2.1 Rigid modules

Since R is Gorenstein, recall that

For $M\in \operatorname {CM} R$ , we say that M is basic if there are no repetitions in its Krull–Schmidt decomposition into indecomposables. We call M a generator if one of these indecomposable summands is R, and we call M rigid if $\operatorname {Ext}^1_R(M,M)=0$ . We say $M\in \operatorname {CM} R$ is maximal rigid if it is rigid, and furthermore it is maximal with respect to this property (see [Reference Iyama and WemyssIW1, 4.1]).

By the Auslander–McKay correspondence for cDV singularities, there is a one-to-one correspondence between flopping contractions $Y\to \operatorname {Spec} R$ , up to R-isomorphism, and basic rigid generators in $\operatorname {CM} R$ [Reference WemyssW, 4.13]. In particular, both sets are finite. For our fixed flopping contraction $f\colon X\to \operatorname {Spec} R$ , the corresponding basic rigid CM generator across the bijection will be denoted N throughout this paper. The contraction algebra of f may then be defined as the stable endomorphism algebra [Reference Donovan and WemyssDW1, Reference Donovan and WemyssDW3].

The set of basic rigid CM generators carries an operation called mutation. Indeed, given such an L, with indecomposable summand $L_i\ncong R$ , there is the so-called exchange sequence

(2.1) $$ \begin{align} 0\to L_i\to\bigoplus_{j\neq i}{L_j}^{\oplus b_{ij}} \to K_i\to 0 \end{align} $$

and

[Reference WemyssW, (A.A)]. Given the summand $K_i$ of $\unicode{x3bd} _iL$ , we can mutate again and obtain $\unicode{x3bd} _i\unicode{x3bd} _iL$ . It is a general fact for isolated cDV singularities that $\unicode{x3bd} _i\unicode{x3bd} _iL\cong L$ [Reference WemyssW, 4.20(1)], and moreover in the second exchange sequence

$$\begin{align*}0\to K_i\to\bigoplus_{j\neq i}{L_j}^{\oplus c_{ij}} \to L_i\to 0 \end{align*}$$

we have $b_{ij}=c_{ij}$ for all $i,j$ [Reference WemyssW, 5.22]. Furthermore, there are equivalences of derived categories

We will abuse notation and notate both as

and refer to them as the mutation functors.

Given our fixed basic rigid CM generator N corresponding to f, write $\operatorname {Mut}_0(N)$ for the set of basic rigid CM generators that can be obtained from N by iteratively mutating at indecomposable summands, noting that, in this paper, we will never mutate at the summand R. Geometrically, across the Auslander–McKay correspondence, this corresponds to all R-schemes that can be obtained from X by iteratively flopping curves.

2.2 Hyperplanes and labels

We fix a decomposition $N=R\oplus N_1\oplus \ldots \oplus N_n$ and will often implicitly declare $N_0=R$ . For every $L\in \operatorname {Mut}_0(N)$ and each indecomposable summand $L_i$ of L, there is an exact sequence

(2.2) $$ \begin{align} 0\to \bigoplus_{j=0}^n N_j^{\oplus a_{ij}}\to \bigoplus_{j=0}^n N_j^{\oplus b_{ij}}\to L_i\to 0 , \end{align} $$

where the first and second terms do not share any indecomposable summands. In the setting where $N\in \operatorname {CM} R$ is maximal rigid, this fact is very well known; in the setting here for rigid generators, we rely instead on [Reference Iyama and WemyssIW2, 9.29, 5.6]. In any case, consider the cone

It is clear that

, which we denote throughout by $C_+$ . Furthermore, the chambers $C_L$ , as L varies over $\operatorname {Mut}_0(N)$ , sweep out the chambers of a simplicial hyperplane arrangement

[Reference Hirano and WemyssHW1, Reference Iyama and WemyssIW2].

As explained by the general Coxeter-style labelling of walls and chambers in [Reference Iyama and WemyssIW2, §9.7], the fixed decomposition $N=R\oplus N_1\oplus \ldots \oplus N_n$ induces an ordering on the summands of all other elements L of $\operatorname {Mut}_0(N)$ such that each local wall crossing has a label $s_i$ and corresponds to mutation at the ith summand. In this way, there is a compatible labelling on the edges of the 1-skeleton of . We illustrate this in an example.

Figure 1

Exchange graph and $1$ -skeleton for a certain two-curve $cD_4$ singularity.

Fixing an ordering of the summands of N not only fixes an ordering of the projective modules of $\Lambda =\operatorname {End}_R(N)$ and, via the pairing between simples and projectives, an ordering of its simples , but also fixes an ordering on the projectives and simples of $\operatorname {End}_R(L)$ for all $L\in \operatorname {Mut}_0(N)$ .

To fix notation, for $L\in \operatorname {Mut}_0(N)$ , suppose that the induced ordering on the summands of L is $L=L_0\oplus L_1\oplus \ldots \oplus L_n$ , where $L_0= R$ . There is an induced ordering on the projectives of $\operatorname {End}_R(L)$ and again via the pairing between simples and projectives, an ordering of its simples . The simples for the contraction algebra $\operatorname {\underline {End}}_R(L)$ are .

2.3 Standard equivalences for simple wall crossings

Let $L\in \operatorname {Mut}_0(N)$ , with associated and contraction algebra . To ease notation, suppose that , with contraction algebra $\mathrm {B}_{\mathrm {con}}$ , and consider the good truncation

(2.3)

which is a complex of $\mathrm {B}_{\mathrm {con}}$ - $\mathrm {A}_{\mathrm {con}}$ bimodules. In this basic rigid setting, it is already known that the complex is a two-sided tilting complex [Reference AugustAu2, 3.3].

Theorem 2.4. Suppose that $L,M \in \operatorname {Mut}_0(N)$ , which correspond to the flopping contractions $X_L\to \operatorname {Spec} R$ and $X_M\to \operatorname {Spec} R$ , say. Then the following are equivalent:

1. 1. $C_L$ and $C_M$ share a codimension one wall in .

2. 2. $M\cong \unicode{x3bd} _iL$ for some $i\neq 0$ .

3. 3. $X_L$ and $X_M$ are related by a flop at a single irreducible curve.

In this case, set

,

and

,

. Then the mutation operations intertwine via the following commutative diagram

where

is the mutation functor, $\operatorname {\mathsf {Flop}}_i$ is the inverse of the Bridgeland–Chen flop functor [Reference BridgelandB1, Reference ChenC],

and

are the standard projective generators of zero perverse sheaves [Reference Van den BerghVdB] and $\operatorname {res}$ is restriction of scalars induced from the ring homomorphisms $\mathrm {A}\to \mathrm {A}_{\mathrm {con}}$ and $\mathrm {B}\to \mathrm {B}_{\mathrm {con}}$ .

2.4 K-theory

Since $\Lambda _{\mathrm {con}}$ is a finite-dimensional algebra, consider the Grothendieck group

which is well-known to be a free abelian group with basis given by the ordered simples

. The notation $\mathsf {G}_0$ is chosen since we are including in the Grothendieck group all modules, whereas $K_0$ often only deals with vector bundles and thus projectives.

Continuing the notation from Theorem 2.4, for any $L\in \operatorname {Mut}_0(N)$ , consider its associated contraction algebra $\mathrm {A}_{\mathrm {con}}=\operatorname {\underline {End}}_R(L)$ and its mutations

. Abusing notation slightly, consider the standard equivalences

where the functor from right to left is induced by the mutation $\unicode{x3bd} _i L \to \unicode{x3bd} _i \unicode{x3bd} _i L \cong L$ . These and their inverses induce the following four isomorphisms on K-theory:

(2.4)

As in Subsection 2.2, write for the ordered simples of $\mathrm {A}_{\mathrm {con}}$ . For lack of suitable alternatives, also write for the correspondingly ordered simples in $\unicode{x3bd} _i\mathrm {A}_{\mathrm {con}}$ .

Lemma 2.5. With the notation in equation (2.1), $\mathsf {F}_i\colon \mathsf {G}_0(\mathrm {A}_{\mathrm {con}})\to \mathsf {G}_0(\unicode{x3bd} _i\mathrm {A}_{\mathrm {con}})$ sends

(2.5)

Proof. The fact that is [Reference WemyssW, 4.15]. Theorem 2.4 then implies that maps, under restriction of scalars, to . It follows that in $\mathop {\mathrm {D}^b}(\mathrm {A}_{\mathrm {con}})$ ; see, for example, [Reference AugustAu2, 6.6]. This establishes the top row.

For the second row, applying $\operatorname {Hom}_R(L,-)$ to equation (2.1) and using the rigidity of L gives an exact sequence

Applying

to this, with $t\neq i$ , yields

Further, it is clear that

is zero if $j\neq t$ and equals $\mathbb {C}$ if $j=t$ . Combining, we see that

is a module, filtered by $b_{it}$ copies of

and one copy of

. Again, the left-hand side of the commutative diagram in Theorem 2.4 then shows that

must also be a module, filtered by $b_{it}$ copies of

and one copy of

. The second row follows.

Remark 2.6. Applying the above to $L=N$ , basing $\mathsf {G}_0(\Lambda _{\mathrm {con}})$ by the ordered simples

, the above transformation given by equation (2.5) assembles into an $n\times n$ matrix, with coefficients in $\mathbb {Z}$ , representing the map $\mathsf {F}_i\colon \mathbb {Z}^n\to \mathbb {Z}^n$ . As is standard in linear algebra, the dual map

where $\mathsf {G}_0(\Lambda _{\mathrm {con}})^*$ has dual basis $\textbf {e}_1,\ldots ,\textbf {e}_n$ , say, is given by the transpose matrix. But the transpose is precisely the transformation

$$\begin{align*}\textbf{e}_t\mapsto \left\{ \begin{array}{cl} \textbf{e}_t& \mbox{if } t\neq i,\\ -\textbf{e}_i+\sum_{j\neq i}b_{ij}\textbf{e}_j & \mbox{if } t=i, \end{array} \right. \end{align*}$$

which is precisely the transformation

seen in moduli tracking [Reference WemyssW, §5] or in K-theory of projectives in [Reference Hirano and WemyssHW2, 3.2]. In particular, it will be convenient to think of the dual basis $\textbf {e}_1,\ldots ,\textbf {e}_n$ as being the basis

of

, where $\operatorname {per}\Lambda $ is the full subcategory of $\mathop {\mathrm {D}^b}(\Lambda )$ consisting of perfect complexes. Then the transformation $\mathsf {F}_i^*$ can be identified with these transformations elsewhere in the literature, and we can use those results freely. Note that the hyperplane arrangement

from Subsection 2.2 is defined in terms of $\textbf {e}_i$ and so naturally lives in $\mathsf {G}_0(\Lambda _{\mathrm {con}})^*$ .

By the above remark, the proof of the following does follow as the dual of [Reference Hirano and WemyssHW2, 3.2]. It is, however, instructive to give a direct proof.

Proof. By equation (2.5), the matrices are controlled by the numbers $b_{ij}$ appearing in the relevant exchange sequences. Say the top $\mathsf {F}_i$ is controlled by numbers $b_{ij}$ , and the bottom $\mathsf {F}_i$ is controlled by numbers $c_{ij}$ . That the two matrices labelled $\mathsf {F}_i$ are the same is simply the statement that $b_{ij}=c_{ij}$ , which has already been explained in Subsection 2.1. Given this fact that $b_{ij}=c_{ij}$ , we see that $\mathsf {F}_i\mathsf {F}_i=\mathrm {Id}$ by simply observing

which is clearly the identity. Applying $\mathsf {F}_i^{-1}$ to each side of the equation $\mathsf {F}_i\mathsf {F}_i=\mathrm {Id}$ gives $\mathsf {F}_i^{-1}=\mathsf {F}_i$ , and all statements follow.

2.5 Groupoids

As in Subsection 2.2, associated to every contraction algebra is a hyperplane arrangement . As is standard, there is an associated graph defined as follows.

By definition, if there is an arrow $a \colon v_1\to v_2$ , then there is a unique arrow $b\colon v_2\to v_1$ with the opposite direction of a. For an arrow $a\colon v_1\to v_2$ , set and .

A positive path of length n in

is a formal symbol

$$\begin{align*}p=a_n\circ \ldots\circ a_2\circ a_1 \end{align*}$$

whenever there exists a sequence of vertices $v_0,\ldots ,v_n$ of

and arrows $a_i\colon v_{i-1}\to v_i$ in

. Set

,

,

, and write $p\colon s(p)\to t(p)$ . If $q=b_m\circ \ldots \circ b_2 \circ b_1$ is another positive path with $t(p)=s(q)$ , we consider the formal symbol

and call it the composition of p and q.

Following [Reference DelucchiD2, p7], let $\sim $ denote the smallest equivalence relation, compatible with morphism composition, that identifies all morphisms that arise as positive minimal paths with the same source and target. Then consider the free category

on the graph

, whose morphisms correspond to directed paths, and its quotient category

called the category of positive paths.

It is a very well known fact [Reference DeligneD1, Reference ParisP1, Reference ParisP3, Reference SalvettiS] (see also [Reference ParisP2, 2.1]) that for any vertex , there is an isomorphism , where . This fact is general and is significantly weaker and easier to establish than statements involving $K(\unicode{x3c0} \,,1)$ . We will use this fact implicitly throughout.

2.6 Composition and K-theory

For any

, say $\unicode{x3b1} ={s_{i_t}}\circ \ldots \circ {s_{i_1}}$ , consider

The following is known and easy to establish just using the tilting order, in the case when the modules are maximal rigid (e.g., if X is smooth). In our more general situation of rigid generators, the same proof works, but it relies on some recent advances in [Reference Iyama and WemyssIW2].

For any two positive minimal paths $\unicode{x3b1} $ and $\unicode{x3b2} $ with the same start and end points, the above proposition shows that there is a functorial isomorphism $F_{\unicode{x3b1} }\cong F_{\unicode{x3b2} }$ . Hence the association $\unicode{x3b1} \mapsto F_{\unicode{x3b1} }$ descends to a functor from . Since $F_{\unicode{x3b1} }$ is already an equivalence, this in turn formally descends to a functor from . Using the same logic, the assignment also descends to a functor from and then a functor from .

Furthermore, for every

, using the notation from the second sentence in Proposition 2.11, the following diagram commutes

(2.6)

just by composition, since by [Reference AugustAu2, 1.1], respectively [Reference AugustAu2, 3.2], both of the following commute:

Later, the following is one of the crucial ingredients in establishing that $\operatorname {Stab}\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ is a covering space. Recall that for the standard derived equivalence $F_{\unicode{x3b1} }$ associated to a path $\unicode{x3b1} $ , the induced map on the K-theory is denoted $\mathsf {F}_{\unicode{x3b1} }$ .

2.7 The dual composition

For each $L \in \operatorname {Mut}_0(N)$ , consider

and recall that

. Choose a positive minimal path $\unicode{x3b2} \colon C_L \to C_+$ , which in turn gives rise to a derived equivalence

This derived equivalence is independent of the choice of positive minimal path, by Proposition 2.11. It induces an isomorphism $\mathsf {K}_0(\operatorname {per}\Lambda _L)\to \mathsf {K}_0(\operatorname {per}\Lambda )$ on the K-theory of perfect complexes, so write

in $\mathsf {K}_0(\operatorname {per}\Lambda )\cong \mathbb {Z}^{n+1}$ , where

and

. Since $\unicode{x3b1} $ is a sequence of mutations that do not involve mutating the zeroth summand, the zeroth summand is fixed at each stage. Hence this isomorphism descends to an isomorphism

Basing the first by

and the second by

, the matrix representing the isomorphism is

for $1\leq i,j\leq n$ . By Remark 2.6, later we will think of these bases as $\textbf {e}^{\prime }_1,\ldots ,\textbf {e}^{\prime }_n$ of $\mathsf {G}_0(\operatorname {\underline {End}}_R(L))^*$ and $\textbf {e}_1,\ldots ,\textbf {e}_n$ of $\mathsf {G}_0(\Lambda _{\mathrm {con}})^*$ , respectively.

3.1 Stability generalities

Throughout this subsection, denotes a triangulated category whose Grothendieck group is a finitely generated free $\mathbb {Z}$ -module.

Proposition 3.1 [Reference BridgelandB2, 5.3].

To give a stability condition on

is equivalent to giving a bounded t-structure

with heart

and a group homomorphism

, called the central charge, such that for all

, the complex number $Z(E)$ lies in the semi-closed upper half-plane

and where furthermore Z must satisfy the Harder–Narasimhan property.

Write for the set of locally-finite stability conditions on . We do not define these here, as below this condition is automatic for all stability conditions on $\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ , since all hearts of bounded t-structures will be equivalent to finite-dimensional modules on some finite-dimensional algebra.

Theorem 3.2 [Reference BridgelandB2, 1.2].

The space

has the structure of a complex manifold, and the forgetful map

is a local isomorphism onto an open subspace of

.

Any triangle equivalence

induces a natural map

defined by

, where $\unicode{x3c6} ^{-1}$ is the corresponding isomorphism on K-theory

induced by the functor

. In this way, the group

of isomorphism classes of autoequivalences of

acts on

.

3.2 t-structures for $\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$

The contraction algebra $\Lambda _{\mathrm {con}}$ is a silting-discrete symmetric algebra [Reference AugustAu1, 3.3, 4.12]. Being symmetric, the technical condition of being silting-discrete is equivalent [Reference Aihara and MizunoAM, 2.11] to there being only finitely many basic tilting complexes between P and $P[1]$ (with respect to the silting order $\leq $ ), for every tilting complex P obtained by iterated irreducible left mutation from the free module $\Lambda _{\mathrm {con}}$ . Geometrically, for each such P, this set is finite since it is in bijection with R-schemes obtained by iterated flops of irreducible curves starting from X, which is well-known to be finite.

This fact has the following remarkable consequence.

Proposition 3.3. Suppose that

is the heart of a bounded t-structure on $\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ . Then

for some $L\in \operatorname {Mut}_0(M)$ and for some

, where

and $F_{\unicode{x3b1} }$ is the derived equivalence from Subsection 2.6 associated to $\unicode{x3b1} $ .

Proof. Since $\Lambda _{\mathrm {con}}$ is silting-discrete, necessarily has finite length [Reference Pauksztello, Saorín and ZvonarevaPSZ]. Furthermore, by the bijections in [Reference Koenig and YangKY, §5], there exists a silting complex T in $\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ such that, in the notation of [Reference Koenig and YangKY, §5.4],

(3.1)

Since $\Lambda _{\mathrm {con}}$ is symmetric, silting equals tilting, so T is a tilting complex. It is already known (see [Reference AugustAu1, 2.12]) that every tilting complex T in $\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ can be obtained as a composition of mutations from $\Lambda _{\mathrm {con}}$ , so say $T\cong \unicode{x3bc} _{\unicode{x3b2} }\Lambda _{\mathrm {con}}$ for some . Set ; then [Reference AugustAu1, 3.10(1)] gives $F_{\unicode{x3b2} }(\unicode{x3bc} _{\unicode{x3b2} }\Lambda _{\mathrm {con}})\cong \mathrm {B}_{\mathrm {con}}$ , and hence $F_{\unicode{x3b2} }(T)\cong \mathrm {B}_{\mathrm {con}}$ .

Thus, applying $F_{\unicode{x3b2} }$ to equation (3.1),

so applying $F_{\unicode{x3b2} }^{-1}=F_{\unicode{x3b2} ^{-1}}$ shows that

. Since

, the result follows.

Recall that inside $\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ are the simples , which base the K-theory $\mathsf {G}_0(\Lambda _{\mathrm {con}})$ . In a similar way, the simple modules of $\operatorname {\underline {End}}_R(L)$ base its Grothendieck group.

The general action of $\operatorname {Auteq}\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ on $\operatorname {Stab}\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ simplifies somewhat if we restrict to those standard equivalences given by

. The functorial assignment $\unicode{x3b1} \to F_{\unicode{x3b1} }$ defines a group homomorphism

and we set $\mathop {\mathsf {PBr}}\nolimits $ to be the image of this homomorphism. Then using Corollary 3.4 to describe the points of $\operatorname {Stab}\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ , the action of $F_{\unicode{x3b2} }\in \mathop {\mathsf {PBr}}\nolimits $ on $\operatorname {Stab}\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ is

(3.2)

since $\mathsf {F}_{\unicode{x3b2} }=\mathop {\mathrm {Id}}\nolimits $ by Proposition 2.12(2).

4.1 A covering map

As in Section 3.2, let $\mathop {\mathsf {PBr}}\nolimits $ be the image of the homomorphism

sending $\unicode{x3b2} \mapsto F_{\unicode{x3b2} }$ . Then $F_{\unicode{x3b2} }\in \mathop {\mathsf {PBr}}\nolimits $ acts on the space of stability conditions via the action in equation (3.2). In this subsection, we will establish that this action is free and properly discontinuous, so that $\operatorname {Stab}\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}}) \to \operatorname {Stab}\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})/\mathop {\mathsf {PBr}}\nolimits $ is a covering map.

The following is one of our main technical results. It establishes a condition under which elements of $\mathop {\mathsf {PBr}}\nolimits $ , inside the autoequivalence group of $\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ , are the identity. The proof is via Fourier–Mukai techniques. Forgetting the ambient geometry is thus a bad idea: it seems extremely difficult to establish the following result in a purely algebraic manner.

Proof. By the assumption, the standard equivalence $F_{\unicode{x3b1} }$ is induced by the one-sided tilting complex $\Lambda _{\mathrm {con}}$ . By the usual lifting argument (see, e.g., [Reference Rouquier and ZimmermannRZ, 2.3]), the bimodule complex defining $F_{\unicode{x3b1} }$ must be isomorphic to ${}_{1}({\Lambda _{\mathrm {con}}})_{\unicode{x3b6} }$ as bimodules, for some algebra automorphism $\unicode{x3b6} \colon \Lambda _{\mathrm {con}}\to \Lambda _{\mathrm {con}}$ . Hence $F_{\unicode{x3b1} }$ is induced by this algebra automorphism.

Since $F_{\unicode{x3b1} }$ induces a Morita equivalence, it must take simples to simples. Furthermore, as $F_{\unicode{x3b1} }$ is the identity on K-theory $\mathsf {G}_0(\Lambda _{\mathrm {con}})$ by Proposition 2.12, $F_{\unicode{x3b1} }$ must fix all simples.

Now consider the commutative diagram

where the left-hand side is equation (2.6) and the right-hand side can be obtained by iterating the right-hand side of Theorem 2.4. Since $F_{\unicode{x3b1} }$ fixes simples, the left-hand commutative diagram implies that

fixes the simples

. By [Reference Van den BerghVdB, 3.5.8], across the right-hand commutative diagram, this in turn implies that $\operatorname {\mathsf {Flop}}_{\unicode{x3b1} }$ fixes the sheaves

, where each $\mathrm {C}_i\cong \mathbb {P}^1$ . Since the flop functor and its inverse both map

to

[Reference BridgelandB1, (4.4)] and commute with the pushdown to $\operatorname {Spec} R$ (see, e.g., [Reference Donovan and WemyssDW1, 7.16]), by the standard Fourier–Mukai argument (see [Reference WemyssW, 4.3], which itself is based on [Reference TodaT]), $\operatorname {\mathsf {Flop}}_{\unicode{x3b1} }\cong h_*$ for some isomorphism $h \colon X\to X$ that commutes with the pushdown. But this isomorphism is the identity on the dense open set obtained by removing the exceptional locus, and hence it must be the identity.

It follows that $\operatorname {\mathsf {Flop}}_{\unicode{x3b1} }\cong \mathop {\mathrm {Id}}\nolimits $ and hence

. Restricting the left-hand commutative diagram to $\operatorname {mod} \Lambda _{\mathrm {con}}$ , we obtain a commutative diagram

where $\operatorname {mod} _{\{1,\ldots ,n\}}\Lambda $ denotes those $\Lambda $ -modules with a finite filtration by the simples

. In particular, $F_{\unicode{x3b1} }$ restricted to $\operatorname {mod} \Lambda _{\mathrm {con}}$ is functorially isomorphic to the identity. Hence $\unicode{x3b6} $ is an inner automorphism (see, e.g., [Reference LinckelmannL, 2.8.16]), which in turn implies that $F_{\unicode{x3b1} }\cong \mathop {\mathrm {Id}}\nolimits $ .

Proof. Consider the open neighbourhood

of x defined by

where $d(-,-)$ is the metric on stability conditions introduced in [Reference BridgelandB2, Section 6]. Suppose that

for some $F_{\unicode{x3b2} }\in \mathop {\mathsf {PBr}}\nolimits $ . We will show that $F_{\unicode{x3b2} }\cong \mathop {\mathrm {Id}}\nolimits $ .

As the two open balls of radius $\frac {1}{4}$ intersect, every must satisfy $d\bigl (y,(F_{\unicode{x3b2} })_*y\bigr )<1$ . Furthermore, the central charges of y and $(F_{\unicode{x3b2} })_*y$ are equal by Proposition 2.12(2) and the top commutative diagram in Lemma 4.1. Using [Reference BridgelandB2, Lemma 6.4], it follows immediately that $y=(F_{\unicode{x3b2} })_*y$ , for every .

In particular, by Corollary 3.4, say for some . Since , the property $(F_{\unicode{x3b2} })_*(x)=x$ implies that , so $F_{\unicode{x3b2} }$ restricts to an equivalence . In turn, this implies that the composition

$$\begin{align*}F_{\unicode{x3b1}^{-1}\unicode{x3b2}\unicode{x3b1}}=F_{\unicode{x3b1}}^{-1}\circ F_{\unicode{x3b2}}\circ F_{\unicode{x3b1}}\colon\mathop{\mathrm{D}^b}(\Gamma_{\mathrm{con}})\to\mathop{\mathrm{D}^b}(\Gamma_{\mathrm{con}}), \end{align*}$$

where , restricts to an equivalence $\operatorname {mod} \Gamma _{\mathrm {con}}\to \operatorname {mod} \Gamma _{\mathrm {con}}$ . It follows that $F_{\unicode{x3b1} ^{-1}\unicode{x3b2} \unicode{x3b1} }(\Gamma _{\mathrm {con}})$ must then be a basic tilting module, given that $\Gamma _{\mathrm {con}}$ is. Since $\Gamma _{\mathrm {con}}$ is symmetric, the only such module is $\Gamma _{\mathrm {con}}$ , so $F_{\unicode{x3b1} ^{-1}\unicode{x3b2} \unicode{x3b1} }(\Gamma _{\mathrm {con}})\cong \Gamma _{\mathrm {con}}$ . By Proposition 4.3 applied to the contraction algebra $\Gamma _{\mathrm {con}}$ , we conclude that $F_{\unicode{x3b1} ^{-1}\unicode{x3b2} \unicode{x3b1} }\cong \mathop {\mathrm {Id}}\nolimits $ , and hence $F_{\unicode{x3b2} }\cong \mathop {\mathrm {Id}}\nolimits $ .

4.2 The regular cover to the complexified complement

In this subsection, we will establish that p induces an isomorphism

(4.4)

Combining with Corollary 4.5 will then prove that equation (4.1) is the universal covering map onto its image, with Galois group $\mathop {\mathsf {PBr}}\nolimits $ . We will establish equation (4.4) in two steps: first by showing that p has image , and then by establishing that equation (4.4) is well-defined and injective.

Our proof makes use of the following key combinatorial result, which is folklore when is an ADE root system. In our mildly more general setting here, the proof, which is basically the same, can be found in [Reference Hirano and WemyssHW2, Appendix A].

The above combinatorics lead directly to the following. In the special case when is an ADE root system, the following result is already implicit in [Reference BridgelandB3] and [Reference TodaT].

The following shows that equation (4.4) is both well-defined and injective.

Lemma 4.8. For $\unicode{x3c3} _1,\unicode{x3c3} _2\in \operatorname {Stab}\mathop {\mathrm {D}^b}(\Lambda _{\mathrm {con}})$ , then

Proof. Note that ( $\Leftarrow $ ) is clear since, by Proposition 2.12(2), the action in equation (3.2) of a pure braid does not affect the central charge of a stability condition.

For ( $\Rightarrow $ ), recall that by Corollary 3.4, we can assume that and , where and . If $p(\unicode{x3c3} _1)=p(\unicode{x3c3} _2)$ , then certainly $Z_1 =Z_2$ since p is simply the forgetful map followed by an isomorphism.

Furthermore, by Corollary 4.2, we see that

, since the intersection contains $p(\unicode{x3c3} _1)=p(\unicode{x3c3} _2)$ . Hence by Proposition 4.6(1), it follows that $s(\unicode{x3b1} )=s(\unicode{x3b2} )$ , and thus we can consider the composition

. Then $F_{\unicode{x3b3} }\in \mathop {\mathsf {PBr}}\nolimits $ and

proving the statement.

Corollary 4.9. The map

induces a homeomorphism

The following is our main result.