simeq E^{p,q}(H^{*},F),$

where $E_{\infty}^{p,q}$ is the limit term of the spectral sequence.

######  Remark 3.5.

Suppose that $R$ is a field and that $E^{*,*}$ converges to $H^{*}$ with a
filtration $F$ bounded below. Since all short exact sequences split this
convergence implies
$H^{p+q}\simeq\bigoplus\limits_{k\in\mathbb{Z}}E_{\infty}^{p+k,q-k}$.

Given a filtered cochain complex $(C,d,F)$ of $R$-modules, the associated
graded $R$-module $H(C,d)$ inherits a filtration

$F^{p}H(C,d)=\mathrm{Im}\big{(}H(\iota):H(F^{p}C,d)\rightarrow H(C,d)\big{)}$

where $\iota$ is the inclusion.

######  Theorem 3.6.

[McC](Th - 2.6) Each filtered cochain complex $(C,d,F)$, with $F$ increasing,
determines a spectral sequence $\\{E_{r}^{*,*},d_{r}\\}$ starting on page 1
with $d_{r}$ of bidegree $(-r,r+1)$ and

$E_{1}^{p,q}\simeq H^{p+q}(F^{p}C/F^{p-1}C)$.

In practice, we build this spectral sequence recursively, starting with $r=0$.
Let $E_{0}^{p,q}=F^{p}C^{p+q}/F^{p-1}C^{p+q}$ and
$d_{0}^{p,q}:E_{0}^{p,q}\rightarrow E_{0}^{p,q+1}$ the differential induced by
the quotient.

Suppose that $E_{r}^{p,q}$ and $d_{r}^{p,q}:E_{r}^{p,q}\rightarrow
E_{r}^{p-r,q+r+1}$ are given. By definition of a spectral sequence,
$E_{r+1}^{p,q}\simeq\mathrm{Ker}(d_{r}^{p,q})/\mathrm{Im}(d_{r}^{p+r,q-r-1})$.
One can then verify that the differential $d$ induces a differential
$d_{r+1}^{p,q}:E_{r+1}^{p,q}\rightarrow E_{r+1}^{p-(r+1),q+(r+1)+1}$.

######  Theorem 3.7.

[McC](Th 3.2) Let $(C,d,F)$ be a filtered cochain complex such that the
filtration is exhaustive and bounded below. Then the associated spectral
sequence starting with $E_{1}^{p,q}\simeq H^{p+q}(F^{p}C/F^{p-1}C)$ converges
to $H(C,d)$, that is,

$E_{\infty}^{p,q}\simeq F^{p}H^{p+q}(C,d)/F^{p-1}H^{p+q}(C,d).$

###  3.2. Application to Drinfeld’s quotients

Morphism spaces in DG quotients of DG categories naturally inherit a
filtration that can be used to define a spectral sequence and compute
homology.

######  Remark 3.8.

Using the notation of Definition 2.9, for all objects $X,Y$ in a DG category
$\mathcal{A}$, the natural filtration

$0=F^{-1}C\subset...\subset F^{p-1}C\subset F^{p}C\subset...\subset
C:=\mathrm{Hom}_{\mathcal{A}/\mathcal{B}}(X,Y)$

given by
$F^{p}C\simeq\bigoplus\limits_{n=0}^{p}\mathrm{Hom}_{\mathcal{A}/\mathcal{B}}^{\langle
n\rangle}(X,Y)$, is exhaustive and bounded below.

Since we are working with DG categories over a field, the previous theorem
combined with Remark 3.5 allow us to express the homology complexes
$H^{*}(\mathrm{Hom}_{\mathcal{A}/\mathcal{B}}(X,Y))$ as
$H^{p+q}(\mathrm{Hom}_{\mathcal{A}/\mathcal{B}}(X,Y))\simeq\bigoplus\limits_{k\in\mathbb{Z}}E_{\infty}^{p+k,q-k}$,
where $E_{r}^{*,*}$ is the spectral sequence associated to the natural
filtration on the $\mathrm{Hom}$ complexes. Note that $\forall p<0,\forall
q\in\mathbb{Z}$, $E_{0}^{p,q}\simeq 0$.

The following proposition will allow us to compute easily the first terms
$E_{1}^{p,q}$.

######  Proposition 3.9.

Let $\mathcal{B}$ be a full subcategory of a DG category $\mathcal{A}$ over a
field $K$, and let $X,Y$ be two objects of $\mathcal{A}$. Let
$\\{E_{r}^{*,*},d_{r}\\}$ be the spectral sequence associated to the natural
filtration of $C=\mathrm{Hom}_{\mathcal{A}/\mathcal{B}}(X,Y)$. Then

$\displaystyle
E_{1}^{p,q}\simeq\bigoplus\limits_{(U_{i})\in\mathcal{B}}\bigoplus\limits_{\begin{subarray}{c}k_{1}+...+k_{p+1}\\\
=\ 2p+q\end{subarray}}H^{k_{p+1}}(\mathrm{Hom}_{\mathcal{A}}(U_{p},Y))\otimes
K[1]\otimes$ $\displaystyle
H^{k_{p}}(\mathrm{Hom}_{\mathcal{A}}(U_{p-1},U_{p}))\otimes...$
$\displaystyle...\otimes K[1]\otimes
H^{k_{1}}(\mathrm{Hom}_{\mathcal{A}}(X,U_{1})).$

Proof:

By definition, $E_{1}^{p,q}=H^{p+q}(F^{p}C/F^{p-1}C)$. Moreover
$F^{p}C/F^{p-1}C$ is isomorphic to

$\bigoplus\limits_{(U_{i})\in\mathcal{B}}\mathrm{Hom}_{\mathcal{A}}(U_{p},Y)\otimes
K[1]\otimes\mathrm{Hom}_{\mathcal{A}}(U_{p-1},U_{p})\otimes...\otimes
K[1]\otimes\mathrm{Hom}_{\mathcal{A}}(X,U_{1})$ as a graded vector space. We
will show that $F^{p}C/F^{p-1}C$ is actually isomorphic to it as a complex,
endowed with the differential coming from the tensor product.

Recall that given complexes $(C^{r},d^{r}),...,(C^{1},d^{1})$, their tensor
product is defined by $(C^{r}\otimes...\otimes C^{1},d^{\otimes})$ where for
an homogeneous element $x_{r}\otimes...\otimes x_{1}$, $d^{\otimes}$ is given
by $d^{\otimes}(x_{r}\otimes...\otimes
x_{1})=\sum\limits_{k=1}^{r}(-1)^{\sum\limits_{l=k+1}^{r}deg(x_{l})}x_{r}\otimes...\otimes
d_{k}(x_{k})\otimes...\otimes x_{1}$.

On one hand for $f_{p}\otimes\epsilon_{p}\otimes
f_{p-1}\otimes...\otimes\epsilon_{1}\otimes f_{0}$ a homogeneous element in
$(\bigoplus\limits_{(U_{i})\in\mathcal{B}}\mathrm{Hom}_{\mathcal{A}}(U_{p},Y)\otimes
K[1]\otimes\mathrm{Hom}_{\mathcal{A}}(U_{p-1},U_{p})\otimes...\otimes
K[1]\otimes\mathrm{Hom}_{\mathcal{A}}(X,U_{1}),d^{\otimes})$,

$\displaystyle d^{\otimes}(f_{p}$ $\displaystyle\otimes\epsilon_{p}\otimes
f_{p-1}\otimes...\otimes\epsilon_{1}\otimes f_{0})$
$\displaystyle=\sum\limits_{k=0}^{p}(-1)^{\sum\limits_{l=k+1}^{p}deg(f_{l})-(p-k)}f_{p}\otimes\epsilon_{p}\otimes...\otimes
d(f_{k})\otimes...\otimes\epsilon_{1}\otimes f_{0}$
$\displaystyle+\sum\limits_{k=1}^{p}(-1)^{\sum\limits_{l=k}^{p}deg(f_{l})-(p-k)}f_{p}\otimes\epsilon_{p}\otimes...\otimes
d(\epsilon_{k})\otimes...\otimes\epsilon_{1}\otimes f_{0}$
$\displaystyle=\sum\limits_{k=0}^{p}(-1)^{\sum\limits_{l=k+1}^{p}deg(f_{l})-(p-k)}f_{p}\otimes\epsilon_{p}\otimes...\otimes
d(f_{k})\otimes...\otimes\epsilon_{1}\otimes f_{0}$

since $K[1]$ is endowed with the zero differential.

On the other hand, for a homogeneous element
$\overline{f_{p}\epsilon_{p}f_{p-1}...\epsilon_{1}f_{0}}$ of
$F^{p}C/F^{p-1}C$,

$\displaystyle d(\overline{f_{p}\epsilon_{p}f_{p-1}...\epsilon_{1}f_{0}})=$
$\displaystyle\sum\limits_{k=0}^{p}(-1)^{\sum\limits_{l=k+1}^{p}deg(f_{l})-(p-k)}\overline{f_{p}\epsilon_{p}...d(f_{k})...\epsilon_{1}f_{0}}$
$\displaystyle+\sum\limits_{k=1}^{p}(-1)^{\sum\limits_{l=k}^{p}deg(f_{l})-(p-k)}\overline{f_{p}\epsilon_{p}...f_{k}f_{k-1}...\epsilon_{1}f_{0}}$
$\displaystyle=\sum\limits_{k=0}^{p}(-1)^{\sum\limits_{l=k+1}^{p}deg(f_{l})-(p-k)}\overline{f_{p}\epsilon_{p}...d(f_{k})...\epsilon_{1}f_{0}}$

since the $f_{p}\epsilon_{p}...f_{k}f_{k-1}...\epsilon_{1}f_{0}$ are in
$F^{p-1}C$. This shows that the differential induced by the quotient coincides
with the one coming from the tensor product, inducing an isomorphism of
complex.

For the moment we have shown an isomorphism of complexes

$\displaystyle E_{1}^{p,q}\simeq
H^{p+q}(\bigoplus\limits_{(U_{i})\in\mathcal{B}}\mathrm{Hom}_{\mathcal{A}}(U_{p},Y)\otimes
K[1]\otimes$
$\displaystyle\mathrm{Hom}_{\mathcal{A}}(U_{p-1},U_{p})\otimes...$
$\displaystyle...\otimes
K[1]\otimes\mathrm{Hom}_{\mathcal{A}}(X,U_{1}),d^{\otimes}).$

We conclude using the Künneth formula for complexes ([W94](Th 3.6.3)). Given
two complexes $P$ and $Q$, there exist for all $n$ a short exact sequence

$0\rightarrow\bigoplus\limits_{i+j=n}H^{i}(P)\otimes H^{j}(Q)\rightarrow
H^{n}(P\otimes
Q)\rightarrow\bigoplus\limits_{i+j=n-1}Tor_{1}^{K}(H^{i}(P),H^{j}(Q))\rightarrow
0.$

Since we are working with vector spaces, the torsion groups vanish and there
is an isomorphism $\bigoplus\limits_{i+j=n}H^{i}(P)\otimes H^{j}(Q)\simeq
H^{n}(P\otimes Q)$. This gives

$\displaystyle
E_{1}^{p,q}\simeq\bigoplus\limits_{(U_{i})\in\mathcal{B}}\bigoplus\limits_{\begin{subarray}{c}h_{1}+...+h_{2p+1}\\\
=p+q\end{subarray}}H^{h_{2p+1}}(\mathrm{Hom}_{\mathcal{A}}(U_{p},Y))\otimes
H^{h_{2p}}(K[1])\otimes$ $\displaystyle
H^{h_{2p-1}}(\mathrm{Hom}_{\mathcal{A}}(U_{p-1},U_{p}))\otimes...$
$\displaystyle...\otimes H^{h_{2}}(K[1])\otimes
H^{h_{1}}(\mathrm{Hom}_{\mathcal{A}}(X,U_{1})).$

Finally since $H^{i}(K[1])$ is non zero if and only if $i=-1$, this gives the
desired formula.

$\square$

##  4\. Admissible dissections and graded pinched gentle algebras

###  4.1. Adapted admissible dissections

In order to simplify computations in $per(\Lambda)$, it will be useful to
choose an adapted set of generators. Recall that graded gentle algebras are in
one-to-one correspondence with marked surfaces with graded admissible
dissection, that is, a quadruple $(S,M,\Delta,G)$ where $S$ is a compact
oriented surface with boundary, $M$ a finite set of
$\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}$
and $\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet$
marked points and punctures, $\Delta$ an admissible
$\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}$-dissection,
and $G$ a grading of the minimal oriented intersections. See for example
[HKK17, OPS18, APS23, LP20] for the statement of this result, and for more
details and examples. Throughout this section, we will use the terminology of
[CJS23](section 1.5). Contrary to the convention adopted in [APS23], we will
draw marked surfaces with admissible dissection in such a way that the
orientation of the arrows of the corresponding quiver will be given by
rotating clockwise around a
$\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}$-point.

By a theorem of [LP20] this correspondence gives rise, in the homologically
smooth case (that is, in the case without
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet$-punctures),
to an equivalence between the derived category of the graded gentle algebra
and the partially wrapped Fukaya category $\mathcal{W}(S,M,\eta(\Delta,G))$ of
the associated marked surface with graded admissible dissection:

######  Theorem 4.1.

[LP20](Theorem 3.2.2) For a homologically smooth graded gentle algebra
$\Lambda$ with associated marked surface with graded admissible dissection
$(S,M,\Delta,G)$, there is a line field $\eta(\Delta,G)$ on $S$ such that
there is an equivalence

$per(\Lambda)\simeq\mathcal{W}(S,M,\eta(\Delta,G))$.

######  Remark 4.2.

In case $\Lambda$ is not homologically smooth, following [OZ22](Definition
3.24, Remark 3.26) we can still embed $per(\Lambda)$ into a partially wrapped
Fukaya category $\mathcal{W}(S_{s},M_{s},\eta(\Delta_{s},G_{s}))$ as follows.
If $\Lambda$ is associated to $(S,M,\Delta,G)$, let $S_{s}$ be the marked
surface obtained by replacing each
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet$-puncture
by a boundary component containing one
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet$-point
and one
$\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}$-point,
and let $M_{s}$ be the new set of marked points and punctures. The collection
$\Delta$ induces an arc system on $S_{s}$ that can be completed into a full
arc system $\Delta_{s}$ by choosing a new
$\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}$-arc
for each new boundary component. Finally a choice of grading $G_{s}$ that
specializes to $G$ on $\Delta$ induces an equivalence between $per(\Lambda)$
and the full subcategory of $\mathcal{W}(S_{s},M_{s},\eta(\Delta_{s},G_{s}))$
generated by $\Delta$. Equivalences for
$\mathcal{W}(S_{s},M_{s},\eta(\Delta_{s},G_{s}))$ induce equivalences for
$per(\Lambda)$ (see [OZ22], Lemma 3.28).

For $\gamma:\mathbb{S}^{1}\rightarrow S$ an immersed curve, we denote by
$w_{\eta(\Delta,G)}(\gamma)$ its winding number with respect to
$\eta(\Delta,G)$ (see [LP20] Definition 1.1.3).

Following [OPS18](Assumption 2.7), we assume that any finite collection of
curves is in minimal position, that is, the number of intersections of each
pair of (not necessarily distinct) curves in this set is minimal in their
respective homotopy class. According to [T08], it follows from [FHS82] and
[NC01] that, up to homotopy, this assumption is always satisfied. Given two
curves $\gamma,\gamma^{\prime}$, denote by $|\gamma\cap\gamma^{\prime}|$ this
minimal number of intersections.

For $\gamma:\mathbb{S}^{1}\rightarrow S$ a simple closed curve, let
$D_{\gamma}:S\rightarrow S$ be the associated Dehn twist along $\gamma$ (see
for instance [FM12] for a definition). For $\delta$ a
$\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}$-arc
on $(S,M)$ intersecting a simple closed curve $\gamma$, each (possibly equal)
end point of $\delta$ give rise to an oriented intersection between $\delta$
and $D_{\gamma}(\delta)$, denoted $\alpha(\delta)$ and $\beta(\delta)$. Recall
that an oriented intersection between to consecutive
$\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}$-arcs
at a
$\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}$-point
is called a minimal oriented intersection.

######  Proposition 4.3.

Let $(S,M,\Delta,G)$ be a graded marked surface and let
$\gamma_{1},...,\gamma_{r}$ be a collection of pairwise non-intersecting
simple closed curves on $S$ with zero winding number. Suppose moreover that
these curves do not enclose a subsurface containing only punctures.

There exists a graded admissible
$\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}$-dissection
$(\Delta^{\prime},G^{\prime})$ on $(S,M)$ such that
$\eta(\Delta^{\prime},G^{\prime})\simeq\eta(\Delta,G)$, and for all
$i\in\\{1,...,r\\}$ there exists $\delta_{i}\in\Delta^{\prime}$ such that:

  * •

The endpoints of $\delta_{i}$ are not
$\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}$-punctures,

  * •

$|\delta_{i}\cap\gamma_{i}|=1$,

  * •

$D_{\gamma_{i}}(\delta_{i})\in\Delta^{\prime}$,

  * •

For all
$\delta\in\Delta^{\prime}\backslash\\{\delta_{i},D_{\gamma_{i}}(\delta_{i})\\}$,
$|\delta\cap\gamma_{i}|=0$,

  * •

$\alpha(\delta_{i})$ and $\beta(\delta_{i})$ are minimal and
$G^{\prime}(\alpha(\delta_{i}))=G^{\prime}(\beta(\delta_{i}))$.

We will call such a dissection $(\Delta^{\prime},G^{\prime})$ an admissible
dissection adapted to the collection $\\{\gamma_{1},...,\gamma_{r}\\}$.

######  Example 4.4.

Consider the graded gentle algebra $\Lambda_{0}$ given by:

${5}$${4}$${0}$${3}$${1}$${2}$$\scriptstyle{a}$$\scriptstyle{b}$$\scriptstyle{c}$$\scriptstyle{d}$$\scriptstyle{f}$$\scriptstyle{e}$

and $|a|=|d|=|f|=0$, $|c|=1$ and $|b|=|e|=-1$. The associated graded marked
surface $S$ is depicted on the left of the following figure:

The figure on the right is an example of adapted admissible dissection (the
grading being zero) to the unique simple closed curve of $S$. Let $S_{1}$ be
this new graded marked surface.

The proof of Proposition 4.3 will rely on the following lemma:

######  Lemma 4.5.

[LP20](Remark 1.2.5) Let $S$ be an oriented surface with non-empty boundary
$\partial S$ decomposing as the disjoint union of connected components
$\partial S=\bigsqcup\limits_{i=1}^{r}\partial_{i}S$.

A line field $\eta$ on $S$ relates its genus $g$ with the winding number of
its boundary components via the formula:

$\sum\limits_{i=1}^{r}w_{\eta(\Delta,G)}(\partial_{i}S)=4-2r-4g$.

Proof of Proposition 4.3: Each simple closed curve $\gamma_{i}$ has zero
winding number, regardless of the choice of orientation.

Let $S^{\prime}$ be the surface obtained by cutting $S$ along each
$\gamma_{i}$. It is the disjoint union of connected surfaces
$S^{\prime}=\bigsqcup\limits_{j=1}^{s}S^{\prime}_{j}$, and each $\gamma_{i}$
give rise to two boundary components of $S^{\prime}$: $\gamma_{i}^{+}$ and
$\gamma_{i}^{-}$. Let us show by contradiction that each $S^{\prime}_{j}$
contains a boundary component of the original surface $S$ (and thus contains
at least one
$\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}$
marked point on this boundary).

Since by hypothesis each $S^{\prime}_{j}$ cannot contain only punctures, each
$S^{\prime}_{j}$ containing a puncture must contain a boundary component of
$S$.

Suppose now that $S^{\prime}_{j}$ does not contain any puncture and that all
its distinct boundary components $\rho_{1}^{j},...,\rho_{r_{j}}^{j}$ are in
$\\{\gamma_{1}^{+},...,\gamma_{r}^{+},\gamma_{1}^{-},...,\gamma_{r}^{-}\\}$.
Let $g_{j}$ be its genus. By Lemma 4.5, after choosing the orientation on each
$\rho_{i}^{j}$ which is compatible with the orientation of $S^{\prime}_{j}$,
we have

$\displaystyle\sum\limits_{i=1}^{r_{j}}w_{\eta(\Delta,G)}(\rho_{i}^{j})=4-2r_{j}-4g_{j}$
$\displaystyle\Leftrightarrow 0=4-2r_{j}-4g_{j}$ $\displaystyle\Leftrightarrow
r_{j}=2\ \text{and}\ g_{j}=0,$

but then $\rho_{1}^{j}$ and $\rho_{2}^{j}$ should be homotopic, a
contradiction.

Thus we can choose an arbitrary
$\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}$
marked point $m_{j}$ in each $S^{\prime}_{j}$.

For all $j\in\\{1,...,s\\}$, and for all $k\in\\{1,...,r_{j}\\}$, let
$\delta_{k}^{j}$ be a simple arc from $\delta_{k}^{j}(0)=m_{j}$ to
$\delta_{k}^{j}(1)=\rho_{k}^{j}(1)$ (after the identification with some
$\gamma_{i}^{\pm}:\mathbb{S}^{1}\rightarrow S$), and such that $\forall k\neq
k^{\prime}\in\\{1,...,r_{j}\\}$,
$|\delta_{k}^{j}\cap\delta_{k^{\prime}}^{j}|=0$.

For each $i\in\\{1,...,r\\}$, this gives rise to a
$\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}$-arc
$\delta_{i}$ on $S$ intersecting $\gamma_{i}$ once (and no other
$\gamma_{i^{\prime}\neq i})$ once and transversely, in the following way: if
$\gamma_{i}^{+}=\rho_{k}^{j}$ and
$\gamma_{i}^{-}=\rho_{k^{\prime}}^{j^{\prime}}$, define $\delta_{i}$ to be the
concatenation
$\delta_{i}=\delta_{k}^{j}.(\delta_{k^{\prime}}^{j^{\prime}})^{-1}$. Moreover,
define $\mu_{i}^{+}=\delta_{k}^{j}.\gamma_{i}.(\delta_{k}^{j})^{-1}$ and
$\mu_{i}^{-}=\delta_{k^{\prime}}^{j^{\prime}}.\gamma_{i}.(\delta_{k^{\prime}}^{j^{\prime}})^{-1}$.
Cutting $S$ along the $\mu_{i}^{\pm}$, we obtain a surface
$\Gamma=\Gamma^{\prime}\bigsqcup(\bigsqcup\limits_{i=1}^{r}\Gamma_{i})$, where
each $\Gamma_{i}$ is an annulus containing $\gamma_{i}$ and whose boundary
components $\mu_{i}^{\pm}$ each have one
$\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}$
marked point. The following figure depicts the situation:

Let $\Gamma^{\prime}=\bigsqcup\limits_{k\in K_{0}}\Gamma_{k}$ be the
decomposition of $\Gamma^{\prime}$ into connected components. We have seen
above that for each $k$ in $K_{0}$, $\Gamma_{k}$ contains at least one
$\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}$
marked point, thus [HKK17](Lemma 3.3) ensures that it admits an admissible
dissection $A_{k}$. We set $\Delta^{\prime}:=(\bigsqcup\limits_{k\in
K_{0}}A_{k})\bigsqcup\\{\delta_{1},...,\delta_{r},D_{\gamma_{1}}(\delta_{1}),...,D_{\gamma_{r}}(\delta_{r})\\}$.
In order to see that this indeed give an admissible dissection on $(S,M)$, we
need to check that each pair of arcs are non-intersecting, which is the case
by construction, that they are pairwise distinct and that they cut $S$ into
polygons, each of which containing exactly one
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet$-point.
We call such a polygon an admissible polygon.

The curves in $\Delta^{\prime}\bigsqcup\\{\gamma_{1},...,\gamma_{r}\\}$ are in
minimal position by assumption. For $k\neq k^{\prime}\in K_{0}$, any
$\delta\in A_{k}$ and $\delta^{\prime}\in A_{k^{\prime}}$ are distinct since
they do not share common endpoints. For all $i$, $\delta_{i}$ is distinct from
$D_{\gamma_{i}}(\delta_{i})$ and for $i\neq j\in\\{1,...,r\\}$,
$\delta_{i}\not\simeq\delta_{j},\delta_{i}\not\simeq
D_{\gamma_{j}}(\delta_{j})$ and $D_{\gamma_{i}}(\delta_{i})\not\simeq
D_{\gamma_{j}}(\delta_{j})$ since they do not have the same minimal relative
position with respect to the set $\\{\gamma_{1},...,\gamma_{r}\\}$.

Let $S_{0}$ be a subsurface cut out by $\Delta^{\prime}$. If all boundary arcs
of $S_{0}$ are contained in some $A_{k}$, then $S_{0}$ is an admissible
polygon since $A_{k}$ is an admissible collection. Otherwise $S_{0}$ is
obtained by gluing an admissible polygon of some $A_{k}$ with a collection of
triangles of the form $(\mu_{i}^{\pm},\delta_{i},D_{\gamma_{i}}(\delta_{i}))$.
The gluing is made along the $\mu_{i}^{\mp}$’s, and since elements of
$\\{\delta_{1},...,\delta_{r},D_{\gamma_{1}}(\delta_{1}),...,D_{\gamma_{r}}(\delta_{r})\\}$
are all distinct, $S_{0}$ is a polygon. Finally since each triangle
$(\mu_{i}^{\pm},\delta_{i},D_{\gamma_{i}}(\delta_{i}))$ contains no
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet$
points, $S_{0}$ is admissible.

Now $\eta(\Delta,G)$ induces a unique grading $G^{\prime}$ on
$\Delta^{\prime}$ such that
$\eta(\Delta^{\prime},G^{\prime})\simeq\eta(\Delta,G)$, and by construction
$\alpha(\delta_{i})$ and $\beta(\delta_{i})$ are minimal for all $i$. The fact
that $G^{\prime}(\alpha(\delta_{i}))=G^{\prime}(\beta(\delta_{i}))$ follows
from $w_{\eta(\Delta,G)}(\gamma_{i})=0$.

$\square$

Proposition 4.3 has the following consequence on the algebraic side:

######  Corollary 4.6.

Let $\Lambda=(KQ/I,|\ .\ |)$ be a graded gentle algebra associated to the
graded marked surface with admissible dissection $(S,M,\Delta,G)$.

Let $B_{1},...,B_{r}$ be a collection of band objects of $per(\Lambda)$ of
parameters $(n_{i}=1,\mu_{i}\in K^{*})$ corresponding to a collection
$\gamma_{1},...,\gamma_{r}$ of simple closed curves on $S$ under the
equivalence $per(\Lambda)\simeq\mathcal{W}(S,M,\eta(\Delta,G))$ or under the
inclusion into $\mathcal{W}(S_{s},M_{s},\eta(\Delta_{s},G_{s}))$ in case
$\Lambda$ is not homologically smooth. Suppose that the $\gamma_{i}$’s are
pairwise non-intersecting and do not enclose a subsurface containing only
punctures.

There exists a graded gentle algebra
$\Lambda^{\prime}=(KQ^{\prime}/I^{\prime},|\ .\ |^{\prime})$ and an
equivalence $\Psi:per(\Lambda)\rightarrow per(\Lambda^{\prime})$ such that
each $\Psi(B_{i})$ is a band object supported by an acyclic graded Kronecker
$(\alpha_{i},\beta_{i})$ of $\Lambda^{\prime}$, and such that $\forall i\neq
j$,
$\\{\sigma(\alpha_{i}),\tau(\alpha_{i})\\}\bigcap\\{\sigma(\alpha_{j}),\tau(\alpha_{j})\\}=\emptyset$.

######  Example 4.7.

Let $\Lambda_{0}$ be the gentle algebra of Example 4.4, and $B$ be a band
object of parameter $(1,\mu)$ of $per(\Lambda)$ corresponding to the only
simple closed curve of $S$. Then Theorem 4.1 shows that $\Lambda_{0}$ is
derived equivalent to the gentle algebra associated the graded marked surface
$S_{1}$. This algebra is the gentle algebra $\Lambda_{1}$ of Example 1.2.

Under this equivalence, $B$ is isomorphic to (a shift of) $(P_{2}\bigoplus
P_{1}[1],\partial=\begin{pmatrix}0&\alpha+\mu\beta\\\ 0&0\end{pmatrix})$.

Proof of Corollary 4.6: A choice of admissible dissection
$(\Delta^{\prime},G^{\prime})$ as in 4.3 gives the desired algebra
$\Lambda^{\prime}$, which is derived equivalent to $\Lambda$ by
[HKK17](Proposition 3.2). The acyclic graded Kroneckers come from the minimal
angles $\\{\alpha(\delta_{i}),\beta(\delta_{i})\\}$, and the description of
the $B_{i}$’s in $per(\Lambda^{\prime})$ in term of one sided twisted
complexes is given by [HKK17](Theorem 4.3).

$\square$

###  4.2. Pinched graded marked surfaces

In this section we establish the correspondence between graded pinched gentle
algebras and pinched marked surfaces with simple admissible dissections, a
generalisation of graded marked surface to surfaces with conical
singularities.

######  Definition 4.8.

  1. $\bullet$

A (topological) oriented surface with conical singularities is a compact
topological space $S$ in which every point admits a neighbourhood homeomorphic
to one of the following:

    1. -

The open unit disk $D$ of $\mathbb{R}^{2}$,

    2. -

The half disk $D\cap H$, where $H$ is the closed upper half plane of
$\mathbb{R}^{2}$,

    3. -

The wedge sum $C_{0}=D\sqcup D^{\prime}/\sim$ of two pointed open disks (see
the left-hand side of Figure 4.1).

Since $S$ is compact, it admits a finite number of boundary components, each
homeomorphic to a circle, and a finite set of points $C$, called the set of
conical singularities, that correspond to the singular point of $C_{0}$. The
surface $S\backslash C$ must be oriented. Elements of $C$ will be represented
by symbols
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\mathbin{\mathchoice{\ooalign{$\displaystyle\circ$\cr$\displaystyle\cdot$\cr}}{\ooalign{$\textstyle\circ$\cr$\textstyle\cdot$\cr}}{\ooalign{$\scriptstyle\circ$\cr$\scriptstyle\cdot$\cr}}{\ooalign{$\scriptscriptstyle\circ$\cr$\scriptscriptstyle\cdot$\cr}}}$.

  2. $\bullet$

A pinched marked surface $\mathcal{S}=(S,C,M\sqcup P)$ is the data of:

    1. -

A surface with conical singularities $(S,C)$,

    2. -

A finite set
$P=P_{\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}}\sqcup
P_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}$
of points in the interior of $S\backslash C$, called punctures and represented
respectively by symbols
$\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}$
and
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet$,

    3. -

A finite set
$M=M_{\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}}\sqcup
M_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}$
of points in the boundary of $S\backslash C$, called marked points and
represented respectively by symbols
$\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}$
and
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet$.
They are required to alternate on each boundary component, and each component
must contain at least one marked point.

  3. $\bullet$

A
$\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}$-arc
is a non contractible curve on $S$ with endpoints in
$M_{\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}}\sqcup
P_{\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}}$.
Every curve will be considered up to homotopy. Note that
$\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}$-arcs
are allowed to go through conical singularities.

  4. $\bullet$

Let $\mathcal{S}_{\sslash}$ be the marked surface obtained from $\mathcal{S}$
by splitting each conical singularity into two
$\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}$-punctures
(see the right-hand side of Figure 4.1). A simple admissible dissection
$\Delta$ on $\mathcal{S}$ is a set of
$\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}$-arcs
that don’t intersect (in the interior of $S$) and that satisfy:

    1. -

For each $c\in C$ there is exactly one arc $\gamma_{c}$ in $\Delta$ that goes
through $c$, it goes through it only once and does not go through other
singularities,

    2. -

The collection $\Delta_{\sslash}$ of arcs on $\mathcal{S}_{\sslash}$ induced
from $\Delta$ by splitting each $\gamma_{c}$ in two, is an admissible
dissection.

  5. $\bullet$

A grading $G$ on a simple admissible dissection $\Delta$ is the data of an
integer for each minimal oriented angle of $\Delta$.

######  Remark 4.9.

There are two other equivalent ways of thinking of pinched marked surfaces.

  1. (1)

First we can think of a pinched marked surface as a marked surface where pairs
of
$\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}$-punctures
have been identified. More precisely, given a marked surface $(S,M\sqcup P)$
and a fixed-point free involution $\iota$ on a subset
$P_{\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}}^{\iota}$
of
$P_{\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}}$,
the associated pinched marked surface $\mathcal{S}$ is obtained by taking the
quotient of $S$ under the identifications $p\sim\iota(p)$ for all $p\in
P_{\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}}^{\iota}$.
Thus the set of conical singularities is the quotient of
$P_{\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}}^{\iota}$
under the action of $\iota$, and the new set of
$\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}$-punctures
is
$P_{\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}}\backslash
P_{\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}}^{\iota}$.
This construction is inverse to the splitting of the conical singularities
$\mathcal{S}_{\sslash}$, where $\iota$ is given by remembering each pairing of
new
$\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}$-punctures.

Under this correspondence, simple admissible dissections of the pinched marked
surface $\mathcal{S}$ are in bijection with admissible dissections on
$(S,M\sqcup P)$ satisfying that each puncture in
$P_{\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}}^{\iota}$
is of valency one. Under this correspondence, the
$\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}$-arcs
$\gamma_{p}$ and $\gamma_{\iota(p)}$ admitting respectively $p$ and $\iota(p)$
in
$P_{\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}}^{\iota}$
as one of their end points, give rise via concatenation to an arc going
through the corresponding conical singularity.

Note that here we must exclude the case where $(S,M\sqcup P)$ is a sphere
without boundary, with one
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet$-puncture
and two
$\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}$-punctures
$p$ and $p^{\prime}$, and with $\iota(p)=p^{\prime}$, since the
$\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}$-arc
in the admissible dissection of $(S,M\sqcup P)$ would descend to a closed
curve on $\mathcal{S}$. This is the only case where this can happen.

Each grading $G$ on $(\mathcal{S},\Delta)$ corresponds to a grading
$G_{\sslash}$ on $(\mathcal{S}_{\sslash},\Delta_{\sslash})$ that associate
zero to each minimal oriented angle at
$\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}$-punctures
arising from conical singularities.

  2. (2)

Similarly each collection of non intersecting simple closed curves on a marked
surface $(S,M\sqcup P)$ gives rise to a pinched marked surface by taking the
topological quotient that identifies each simple closed curve with a point.

The next proposition follows easily from the correspondence established in
[HKK17, BCS18, PPP19, LP20] for gentle algebras. A quiver with relation
$(Q,I)$ as in Definition 1.6 will be called a pinched gentle quiver.

######  Proposition 4.10.

Graded pinched gentle quivers are in one to one correspondence with pinched
marked surfaces with a graded simple admissible dissection.

Proof: Let $(Q,I,|\ .\ |)$ be a graded pinched gentle quiver with set of
vanishing loops $Q_{1}^{p}$. Let $\gamma_{v}$ be a loop in $Q_{1}^{p}$
attached at vertex $v$, and let $\alpha^{+},\alpha^{-},\beta^{+},\beta^{-}$ be
the (possibly identified or equal to zero) arrows given in Definition 1.6. By
Remark 1.7, we can assume that at least one of $\alpha^{+},\alpha^{-}$, and at
least one of $\beta^{+},\beta^{-}$ is non-zero. Let $(Q^{\prime},I^{\prime},|\
.\ |^{\prime})$ be the graded quiver with relations obtained by resolving each
loop in $Q_{1}^{p}$ in the following way:

${{\tilde{3}}}$${{\tilde{0}}}$${{\tilde{3}}}$${{\tilde{0}}}$${1}$${{1_{\alpha}}}$${{1_{\beta}}}$${0}$${3}$${0}$${3}$$\scriptstyle{\alpha^{-}}$$\scriptstyle{\beta^{-}}$$\scriptstyle{\beta^{+}}$$\scriptstyle{\alpha^{+}}$$\scriptstyle{(Q,I,|\
.\
|)}$$\scriptstyle{\alpha^{-}}$$\scriptstyle{\alpha^{+}}$$\scriptstyle{\beta^{-}}$$\scriptstyle{\beta^{+}}$$\scriptstyle{(Q^{\prime},I^{\prime},|\
.\
|^{\prime})}$$\scriptstyle{\gamma}$$\scriptstyle{\gamma_{\alpha}}$$\scriptstyle{\gamma_{\beta}}$

Each loop $\gamma$ in $Q_{1}^{p}$ gives rise to two degree zero loops
$\gamma_{\alpha}$ and $\gamma_{\beta}$ attached on different vertices, and
satisfying the relations
$\alpha^{+}\gamma_{\alpha}=\gamma_{\alpha}\alpha^{-}=0=\beta^{+}\gamma_{\beta}=\gamma_{\beta}\beta^{-}$.
Thus $(Q^{\prime},I^{\prime},|\ .\ |^{\prime})$ is a graded gentle quiver and
thus corresponds to a marked surface with graded admissible dissection
$(S,M\sqcup P,\Delta,G)$. Moreover for each $\gamma$ in $Q_{1}^{p}$, the set
of couples $(\gamma_{\alpha},\gamma_{\beta})$ give the data of a partial
coupling on the
$\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}$-punctures
of $(S,M\sqcup P)$. By Remark 4.9 (1), this gives rise to a unique pinched
marked surfaces with graded simple admissible dissection.

Conversely, starting from a pinched marked surfaces with graded simple
admissible dissection $\mathcal{S}$, one can split its conical singularities
to obtain $\mathcal{S}_{\sslash}$ as in Remark 4.9 (1). One gets an associated
graded gentle quiver and can do the inverse procedure of the last figure to
obtain a graded pinched gentle quiver. By definition the two constructions are
inverse from each other.

$\square$

######  Example 4.11.

Consider the pinched gentle algebra ${\Lambda_{1}}_{(\alpha,\beta)}$ of
Example 1.2 (it corresponds to the left quiver with relations of the last
figure). The corresponding pinched marked surface, depicted on the left part
of the following figure, is homeomorphic to the conical singularity $C_{0}$.
The right part illustrates the marked surface with admissible dissection
obtained by splitting the conical singularity.

Figure 4.1.  A pinched marked surface with graded simple admissible
dissection,
and the splitting of its conical singularity.

######  Remark 4.12.

Simple admissible dissections are expected to be a special case of a more
general notion of admissible dissection on pinched marked surfaces. This
notion should induce a bijection with generating sets of the bounded derived
category of the corresponding graded pinched gentle algebra. For example, one
could consider dissections admitting several arcs going through a conical
singularity.

The localization described in Theorem 1.10 corresponds at the level of pinched
surfaces to the contraction of the simple closed curve. More precisely, we
have the following proposition:

######  Proposition 4.13.

Let $\Lambda$ be a graded pinched gentle algebra with a graded Kronecker
$(\alpha,\beta)$, let $\mathcal{S}$ be its associated pinched marked surface
and let $\gamma$ be the simple closed curve on $\mathcal{S}$ corresponding to
$(\alpha,\beta)$. (Note that by definition of a graded Kronecker, $\gamma$
does not go through a conical singularity).

The pinched marked surface $\mathcal{S}_{(\alpha,\beta)}$, associated to the
pinching $\Lambda_{(\alpha,\beta)}$ of $\Lambda$ at $(\alpha,\beta)$, is
homeomorphic to the quotient of $\mathcal{S}$ that identifies all points in
$\gamma$.

Proof: The pinched marked surface obtained by identifying all points in
$\gamma$ is the same as the one obtained by:

(i) Splitting $\mathcal{S}$ at $\gamma$ to obtain a pinched marked surface
with new boundary components $\gamma_{\alpha}$ and $\gamma_{\beta}$,

(ii) contracting $\gamma_{\alpha}$ and $\gamma_{\beta}$ to obtain two
$\color[rgb]{0.0546875,0.58984375,0.0078125}\definecolor[named]{pgfstrokecolor}{rgb}{0.0546875,0.58984375,0.0078125}{\circ}$-punctures
$p_{\alpha}$ and $p_{\beta}$,

(iii) gluing $p_{\alpha}$ and $p_{\beta}$ to create a conical singularity.

Splitting the other conical singularities of the pinched marked surface
$\mathcal{S}^{\prime}$ obtained at step (ii) yields exactly the one used in
the proof of Proposition 4.10 to define $\mathcal{S}_{(\alpha,\beta)}$.

$\square$

######  Remark 4.14.

Up to derived equivalence we can suppose that $(\alpha,\beta)$ is in degree
zero. In this case, the pinching of $\mathcal{S}$ at $\gamma$ sends the graded
simple admissible dissection associated to $\Lambda$ to the graded simple
admissible dissection associated to $\Lambda_{(\alpha,\beta)}$ (with the
identification of the arcs corresponding to $\sigma(\alpha)$ and
$\tau(\alpha)$).

We now state a useful result on derived equivalences for graded pinched gentle
algebras. For a closed curve $\gamma$ on a pinched marked surface with simple
admissible dissection $(\mathcal{S},\Delta,G)$ that does not go through a
singularity, we define its winding number to be
$w_{\eta(\Delta_{\sslash},G_{\sslash})}(\gamma)$.

######  Proposition 4.15.

Let $\Lambda$ be a graded pinched gentle algebra corresponding to a pinched
marked surface with graded simple admissible dissection
$(\mathcal{S},\Delta,G)$. Let $\gamma$ be a simple closed curve on
$\mathcal{S}$ with zero winding number that does not go through a singularity
nor encloses a subsurface containing only punctures or singularities.

There exists a graded simple admissible dissection
$(\Delta^{\prime},G^{\prime})$ on $\mathcal{S}$ that is adapted to $\gamma$
(see Proposition 4.3), and such that $per(\Lambda)\simeq
per(\Lambda^{\prime})$ where $\Lambda^{\prime}$ is the graded pinched gentle
algebra associated to $(\mathcal{S},\Delta^{\prime},G^{\prime})$.

We can then use this new description of the perfect derived category to apply
Proposition 1.11 and take the pinching of $\Lambda^{\prime}$ at the graded
Kronecker $(\alpha,\beta)$ corresponding to $\gamma$. By Proposition 4.13,
$\Lambda^{\prime}_{(\alpha,\beta)}$ gives a graded simple admissible
dissection on the pinched marked surface obtained by contracting $\gamma$ in
$\mathcal{S}$.

Proof: Let $\Gamma$ be the graded gentle algebra obtained by replacing each
vanishing loop $\delta_{i}$ in $Q_{1}^{p}$ by a zero graded Kronecker
$(\alpha_{i},\beta_{i})$ in such a way that $\Gamma$ satisfies
$\Gamma_{(\alpha_{i},\beta_{i})_{i}}=\Lambda$. By Proposition 1.11, each
choice of band objects $B_{i}$ supported on $(\alpha_{i},\beta_{i})$ gives an
equivalence $per(\Lambda)\simeq per(\Gamma)/thick(B_{i})_{i}$.

Let $(\mathcal{S}(\Gamma),\Delta(\Gamma),G(\Gamma))$ be the marked surface
with admissible dissection associated to $\Gamma$, and for all $i$ let
$\gamma_{i}$ be the simple closed curve corresponding to
$(\alpha_{i},\beta_{i})$. The curve $\gamma$ is lifted to a curve on
$\mathcal{S}$ with zero winding number, and it does not intersect the
$\gamma_{i}$’s. Moreover this collection of closed curves does not enclose a
subsurface containing only punctures, so we can apply Proposition 4.3. Let
$(\Delta(\Gamma)^{\prime},G(\Gamma)^{\prime})$ be the resulting graded
admissible dissection, let $\Gamma^{\prime}$ be the associated graded gentle
algebra, and let $(\alpha^{\prime}_{i},\beta^{\prime}_{i})_{i},(\alpha,\beta)$
be the new graded Kroneckers.

Setting
$\Lambda^{\prime}=\Gamma^{\prime}_{(\alpha^{\prime}_{i},\beta^{\prime}_{i})_{i}}$
we get $per(\Lambda^{\prime})\simeq
per(\Gamma^{\prime})/thick(B_{i})_{i}\simeq per(\Gamma)/thick(B_{i})_{i}\simeq
per(\Lambda)$. By Remark 4.14, the graded simple admissible dissection
associated to $\Lambda^{\prime}$ corresponds to the image of
$(\Delta(\Gamma)^{\prime},G(\Gamma)^{\prime})$ under the pinching at each
$\gamma_{i}$, and thus is adapted to $\gamma$.

$\square$

##  5\. Formality of the quotient

In this section we prove a technical result, namely the formality of
$(\mathcal{A}/\mathcal{B})_{\circ}:=full_{\mathcal{A}/\mathcal{B}}(\mathcal{P}(\Lambda)_{0})$,
using the notations of Setting 2.13. We place ourselves in this setting:
$\Lambda=(KQ/\langle I\rangle,|\ .\ |)$ is a graded pinched gentle algebra and
$B$ is a band object of parameter $\mu\in K^{*}$ in $per(\Lambda)$, supported
by an acyclic graded Kronecker $(\alpha,\beta)$. We set
$\omega:=\alpha+\mu\beta$ and $1:=\sigma(\alpha),2:=\tau(\alpha)$.

###  5.1. The quasi-equivalence

######  Lemma 5.1.

There is a DG equivalence

$(\mathcal{A}/\mathcal{B})_{\circ}\simeq\mathcal{P}(K\widehat{Q}/\langle
I\rangle,\widehat{|\ .\ |},\widehat{d})$,

where:

  1. -

$(K\widehat{Q}/\langle I\rangle,\widehat{|\ .\ |})$ is obtained by adding to
$Q$ an arrow $\gamma$ from 1 to 2 of degree $|\alpha|-2$, loops $\epsilon_{1}$
and $\epsilon_{2}$ at 1 and 2 of degree $-1$, and an arrow $\delta$ from 2 to
1 of degree $-|\alpha|$:

${1}$${2}$$\scriptstyle{\epsilon_{1}}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\scriptstyle{\gamma}$$\scriptstyle{\epsilon_{2}}$$\scriptstyle{\delta}$

  1. -

$\widehat{d}$ is defined by setting $\widehat{d}(\epsilon_{1})=\delta\omega-
e_{1}$, $\widehat{d}(\epsilon_{2})=(-1)^{q}(e_{2}-\omega\delta)$,
$\widehat{d}(\gamma)=(-1)^{q+1}(\omega\epsilon_{1}+\epsilon_{2}\omega)$, and
by sending every other arrows to zero.

Proof: Recall that $\epsilon$ is the new generator added to
$\mathrm{End}_{\mathcal{A}}(B)$ when taking the quotient.

Let $i,j\in Q_{0}$. Using the bracket of Notation 2.2, let
$[e_{1}):=[e_{1}]\in\mathrm{Hom}_{\mathcal{A}}(P_{1},B)$,
$[e_{2}):=[e_{2}]\in\mathrm{Hom}_{\mathcal{A}}(P_{2},B)$,
$(e_{1}]:=[e_{1}]\in\mathrm{Hom}_{\mathcal{A}}(B,P_{1})$ and
$(e_{2}]:=[e_{2}]\in\mathrm{Hom}_{\mathcal{A}}(B,P_{2})$. Every morphism
$\rho=\begin{pmatrix}\rho_{2}\\\
\rho_{1}\end{pmatrix}\in\mathrm{Hom}_{\mathcal{A}}(P_{i},B)$ factors as
$\rho=(e_{2}]\rho_{2}+(e_{1}]\rho_{1}$ and dually for
$\rho^{\prime}\in\mathrm{Hom}_{\mathcal{A}}(B,P_{j})$.

The equivalence is then given by sending $\mathrm{Hom}_{\Lambda}(P_{i},P_{j})$
to itself via $\Lambda\hookrightarrow K\widehat{Q}/\langle I\rangle$ for all
$i,j$, and $(e_{1}]\epsilon[e_{1})$ to $\epsilon_{1}$,
$(e_{2}]\epsilon[e_{2})$ to $\epsilon_{2}$, $(e_{1}]\epsilon[e_{2})$ to
$\delta$ and $(e_{2}]\epsilon[e_{1})$ to $\gamma$.

$\square$

######  Definition 5.2.

Recall that the localization $\Lambda[\omega^{-1}]$ of $\Lambda$ at $\omega$
has been introduced in Definition 1.1. Using notations of Lemma 5.1, we define
the morphism of DG algebras $\phi:(K\widehat{Q}/\langle I\rangle,\widehat{|\
.\ |},\widehat{d})\rightarrow\Lambda[\omega^{-1}]$ in the following way:

Let $\phi:Q_{0}\sqcup\widehat{Q}_{1}\rightarrow K\widetilde{Q}$ be given by
$\phi_{|Q_{0}\sqcup Q_{1}\sqcup\\{\delta\\}}=id$,
$\phi(\epsilon_{1})=\phi(\epsilon_{2})=\phi(\gamma)=0$. It extends to an
epimorphism of graded algebra $\phi:K\widehat{Q}\rightarrow K\widetilde{Q}$,
and to an epimorhism of DG algebra $\phi:(K\widehat{Q}/\langle
I\rangle,\widehat{|\ .\ |},\widehat{d})\rightarrow\Lambda[\omega^{-1}]$ since
$\phi(I)\subset\widetilde{I}$ and
$\phi(\widehat{d}(\epsilon_{1})),\phi(\widehat{d}(\epsilon_{2})),\phi(\widehat{d}(\gamma))\in\langle\widetilde{I}\rangle$.

The next subsection will be devoted to the proof of the following proposition.

######  Proposition 5.3.

The functor
$\mathcal{P}(\phi):(\mathcal{A}/\mathcal{B})_{\circ}\rightarrow\mathcal{P}(\Lambda[\omega^{-1}])$
induced by the morphism $\phi$ of Definition 5.2 is a quasi-equivalence.

###  5.2. Computation of $\mathrm{H^{*}}(\mathcal{A}/\mathcal{B})_{\circ}$

######  Lemma 5.4.

$H^{*}\mathrm{End}_{\mathcal{A}}(B)\ \simeq K[x]/(x^{2})$, where x is of
degree 1

Proof: For a graded pinched gentle algebra $\Lambda=(KQ/\langle I\rangle,|\ .\
|)$ and $i\neq j\in Q_{0}$, a basis of $e_{j}\Lambda e_{i}\simeq
e_{j}\Lambda^{g}e_{i}$ is given by paths in $(Q^{g},I^{g})$. Let us give a
basis of $e_{1}\Lambda e_{2}$ in case it is non-empty.

First there can be no path from 2 to 1 of the form $\rho^{\prime}\alpha\rho$
or $\rho^{\prime}\beta\rho$. Indeed, if $\rho^{\prime}\alpha\rho\in
e_{1}\Lambda e_{2}$ then the acyclicity of $(\alpha,\beta)$ ensures that
$\rho\alpha\in I^{g}$ and $\alpha\rho^{\prime}\in I^{g}$, but then
$\rho\beta\notin I^{g}$ and $\beta\rho^{\prime}\notin I^{g}$, and thus
$\rho^{\prime}\alpha\rho\beta$ is an oriented cycle, a contradiction.

Now let $\rho$ be a non-zero path in $e_{1}\Lambda e_{2}$ not passing through
$\alpha$ or $\beta$. We can suppose without loss of generality that
$\rho\alpha\notin I^{g}$. The acyclicity of $(\alpha,\beta)$ implies that
$\alpha\rho\in I^{g}$ and thus $\beta\rho\notin I^{g}$.

There can be no other path in $e_{1}\Lambda e_{2}$ different from $\rho$.
Indeed suppose that $\rho^{\prime}$ is such a path. Then
$\rho^{\prime}\beta\in I^{g}$, otherwise $\rho^{\prime}\beta\rho\in
e_{1}\Lambda e_{2}$ is a contradiction. Since in a gentle bound quiver, every
path $\gamma$ belongs to a unique maximal path $\hat{\gamma}$, we have
$\hat{\rho}=\hat{\alpha}=\hat{\rho^{\prime}}$. But then we can suppose that
$\rho=\rho^{\prime\prime}\rho^{\prime}$, and thus
$\rho=\rho^{\prime\prime\prime}\alpha\rho^{\prime}$ or
$\rho=\rho^{\prime\prime\prime}\beta\rho^{\prime}$, a contradiction.

In conclusion, if $e_{1}\Lambda e_{2}\neq\emptyset$ then $e_{1}\Lambda
e_{2}=\langle\rho\rangle$ where $\rho$ is a path which we can suppose verify
$\rho\alpha,\beta\rho\notin I$. For the rest this proof, let $\rho=0$ if
$e_{1}\Lambda e_{2}$ is zero.

We can deduce:

(1) $e_{1}\Lambda e_{1}=\langle e_{1},\rho\alpha\rangle,\ \ e_{2}\Lambda
e_{1}=\langle\alpha,\beta,\beta\rho\alpha\rangle,\ \ e_{1}\Lambda
e_{2}=\langle\rho\rangle,\ \ e_{2}\Lambda e_{2}=\langle
e_{2},\beta\rho\rangle,$

which gives the basis
$\mathrm{End}_{\mathcal{A}}(B)=\langle[e_{1}],[e_{2}],[\alpha],[\beta],[\rho],[\rho\alpha],[\beta\rho],[\beta\rho\alpha]\rangle$,
with

$|[e_{1}]|=|[e_{2}]|=0,|[\alpha]|=|[\beta]|=1,|[\rho]|=|\rho|+q-1$.

Dropping without ambiguity the bracket notation, the differential is given by:

$\displaystyle d(e_{1})=\omega=-d(e_{2}),\ \ $ $\displaystyle\hskip
28.45274pt\ \ \ \
d(\rho)=\omega\rho-(-1)^{|[\rho]|}\rho\omega=\mu\beta\rho-(-1)^{|[\rho]|}\rho\alpha,$
$\displaystyle d(\alpha)=d(\beta)=0,\ \ \ \ \ $ $\displaystyle\hskip
28.45274ptd(\beta\rho)=(-1)^{|[\rho]|}\beta\rho\alpha,\ \
d(\rho\alpha)=\mu\beta\rho\alpha,\ \ d(\beta\rho\alpha)=0,$

which gives the decomposition as complexes

$\mathrm{End}_{\mathcal{A}}(B)\ \simeq\
\langle[e_{1}],[e_{2}],[\alpha],[\beta]\rangle\bigoplus\langle[\rho],[\rho\alpha],[\beta\rho],[\beta\rho\alpha]\rangle\
=:\ E_{0}\bigoplus E_{1}$.

Now $E_{1}$ is acyclic since $\mathrm{Ker}\
d_{|E_{1}}=\langle\rho\alpha-(-1)^{|[\rho]|}\mu\beta\rho\rangle\bigoplus\langle\beta\rho\alpha\rangle=\mathrm{Im}\
d_{|E_{1}}$, and

(2) $H^{*}\mathrm{End}_{\mathcal{A}}(B)\ \simeq\ H^{*}E_{0}\ \simeq\ \langle
id=e_{1}+e_{2}\rangle^{0}\bigoplus\langle\alpha,\beta\ |\ \omega\rangle^{1}.$

$\square$

A direct computation shows that:

######  Lemma 5.5.

Using the notations of (1),

  1. $\bullet$

$H^{*}\mathrm{Hom}_{\mathcal{A}}(P_{1},B)\simeq\langle\alpha,\beta|\omega\rangle$,

  2. $\bullet$

$H^{*}\mathrm{Hom}_{\mathcal{A}}(P_{2},B)\simeq\langle e_{2}\rangle$,

  3. $\bullet$

$H^{*}\mathrm{Hom}_{\mathcal{A}}(B,P_{1})\simeq\langle e_{1}\rangle$,

  4. $\bullet$

$H^{*}\mathrm{Hom}_{\mathcal{A}}(B,P_{2})\simeq\langle\alpha,\beta|\omega\rangle$.

######  Lemma 5.6.

For all $i\in Q_{0}\backslash\\{1,2\\}$,
$H^{*}\mathrm{Hom}_{\mathcal{A}}(P_{i},B)\simeq 0\simeq
H^{*}\mathrm{Hom}_{\mathcal{A}}(B,P_{i})$.

Proof: Let $e_{1}\Lambda e_{i}\simeq
e_{1}\Lambda^{g}e_{i}=\langle\rho_{1},...,\rho_{r}\rangle$ and $e_{2}\Lambda
e_{i}\simeq e_{2}\Lambda^{g}e_{i}=\langle\gamma_{1},...,\gamma_{s}\rangle$ be
two bases where each $\rho_{k},\gamma_{l}$ is a path in $(Q^{g},I^{g})$. We
use the same notation to designate the induced basis of
$\mathrm{Hom}_{\mathcal{A}}(P_{i},B)$.

Recall that $B$ is of the form $(P_{2}[|\alpha|]\bigoplus
P_{1}[1],\partial=\begin{pmatrix}0&\alpha+\mu\beta\\\ 0&0\end{pmatrix})$ for
some $\mu\in K^{*}$, and thus for $f$ in $\mathrm{Hom}_{\mathcal{A}}(P_{i},B)$
the differential is given by $df=(\alpha+\mu\beta)f$ (see Definition 2.1).

Since $i\neq 1$, one can choose an ordering of the $\rho_{l}$’s and
$r^{\prime}\in\\{0,...,r\\}$ such that $\forall l\in\\{1,...,r^{\prime}\\},\
\beta\rho_{l}\in I$ and $\forall l\in\\{r^{\prime}+1,...,r\\},\
\alpha\rho_{l}\in I$. So for $(\lambda_{l}),(\mu_{h})\in K^{r+s}$,

$\displaystyle
d(\sum\limits_{l=1}^{r}\lambda_{l}\rho_{l}+\sum\limits_{h=1}^{s}\mu_{h}\gamma_{h})=0\
$ $\displaystyle\Leftrightarrow\
(\alpha+\mu\beta)(\sum\limits_{l=1}^{r^{\prime}}\lambda_{l}\rho_{l}+\sum\limits_{r^{\prime}+1}^{r}\lambda_{l}\rho_{l})+(\alpha+\mu\beta)(\sum\limits_{h=1}^{s}\mu_{h}\gamma_{h})=0$
$\displaystyle\Leftrightarrow\
\sum\limits_{l=1}^{r^{\prime}}\lambda_{l}\alpha\rho_{l}+\sum\limits_{r^{\prime}+1}^{r}\lambda_{l}\mu\beta\rho_{l}=0$
$\displaystyle\Leftrightarrow\ \forall l\in\\{1,...,r\\},\ \lambda_{l}=0,$

where the last equivalence comes from the fact that in a gentle algebra
distinct paths correspond to distinct elements, and thus the
$\alpha\rho_{l}$’s and the $\beta\rho_{l}$’s are pairwise distinct. We deduce
that $\mathrm{Ker}\ d=\langle\gamma_{1},...,\gamma_{s}\rangle$.

Now since $i\neq 2$, one can choose an ordering of the $\gamma_{h}$’s and
$s^{\prime}\in\\{0,...,s\\}$ such that $\forall
h\in\\{1,...,s^{\prime}\\},\exists l_{h}\in\\{1,...,r\\},\
\gamma_{h}=\alpha\rho_{l_{h}}$ and $\forall
h\in\\{s^{\prime}+1,...,s\\},\exists l_{h}\in\\{1,...,r\\},\
\gamma_{h}=\beta\rho_{l_{h}}$. Thus for all $(\mu_{h})\in K^{s}$,

$\sum\limits_{h=1}^{s}\mu_{h}\gamma_{h}=\sum\limits_{h=1}^{s^{\prime}}\mu_{h}\alpha\rho_{l_{h}}+\sum\limits_{h=s^{\prime}+1}^{s}\frac{\mu_{h}}{\mu}\mu\beta\rho_{l_{h}}=d(\sum\limits_{h=1}^{s^{\prime}}\mu_{h}\rho_{l_{h}}+\sum\limits_{h=s^{\prime}+1}^{s}\frac{\mu_{h}}{\mu}\rho_{l_{h}})$.

We can deduce $\mathrm{Ker}\ d\simeq\mathrm{Im}\ d$ and
$H^{*}\mathrm{Hom}_{\mathcal{A}}(P_{i},B)=0$. The other case is dual.

$\square$

######  Corollary 5.7.

If i or j does not belong to $\\{1,2\\}$, then
$H^{*}\mathrm{Hom}_{\mathcal{A}/\mathcal{B}}(P_{i},P_{j})\simeq\mathrm{Hom}_{\Lambda}(P_{i},P_{j})$.

Proof: Let $(i,j)$ be such a pair. Following the notations of Remark 3.8,
$H^{k}\mathrm{Hom}_{\mathcal{A}/\mathcal{B}}(P_{i},P_{j})\simeq\bigoplus\limits_{l\in\mathbb{Z}}E_{\infty}^{k-l,l}$.

Let $q\in\mathbb{Z}$. Since $H^{*}\mathrm{Hom}_{\mathcal{A}}(P_{i},B)\simeq 0$
or $H^{*}\mathrm{Hom}_{\mathcal{A}}(B,P_{j})\simeq 0$ by Lemma 5.6,
Proposition 3.9 shows that $\forall p>0$, $E_{1}^{p,q}\simeq 0$. Thus $\forall
p\neq 0$, $E_{\infty}^{p,q}\simeq 0$.

Moreover since for all $r>1$ the codomain of $d_{r}^{0,q}$ and the domain of
$d_{r}^{r,q-r-1}$ are zero, we have $E_{\infty}^{0,q}\simeq
E_{1}^{0,q}\simeq\mathrm{Hom}_{\Lambda}(P_{i},P_{j})^{q}$.

$\square$

######  Proposition 5.8.

Let $(i,j)\in\\{(1,1),(1,2),(2,1),(2,2)\\}$ and $E^{*,*}$ the spectral
sequence on $\mathrm{Hom}_{\mathcal{A}/\mathcal{B}}(P_{i},P_{j})$.

Let $H^{*}\mathrm{Hom}_{\mathcal{A}}(P_{i},B)=:\langle t\rangle$,
$H^{*}\mathrm{Hom}_{\mathcal{A}}(B,P_{j})=:\langle z\rangle$ and
$H^{*}\mathrm{End}_{\mathcal{A}}(B)=:\langle id_{B}\rangle\bigoplus\langle
y\rangle$ be the basis given in Lemma 5.5 and in (2), and let $a:=|z|+|t|-1$

  1. $\bullet$

$\forall q\in\mathbb{Z}$,
$E_{\infty}^{0,q}\simeq\mathrm{Hom}_{\Lambda}(P_{i},P_{j})^{q}$,

  2. $\bullet$

$\forall p>0$, $E_{\infty}^{p,a-p}\simeq\langle
z\epsilon(y\epsilon)^{p-1}t\rangle$,

  3. $\bullet$

$\forall p\neq 0,\forall q\neq a-p$, $E_{\infty}^{p,q}\simeq 0$.

Proof: We have the following relations: $zy=yt=y^{2}=zt=0$. Let $p>0$. For
$q\in\mathbb{Z}$, since by Proposition 3.9

$E_{1}^{p,q}\simeq\bigoplus\limits_{\begin{subarray}{c}k_{1}+...+k_{p+1}\\\ =\
2p+q\end{subarray}}H^{k_{p+1}}(\mathrm{Hom}_{\mathcal{A}}(B,P_{j}))\otimes
K[1]\otimes H^{k_{p}}(\mathrm{End}_{\mathcal{A}}(B))\otimes...\otimes
K[1]\otimes H^{k_{1}}(\mathrm{Hom}_{\mathcal{A}}(P_{i},B)),$

we have

$\displaystyle E_{1}^{p,q}\neq 0$
$\displaystyle\Leftrightarrow\exists(k_{i})\in\\{0,1\\}^{p-1},\
|z|+\sum\limits_{i}k_{i}+|t|=2p+q$ $\displaystyle\Leftrightarrow 0\leq
2p+q-|z|-|t|\leq p-1$ $\displaystyle\Leftrightarrow a+1-2p\leq q\leq a-p.$

For such $p$ and $q$, a basis of $E_{1}^{p,q}$ is given by

$E_{1}^{p,q}=\langle\\{z\epsilon^{n_{0}}\Pi_{l=1}^{r}(y\epsilon^{n_{l}})t\ |\
r\geq 0,\ \forall l\in\\{0,...,r\\}.\ n_{l}\geq 1,\
\sum\limits_{l=0}^{r}n_{l}=p\ \mathrm{and}\ |z|+r-p+|t|=p+q\\}\rangle$,

where $\epsilon$ is a generator of $K[1]$. Moreover,

$\displaystyle
d_{1}^{p,q}(z\epsilon^{n_{0}}\Pi_{l=1}^{r}(y\epsilon^{n_{l}})t)$
$\displaystyle=\sum\limits_{i=0}^{r}(-1)^{|z(\Pi_{l=1}^{i}\epsilon^{n_{l-1}}y)|}z(\Pi_{l=1}^{i}\epsilon^{n_{l-1}}y)d(\epsilon^{n_{i}})(\Pi_{l=i+1}^{r}y\epsilon^{n_{l}})t$
$\displaystyle=\sum\limits_{\begin{subarray}{c}i=0\\\ n_{i}\neq 1\
\mathrm{odd}\end{subarray}}^{r}(-1)^{|z(\Pi_{l=1}^{i}\epsilon^{n_{l-1}}y)|}z(\Pi_{l=1}^{i}\epsilon^{n_{l-1}}y)\epsilon^{n_{i}-1}(\Pi_{l=i+1}^{r}y\epsilon^{n_{l}})t,$

since $\forall n\geq 1.\
d(\epsilon^{n})=\sum\limits_{i=1}^{n}(-1)^{|\epsilon^{i-1}|}\epsilon^{n-1}=0$
if $n$ even and $\epsilon^{n-1}$ otherwise.

We first compute the second page of the spectral sequence.

For $q\in\mathbb{Z}$, $d_{1}^{0,q}=0$ and $E_{1}^{1,q}\neq 0\Leftrightarrow
q=a-1$. Since $d_{1}^{1,a-1}(z\epsilon t)=(-1)^{|z|}zt=0$, $E_{2}^{0,q}\simeq
E_{1}^{0,q}.$

For all $p>0$, $d_{1}^{p,a-p}(z\epsilon(y\epsilon)^{p-1}t)=0$, and for all
$\rho=z\epsilon^{n_{0}}\Pi_{l=1}^{r}(y\epsilon^{n_{l}})t\in
E_{1}^{p+1,a-p-2}$, $d_{1}^{p+1,a-p-2}(\rho)=0$ since $n_{l}=1$ or $2$. Thus
$E_{2}^{p,a-p}\simeq E_{1}^{p,a-p}$.

We regroup the remaining cases by complexes. For each $b>0$, let
$C_{b}^{\bullet}$ be defined by $\forall i\geq a,\ C_{b}^{i}=0$, $\forall
i<a,\ C_{b}^{i}=E_{1}^{b+(a-i),a-b-2(a-i)}$ and $\forall i<a,\
d_{C}^{i}=d_{1}^{b+(a-i),a-b-2(a-i)}$. In this way, $\forall p>0$ and $\forall
q\in\\{a+1-2p,...,a-p-1\\}$, $E_{1}^{p,q}$ is equal to $C_{2p+q-a}^{p+q}$. Let
us show that each $C_{b}^{\bullet}$ is acyclic by defining morphisms
$\Psi:C_{b}^{\bullet}\rightarrow C_{b}^{\bullet}[-1]$ satisfying
$d_{C_{b}^{\bullet}}\circ\Psi+\Psi\circ
d_{C_{b}^{\bullet}}=id_{C_{b}^{\bullet}}$.

For $i<a$ and $\rho=z\epsilon^{n_{0}}\Pi_{l=1}^{r}(y\epsilon^{n_{l}})t$ a
basis element of $C_{b}^{i}$, let $X=\\{l\in\\{0,...,r\\}|\ n_{l}\ \mathrm{is\
odd\ and\ different\ from}\ 1\\}$, $Y=\\{l\in\\{0,...,r\\}|\ n_{l}\
\mathrm{is\ even}\\}$ and $\lambda=\frac{1}{|X|+|Y|}$.

Define $\Psi$ on paths by $\Psi(\rho)=\lambda\sum\limits_{i\in
Y}(-1)^{|z(\Pi_{l=1}^{i}\epsilon^{n_{l-1}}y)|}z(\Pi_{l=1}^{i}\epsilon^{n_{l-1}}y)\epsilon^{n_{i}+1}(\Pi_{l=i+1}^{r}y\epsilon^{n_{l}})t$.

Then $\rho$ appears with coefficient $\lambda|Y|$ in $d\circ\Psi(\rho)$ and
with coefficient $\lambda|X|$ in $\Psi(\rho)\circ d$. For any other path
$\rho^{\prime}\neq\rho$ appearing with coefficient $\mu$ in
$d\circ\Psi(\rho)$, it appears with coefficient $-\mu$ in $\Psi\circ d(\rho)$
and vice-versa. Thus $d\circ\Psi+\Psi\circ d=\rho$ holds.

Finally the values of $E_{\infty}$ correspond to the second page since
$\forall p,q\in\mathbb{Z},\forall r\geq 2,d_{r}^{p,q}=0$.

$\square$

Proof of Proposition 5.3: Since $\mathcal{P}(\Lambda[\omega^{-1}])$ has zero
differential, it suffices to show that
$H^{*}\mathcal{P}(\phi):H^{*}\mathrm{Hom}_{(\mathcal{A}/\mathcal{B})_{\circ}}(P_{i},P_{j})\rightarrow\mathrm{Hom}_{\mathcal{P}(\Lambda[\omega^{-1}])}(P_{i},P_{j})$
is an isomorphism for all $i,j$.

If $i$ or $j$ is different from 1 or 2, the relations in
$\Lambda[\omega^{-1}]$ are such that $e_{j}\Lambda[\omega^{-1}]e_{i}\simeq
e_{j}\Lambda e_{i}$. Thus $H^{*}\mathcal{P}(\phi)$ is an isomorphism by
Corollary 5.7.

For $(i,j)\in\\{(1,1),(1,2),(2,1),(2,2)\\}$, Remark 3.8 and Proposition 5.8
give the isomorphism:

$H^{*}\mathrm{Hom}_{\mathcal{A}/\mathcal{B}}(P_{i},P_{j})\simeq\mathrm{Hom}_{\Lambda}(P_{i},P_{j})\bigoplus\langle
z\epsilon(y\epsilon)^{n}t\ |\ n\geq 0\rangle.$

Let us consider the case $(i,j)=(1,1)$. On one hand, the relations in
$\Lambda[\omega^{-1}]$ allow to choose the following basis:
$\mathrm{Hom}_{\mathcal{P}(\Lambda[\omega^{-1}])}(P_{1},P_{1})=\mathrm{Hom}_{\Lambda}(P_{1},P_{1})\bigoplus\langle\delta(\alpha\delta)^{n}\alpha\
|\ n\geq 0\rangle$. On the other hand, if we choose the following
representatives: $t=\alpha,y=\alpha,z=e_{1}$, the equivalence of Lemma 5.1
gives the rewriting $z\epsilon(y\epsilon)^{n}t=e_{1}\epsilon(e_{2}\alpha
e_{1}\epsilon)^{n}e_{2}\alpha=\delta(\alpha\delta)^{n}\alpha$, and