On the finiteness of the derived equivalence classes of some stable endomorphism rings

We prove that the stable endomorphism rings of rigid objects in a suitable Frobenius category have only finitely many basic algebras in their derived equivalence class and that these are precisely the stable endomorphism rings of objects obtained by iterated mutation. The main application is to the Homological Minimal Model Programme. For a 3-fold flopping contraction f:X→SpecR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f :X \rightarrow {\mathrm{Spec}\;}\,R$$\end{document}, where X has only Gorenstein terminal singularities, there is an associated finite dimensional algebra Acon\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{{\text {con}}}$$\end{document} known as the contraction algebra. As a corollary of our main result, there are only finitely many basic algebras in the derived equivalence class of Acon\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_{\text {con}}$$\end{document} and these are precisely the contraction algebras of maps obtained by a sequence of iterated flops from f. This provides evidence towards a key conjecture in the area.


Introduction
This paper focuses on a fundamental problem in homological algebra: given a basic algebra A, find all the basic algebras B such that A and B have equivalent derived categories. We will give a complete answer to this question for a class of finite dimensional algebras arising from suitable Frobenius categories.
By a well known result of Rickard [33], the above problem is equivalent to first finding all the tilting complexes over A and then computing their endomorphism rings. One approach to the first of these problems is to use mutation; an iterative procedure which produces new tilting complexes from old. The naive hope is that starting from a given tilting complex, all others can be reached using mutation. However, for a general algebra, tilting complexes do not behave well enough for this to work. There are two key problems: (1) The mutation procedure does not always produce a tilting complex.
(2) It is often possible to find two tilting complexes which are not connected by any sequence of mutations.
B Jenny August jennyaugust@mpim-bonn.mpg.de 1 Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany Both problems have motivated results in the literature; the first prompting the introduction of the weaker notion of silting complexes [3,27] and the second resulting in restricting to a class of algebras known as tilting-discrete algebras [2,4]. In this paper, we will combine these ideas to provide a class of examples of finite dimensional algebras for which the derived equivalence class can be completely determined.

Algebraic setting and results
From an algebraic perspective, the algebras we consider arise in cluster-tilting theory, where the objects of study are rigid objects in some category C and the algebras of interest are their endomorphism algebras. More precisely, let E be a Frobenius category such that its stable category C = E is a k-linear, Hom-finite, Krull-Schmidt, 2-Calabi-Yau triangulated category with shift functor denoted . This ensures E has the conditions usually imposed for cluster-tilting theory, but we shall add two additional assumptions: (1) E has at least one but a finite number of maximal rigid objects, and; (2) 2 ∼ = id.
Utilizing the strong links between rigid objects and the silting theory of their endomorphism algebras developed in [1] leads to the following, which is our main result. For maximal rigid objects, the conditions on E above ensure that any two such objects are connected by a sequence of mutations and so we obtain the following corollary of Theorem 1.1.
sequence of codimension two modifications called flops. For this reason, maps which give rise to flops, known as flopping contractions, are of particular interest.
For a 3-fold flopping contraction f : X → X con , Donovan-Wemyss introduced an algebraic invariant, known as the contraction algebra of f . If the map contracts a single curve, this finite dimensional algebra recovers all the previously known numerical invariants [11,17] and has even been used to prove that those numerical invariants can not classify flops completely [7]. It is further conjectured by Donovan-Wemyss that, in the setting of smooth minimal models, the contraction algebra, or more precisely its derived category, completely determines the geometry.

Conjecture 1.
3 Suppose that f : X → Spec R and g : Y → Spec S are smooth minimal models of complete local isolated cDV singularities with associated contraction algebras A con and B con . Then R ∼ = S if and only if A con and B con are derived equivalent.
The 'only if' direction is already known to be true by combining results from [38] with the main result of [15], but the 'if' direction remains a key open problem in the Homological Minimal Model Programme. This conjecture is the main motivation for studying the derived equivalence classes of contraction algebras.
The connection between flops and cluster-tilting theory has been explored in [38], where it is shown that, for certain flopping contractions f : X → Spec R, the contraction algebra is the stable endomorphism algebra of a rigid object in some Frobenius category associated to R. In particular, the setting there satisfies the conditions of Theorem 1.1 and thus, we obtain the following result. Theorem 1.4 (Theorem 4.12) Suppose that f : X → Spec R is a 3-fold flopping contraction where R is complete local and X has at worst Gorenstein terminal singularities. Writing A con for the associated contraction algebra, the following statements hold.
(1) The basic algebras in the derived equivalence class of A con are precisely the contraction algebras of flopping contractions obtained by iterated flops from f . (2) There are only finitely many basic algebras in the derived equivalence class of A con .
Minimal models are a special case where the contraction algebras correspond to maximal rigid objects, and this leads to the following analogue of Corollary 1.2. In many ways, this corollary can be viewed as the 3-fold analogue of [4, 5.1]. Corollary 1.5 (Corollary 4.13) Suppose that f : X → Spec R is a minimal model of a complete local isolated cDV singularity and write A con for the associated contraction algebra. Then the following statements hold.
(1) The basic algebras in the derived equivalence class of A con are precisely the contraction algebras of the minimal models of Spec R. (2) There are only finitely many basic algebras in the derived equivalence class of A con .
This provides a large class of algebras with finite derived equivalence classes. Even in the more restricted setting of Conjecture 1.3, combining [30, 5.5] and [38, 4.10(2)] shows that the quivers of the associated contraction algebras form seven infinite families.
In the process of proving the above, we also establish the following. Theorem 1.6 (Theorem 4.14) Suppose that f : X → Spec R and g : Y → Spec S are minimal models of complete local isolated cDV singularities with associated contraction algebras A con and B con . If A con and B con are derived equivalent then there is a bijection {minimal models of Spec R} ↔ {minimal models of Spec S}.

Further, the bijection preserves both iterated flops and contraction algebras.
This gives the first concrete evidence towards Conjecture 1.3, as it shows that Spec R and Spec S must at least have the same number of minimal models and further that the simple flops graphs of their minimal models must be isomorphic.

Conventions
Throughout, k will denote an algebraically closed field of characteristic zero. For a ring A, we denote the category of finitely generated right modules by mod A. For M ∈ mod A, we let addM be the full subcategory consisting of summands of finite direct sums of copies of M and we let projA := add A be the category of finitely generated projective modules. Finally, K b (projA) will denote the homotopy category of bounded complexes of finitely generated projectives and D b (A) := D b (mod A) will denote the bounded derived category of mod A.

Silting theory
In this subsection we recall silting and tilting theory for a finite dimensional k-algebra . Note that for such a , the bounded homotopy category K b (proj ) is a k-linear, Hom-finite, Krull-Schmidt triangulated category.
We will write silt (respectively tilt ) for the set of isomorphism classes of basic silting (respectively tilting) complexes in K b (proj ). Note that ∈ tilt and that all T ∈ silt have the same number of indecomposable summands [3, 2.8].

Derived equivalences
Tilting objects are of interest due to the following well-known theorem, connecting them with derived equivalences. Theorem 2.2 [33] For each tilting complex T : Moreover, a basic finite dimensional algebra is derived equivalent to if and only if there exists T ∈ tilt such that ∼ = End (T ).
If we restrict to standard derived equivalences, the connection with tilting theory becomes even stronger.
for some complex T of --bimodules. In this case, we call T a two-sided tilting complex.
It is shown in [34, 4.1] that the composition of standard equivalences is again standard and further, the inverse of a standard equivalence is also standard. Moreover, for any tilting complex T ∈ D b ( ), it is shown in [25] that there exists a two-sided tilting complex T such that T ∼ = T in D b ( ). This induces a standard equivalence which maps T → End (T ). It was further shown in [26, 2.1] that such a T is unique in a suitable sense. For our purposes, the following suffices.
Proposition 2.4 [35, 2.3] Suppose that , and are k-algebras and that T (respectively T ) is a two-sided tilting complex of --bimodules (respectively --bimodules) with End (T ) ∼ = End (T ). Then T ∼ = T if and only if there exists an isomorphism γ : → such that in the derived category of -bimodules.
In particular, for any tilting complex there is a unique (up to algebra isomorphism) standard equivalence induced by the tilting complex.

Mutation
Although silting complexes do not necessarily induce derived equivalences, the advantage of silting over tilting complexes is that they have a well-behaved notion of mutation. To define this we require the following.
Suppose that D is an additive category and S is a class of objects in D.
(1) A morphism f : X → Y is called a right S-approximation of Y if X ∈ S and the induced morphism Hom(Z , X ) → Hom(Z , Y ) is surjective for any Z ∈ S. (2) A morphism f : X → Y is said to be right minimal if for any g : X → X such that f • g = f , then g must be an isomorphism.
There is also the dual notion of a (minimal) left S-approximation.
Definition 2.5 [3, 2.31] Let P ∈ K b (proj ) be a basic silting complex for , and write P : where f is a minimal left add(P/P i )-approximation of P i . Then μ i (P) := (P/P i ) ⊕ Q i is also a silting complex, known as the left mutation of P with respect to P i .
Right mutation is defined dually and is denoted μ −1 i as it is inverse to left mutation [3, 2.33]. 1 For general , the mutation of a tilting complex may not be a tilting complex. However, it is well known (see e.g. [3, 2.8]) that if is a symmetric algebra, i.e. ∼ = Hom k ( , k) as -bimodules, then any silting complex is a tilting complex and hence any mutation of a tilting complex is again a tilting complex. This will be the case in our setting later.
To help control mutations, Aihara-Iyama introduced a partial order on silt [3], which generalised the partial order on tilting modules from [32].

Definition 2.6
Let P and Q be silting complexes for . If Hom (P, Q[i]) = 0 for all i > 0, then we say P ≥ Q. Further, we write P > Q if P ≥ Q and P Q.
This order can determine whether two silting complexes are related by mutation.
Theorem 2.7 [2, 3.5] Suppose that T , U are two basic silting complexes for with T ≥ U . If there are only finitely many basic silting complexes P such that T ≥ P ≥ U , then U is obtained by iterated left mutation from T or equivalently, T is obtained by iterated right mutation from U .
The following result, which is implicit in the literature, will be useful for tracking silting complexes through derived equivalences. (1) F maps silting complexes to silting complexes.
(3) If P is a silting complex for , then Proof (1) A standard result of Rickard [33, 6.2] states that F restricts to an equivalence Using that equivalences are fully faithful and must preserve generators easily establishes that the properties of silting complexes are also preserved. (2) Again, this is a simple consequence of the fully faithful property of equivalences.
(3) Since F is a triangle equivalence, it preserves both triangles and minimal left/right approximations. Thus, the exchange triangle defining μ i (P) is mapped under F to the exchange triangle defining μ i (F(P)) and so the first statement, and similarly the second, follow.

Two-term silting complexes
Two-term silting complexes are an important class of silting complexes that have connections with cluster-tilting theory. We recall their definition here.

Definition 2.9
A presilting complex P ∈ K b (proj ) is called two-term if the terms are zero in every degree other than 0 and −1, or equivalently by [2, 2.9], if ≥ P ≥ [1].
We denote the set of isomorphism classes of basic two-term silting (respectively presilting, tilting) complexes as 2-silt (respectively 2-presilt , 2-tilt ). The following shows that mutation of two-term silting complexes is particularly well behaved. Proposition 2.10 [1, 3.8] Suppose P and Q are basic two-term silting complexes for . Then P and Q are related by a single mutation if and only if they differ by exactly one indecomposable summand.

Definition 2.11
A finite dimensional algebra is said to be silting-discrete if for any P ∈ silt and any > 1, the set is finite. Further, is called 2-silting-finite if 2-silt P is a finite set for any P ∈ silt .
Notice that 2-silt = 2-silt . The key advantage of silting-discrete algebras is the following, first shown in [2, 3.9] but provided here with a proof for convenience.

Proposition 2.12
If is a silting-discrete finite dimensional algebra then any silting complex T ∈ K b (proj ) can be obtained from by finite iterated mutation.
Proof Choose a silting complex T . By [2, 2.9], there exists integers m > n such that [n] ≥ T ≥ [m]. We now split the proof into two cases: when n ≥ 0, and when n < 0.
If n ≥ 0, the set {Q ∈ silt | ≥ Q ≥ T } ⊆ (m + 1)-silt and hence is finite by the silting-discrete assumption. Thus, by Theorem 2.7, T is obtained from by iterated left mutation. If is also finite, showing T can be obtained from [n] by iterated left mutation, again using Theorem 2.7. Combining these mutation sequences proves the result.
The following result establishes equivalent conditions for an algebra to be silting-discrete. [4, 2.4] Let be a finite dimensional algebra. Then the following are equivalent:

Theorem 2.13
is 2-silting-finite. (3) 2-silt P is a finite set for any silting complex P which is given by iterated left mutation from .
Write rig C for the set of basic rigid objects in C and write mrig C for the set of basic maximal rigid objects (both taken up to isomorphism). Further, if S is a collection of objects in C, we write rig S for the intersection S ∩ rig C. The mutation of these objects is defined similarly to the mutation of silting complexes.

Definition 2.15 Suppose that
is also a rigid object, known as the left mutation of M with respect to M i . We call the triangle an exchange triangle.
Right mutation is defined dually and we denote it by ν −1 i . As with silting complexes, right and left mutation are inverse operations.

Lemma 2.16 For any rigid object M
This shows that g i is a right add(M/M i )-approximation. There is an isomorphism where g is right minimal and W , Z ∈ add(M/M i ). Completing these maps to triangles gives and hence by the uniqueness of cocones, Hence, Z ∼ = 0 and so g i is right minimal, as required. The other direction is a dual argument.
For maximal rigid objects M and N , it is shown in [39, 3.3], generalising [22, 5.3] for cluster-tilting objects, that M is a mutation of N if and only if M and N differ by exactly one indecomposable summand. An easy consequence of this is that left and right mutation must coincide in this case.
For a rigid object M ∈ C, we will write mut(M) for the collection of basic rigid objects which can be obtained from M by iterated left or right mutation. The relationship between cluster-tilting theory and silting theory relies on the following subcategory of C.

Definition 2.17
Given M ∈ C, define M * M to be the full subcategory of C consisting of the objects N such that there exists a triangle It is easy to show that if M is rigid, then for any triangle such as (2.B), g is a right add(M)approximation and further, by possibly changing M 0 and M 1 , we can choose g to be minimal. The following theorem is a slight generalisation of [1, 4.7], similar to that of [10, 3.2].
such that g is a minimal right add(M)-approximation. Applying Hom C (M, −) to f gives the corresponding 2-term presilting complex.
Proof As the proof is so similar to [1, 4.7], just replacing C with M * M throughout, we only give a sketch of the proof and highlight where adaptations are required.
It is well known (see e.g. [29, 2.3]) that the functor Hom C (M, −) induces an equivalence add(M) ∼ − → proj . This can be used exactly as in [1, 4.6] to show that the map described in the statement gives a bijection where both sides are taken up to isomorphism. The claim that a rigid object N ∈ M * M is sent to a presilting complex then follows exactly as in [1, 4.7], making repeated use of the equivalence add(M) ∼ − → proj . Finally, given a two-term presilting complex P, this is necessarily of the form The proof then proceeds exactly as stated in [1, 4.7]. (1) Since the bijection preserves the number of summands, rigid objects in M * M with the same number of summands as M must correspond to silting complexes.

Remark 2.19
(2) If M is maximal rigid, it is shown in [39, 2.5] that all rigid objects of C lie in M * M and so this bijection restricts to Further, the bijection respects mutation in this case as mutation on both sides corresponds to differing by exactly one indecomposable summand. (3) If C has only finitely many maximal rigid objects, Theorem 2.18 can be used along with [1, 2.38, 3.9], to show that, for any maximal rigid object M, mut(M) = mrig C. This is exactly as in [1, 4.9], but where we replace cluster-tilting with maximal rigid.
Theorem 2.18 can also be used to show the following, which will show that if there are only finitely many basic maximal rigid objects in some category, there must also be only finitely many basic rigid objects.

Lemma 2.20
Let C be a k-linear Hom-finite, Krull-Schmidt, 2-CY triangulated category. If there exists a maximal rigid object in C, then any basic rigid object M ∈ C is a direct summand of some basic maximal rigid object.
Proof Let N ∈ C be a basic maximal rigid object and write := End C (N ). Let φ denote the corresponding bijection from Theorem 2.18. Now take any basic rigid object M ∈ C. By [39, 2.5], M is contained in rig (N * N ) and thus φ(M) ∈ 2-presilt . Hence, by [8, 3.1], there exists P ∈ 2-presilt such that φ(M) ⊕ P is a two-term silting complex for . Mapping back across the bijection, and using Remark 2.19 (2), shows that M ⊕ φ −1 (P) is maximal rigid object which gives the result.

Main results
The following setup will be used throughout this section. Setup 3.1 Suppose E is a Frobenius category such that its stable category E is a k-linear, Hom-finite, Krull-Schmidt, 2-CY triangulated category with shift functor . Additionally assume that: • 2 is isomorphic to the identity functor on E.
• E has at least one but only finitely many basic maximal rigid objects.

Remark 3.2 (1)
The first additional assumption means E is a 0-Calabi-Yau category, but as we wish to view E as a category with cluster-tilting theory plus some additional conditions, we write the assumptions as above. (2) By Lemma 2.20, the second assumption further implies that E has only finitely many basic rigid objects.
A large source of examples of this setup will come from a geometric setting, described in Sect. 4. For an infinite family of examples, see Sect. 4.5.
The first additional assumption gives two important results, the first of which is the following. This was first proved in the setting of hypersurface singularities, but the proof works generally. The key advantage of this result is that it means any silting complex for these endomorphism algebras will in fact be a tilting complex. In particular, the left or right mutation of any tilting complex at any summand, as in Definition 2.5, will again be a tilting complex. This leads to the second important consequence of our assumption on .
Then P is isomorphic to the tilting complex μ i and there is a ring isomorphism Proof The fact that P is a tilting complex and has the required endomorphism ring will follow directly from [15, 4.1] if we can show that the conditions there hold. In particular, we need to show that, for any j ∈ Z, the maps and are surjective. Using the assumption 2 ∼ = id, we need only consider j = 0, 1. When j = 0 this follows as both f i and g i are add(M/M i )-approximations ( f i by definition and g i by Lemma 2.16). When j = 1, rigidity of M and ν i M show that all the terms are zero and hence the maps are surjective as required. Finally, we see that P must be μ i by Proposition 2.10, as P is a two-term silting complex differing from by precisely the i th summand.
Combining Proposition 3.4 and Theorem 2.2 shows that, for any basic rigid object M, there is a derived equivalence By iterating this result, it is easy to see that the stable endomorphism algebras of the elements in the set mut(M) are all derived equivalent algebras. Using the above two results allows us to mutate freely and to investigate how mutation of rigid objects and mutation of tilting complexes interact. Under the setup of 3.1, fix a basic rigid object M := n i=1 M i ∈ E and let := End E (M). Note that each sequence of mutations from M produces a rigid object N ∈ mut(M) along with an induced ordering of the summands of N . To record this data, we introduce the following directed graph which we denote mut(M): • the vertices are ordered tuples N = (N 1 , . . . , N n ), where N := n i=1 N i ∈ mut(M) and the order of the summands is induced by some sequence of mutations from M. Two tuples N and N are identified precisely when N i ∼ = N i for each i ∈ {1, . . . , n}; • there is an arrow s i : N → N if N and N differ in precisely the i th summand, and N ∼ = ν i N .
Note that each vertex has n arrows s 1 , . . . , s n starting at the vertex (corresponding to left mutations) and n arrows s 1 , . . . , s n incident at the vertex (corresponding to right mutations).
Note that if M and N are related by mutation, then mut(M) and mut(N ) will be isomorphic graphs. In particular, for any maximal rigid object, the mutation graph will be the same and so we just denote this mrig(E).
The idea behind the results in this paper is that combinatorial paths in mut(M) will control not only mutation of rigid objects but also tilting complexes of .   Not only will paths in mut(M) correspond to mutation sequences, but also to derived equivalences between the algebras of interest. Combining Proposition 3.4 with Theorem 2.2, for each arrow s i : N → N in mut(M) there exists a derived equivalence where, giving End E (N ) the induced ordering Hom E (N , N 1 ), . . . , Hom E (N , N n ) from N , the jth summand of μ i (End E (N )) maps to Hom E (N , N j ). Fix such a derived equivalence F i for each arrow s i . For example, we could choose F i to be the unique (up to algebra isomorphism) standard derived equivalence associated to μ i (End E (N )), as discussed in Sect. 2.1.1.  ν α N )).

Example 3.9
Consider the path α := s 1 s 1 s 2 s −1 1 from Example 3.7. Using that right and left mutation are equal for maximal rigid objects, the object at the end of the path is as in the diagram. Similarly, but with no cancellation now as left and right mutation are different, The following is the main technical result of this section.
Proof Note that (2) follows easily from part (1) since F α is an equivalence. We will prove part (1) by induction on m.
Base Case m = 1: If α := s i , then the result follows from our choice of the F i . If α := s −1 i for s i : ν −1 i N → N , then the assumption given on F i is that But then, , as required.

Inductive
Step: Let β := s ε m−1 i m−1 . . . s ε 1 i 1 so by the inductive hypothesis, F β (μ β ( )) ∼ = End E (ν β N ). Then, where the last isomorphism holds by applying the base case to the path s ε m i m from ν β N to ν α N .
This has the following easy corollary. Using this result, to completely determine the basic algebras in the derived equivalence class, we just need to show that every basic tilting complex for such a can be obtained by iterated right or left mutation from . By Lemma 2.12, this will follow from showing is silting-discrete. By definition, the left hand side is 2-silt T , while the right hand side is 2-silt . But since is also the endomorphism algebra of a rigid object in E, the first argument shows 2-silt must be finite and hence so is 2-silt T .
The following is our main result. Part (1) is a consequence of Theorem 3.12 showing how to view all tilting complexes combinatorially in mut(M) and part (2) is a consequence of the fact this viewpoint allows us to control the endomorphism rings of the tilting complexes via Proposition 3.10.  Recall from Remark 2.19(3) that, as there are only finitely many basic maximal rigid objects in E, mut(M) = mrig E for any maximal rigid object M ∈ E. Therefore, when restricted to maximal rigid objects, part (2) of Corollary 3.13 specialises to the following.
using Proposition 3.10. If the F i are standard equivalences, then F α must also be and hence F α is the unique (up to algebra isomorphism) standard equivalence associated to the tilting complex μ α . (2) Take any standard equivalence F : D b ( ) → D b ( ). Then F −1 ( ) is a tilting complex for and so by Propositions 2.12 and 3.12, F −1 ( ) ∼ = μ α for some path α in mut(M) starting at M. By part (1), F α is also a standard equivalence induced by μ α and hence F and F α must be the same up to an algebra isomorphism by Proposition 2.4.
In this way, we can say the diagram in Example 3.15 not only contains all the basic members of the derived equivalence class but also all the standard equivalences between them (up to algebra isomorphism) and that these correspond precisely to the paths. In this way, we have a complete picture of the derived equivalence class of these algebras.

Remark 3.17
In this setting, there is no way of telling when two paths give rise to the same derived equivalence. To be able to say this, we would need to define explicit standard equivalences and then understand how to compose them. This idea is explored in [6] where additional structure coming from the geometric setting is used in order to choose the F i explicitly.

Geometric application
In this section, we apply the results from the previous section to certain invariants of 3-fold flopping contractions, known as contraction algebras.

Complete local setup
For our purposes, a 3-fold flopping contraction is a projective birational morphism f : X → X con between Gorenstein normal C-schemes of dimension three satisfying R f * O X = O X con , and which further is an isomorphism in codimension one. When the base of the flopping contraction is affine and complete local, contraction algebras have a very explicit construction and so we begin with this case. Setup 4.1 Take f : X → Spec R to be a 3-fold flopping contraction where R is complete local and X has at worst Gorenstein terminal singularities.
The condition on X in this setup is satisfied when X is smooth but also allows for some mild singularities. Moreover, the condition also forces Spec R to have at worst Gorenstein terminal singularities and hence to be an isolated cDV singularity [31]. u, v, x)) is a Du Val surface singularity and g is arbitrary.
We will sometimes wish to assume further that X is Q-factorial (see [38, §2] for a definition) but will only do so if explicitly stated. For a cDV singularity, it is well known that there are only finitely many minimal models [24] and one goal of the Homological Minimal Model Programme was to provide an algorithm that can produce all the minimal models from a given one, similar to how all maximal rigid objects can be obtained via iterated mutation from a given maximal rigid object in our previous setting. The key observation is that the maximal Cohen-Macaulay modules of R link the two settings. The following summary theorem asserts that E := CM R satisfies all but the last condition of Setup 3.1. For details, and full references, see e.g. [9, §1].

Proposition 4.5 If R is a complete local isolated cDV singularity, CM R is a Frobenius category. Moreover, the stable category CM R is a Krull-Schmidt, Hom-finite, 2-Calabi-
Yau triangulated category with shift functor satisfying 2 ∼ = id.
In particular, rigid objects in CM R can be defined as in Definition 2.14. Moreover, in Theorem 4.11, we will see that CM R also satisfies the last condition of Setup 3.1 and thus we will be able to apply our main results to this setting. However, for this to be useful in studying the geometry, the rigid objects need to have connection with the geometry. This link comes about via the contraction algebras; an invariant of 3-fold flops introduced by Donovan-Wemyss [11,13].

Construction of the contraction algebra
In the setting of 4.1, the contraction algebra attached to f has a very explicit construction, detailed in [12, 3.5], but provided here for convenience.
It is well known that in the setup of 4.1, Spec R has a unique singular point m and the preimage C := f −1 (m) consists of a chain of curves. In particular, giving C the reduced scheme structure, we have For each i, let L i be the line bundle on X such that L i · C j = δ i j . If the multiplicity of C i is equal to 1, set M i := L i . Otherwise, define M i to be given by the maximal extension

Definition 4.6
The contraction algebra associated to f is defined to be denotes the ideal of all morphisms which factor through add R.

Remark 4.7
The contraction algebra A con can also be defined as the representing object of a certain deformation functor of the curves in C, but we will not need this here.

Summary of results from the homological minimal model programme
The following key proposition provides the link between contraction algebras and clustertilting theory.

Proposition 4.8
Assuming the setup of 4.1 and with notation as above, the following hold.
(1) Each N i lies in CM R and further, N := n i=1 N i is a rigid object in CM R.
(2) f is a minimal model if and only if N := n i=1 N i is maximal rigid in CM R.

Proof
(1) As f : X → Spec R is crepant, [19, 4.14] shows that End R (R⊕ N ) lies in CM R and hence, as N i ∼ = Hom R (R, N i ) is a direct summand, N i also lies in CM R. Thus, N is a modifying module in the terminology of [18, 4.1] and hence, by [18, 5.12], satisfies Ext 1 R (N , N ) = 0 which is equivalent to N being rigid in CM R.
(2) By [19, 4.16], f is a minimal model if and only if R ⊕ N is a maximal modifying module and thus if and only if N is a maximal rigid object in CM R by [18, 5.12].
This proposition shows that the construction in Sect Since N := n i=1 N i is a rigid object in CM R, we can choose any summand N i to mutate at and produce a new rigid object ν i N . Alternatively, consider the curve C i in the exceptional locus corresponding to the summand N i . Since f : X → Spec R is a flopping contraction and R is complete local, choosing any such C i , it is possible to factorise f as where g(C j ) is a single point if and only if j = i. For any such factorisation, there exists a certain birational map g + : X + → X con , satisfying some technical conditions detailed in [38, 2.6], which fits into a commutative diagram where φ is a birational equivalence (see e.g. [28, p25] or [36, §2]). We call f + : X + → Spec R the simple flop of f at the curve C i . Since f + is again a flopping contraction, it also has an associated contraction algebra, constructed as in Sect. 4.2.
We can therefore consider two algebras obtained by 'mutation' from f : (1) The algebra End R (ν i N ) obtained by mutating N at summand N i .
(2) The contraction algebra End R (M) associated to f + , where f + is the simple flop of f at the curve C i .
The following proposition shows that these algebras are isomorphic.
Proposition 4.9 [38, 4.20(1)] Under the setup of 4.1, suppose that the contraction algebra associated to f is End R (N ). If f + is the flop of f at C i , then the contraction algebra of f + is End R (ν i N ).
Since f + will have the same number of curves in the exceptional locus as f , we can consider iterated flops. (1) f i 1 is the simple flop of the f at curve C i 1 . ( We call the sequence (i 1 , . . . , i m ) a mutation sequence.
Repeated use of Proposition 4.9 shows that the contraction algebra of f i m ...i 1 is given by In other words, the maps (4.A) and (4.B) respect mutation. Using this, and the fact there are only finitely many minimal models, it is possible to show that the map in (4.B) is surjective [38, 4.10]. Further, both maps can be seen to be injective [38, 4.4] which gives the following result. Since Spec R has only finitely many minimal models, a direct consequence of Theorem 4.11 is that CM R must have only finitely many maximal rigid objects.

New results
Combining Proposition 4.5 and the remark after Theorem 4.11 shows that CM R and its stable category satisfy the conditions in 3.1. Thus we can apply the results of the previous section to obtain the following, which is our main geometric result. (1) A con is a silting-discrete algebra.
(2) The endomorphism algebra of the tilting complex T := μ ε m i m . . . μ ε 1 i 1 A con is isomorphic to the contraction algebra of f i m ...i 1 .
(3) The basic algebras derived equivalent to A con are precisely the contraction algebras of flopping contractions g : Y → Spec R, obtained by a sequence of iterated flops from f . In particular, there are only finitely many such algebras.
Proof Recall that Spec R is a complete local isolated cDV singularity and thus, by Proposition 4.5 and the comments after Theorem 4.11, CM R and its stable category satisfy the conditions in 3.1.
(1) As N is rigid in CM R by Proposition 4.8, the algebra is silting-discrete by Theorem 3.12.
(2) By Proposition 3.10, there is an isomorphism . However, in this setting right and left mutation are equal by [38, 2.25] and thus which is the contraction algebra of f i m ...i 1 , by repeated use of Proposition 4.9.
(3) Combining part (1), Proposition 2.12 and part (2) shows that the endomorphism algebra of any basic tilting complex for A con is isomorphic to the contraction algebra of some flopping contraction obtained from f by a sequence of iterated flops. Applying Theorem 2.2 gives the result.
In the special case of minimal models, it is well known that any two minimal models are connected by a sequence of simple flops [23]. Thus, part (3) of Theorem 4.12 reduces to the following.

Corollary 4.13
Let f : X → Spec R be a minimal model of a complete local isolated cDV singularity. Writing A con for the associated contraction algebra, the basic algebras derived equivalent to A con are precisely the contraction algebras of minimal models of Spec R.
Recall from Conjecture 1.3, that it is expected that the derived category of the contraction algebras of Spec R completely controls the geometry. The following shows that, to some extent, this is true. Theorem 4.14 Suppose that f : X → Spec R and g : Y → Spec S are minimal models of complete local isolated cDV singularities with associated contraction algebras A con and B con . If A con and B con are derived equivalent then there is a bijection {minimal models of Spec R} ←→ {minimal models of Spec S}.

Further, the bijection preserves both mutation and contraction algebras.
Proof Let M ∈ CM R be the maximal rigid object associated to f and let N ∈ CM S be the maximal rigid object associated to g so that A con := End R (M) and B con := End S (N ).
Fix an ordering C 1 , . . . , C n of the curves in the exceptional locus of f , which fixes an ordering on the decomposition M ∼ = n i=1 M i such that M i corresponds to curve C i via the construction in Sect. 4.2.
Since B con is basic and derived equivalent to A con , Corollary 4.13 shows that B con must be isomorphic to a contraction algebra for some minimal model of Spec R. In particular, as any two minimal models are connected by a sequence of flops B con must be the contraction algebra of  (N , N i ) as projective B con -modules. This also fixes a labelling D 1 , . . . , D n of the curves in the exceptional locus of g so that N i corresponds to D i via the construction in Sect. 4 Thus for any mutation sequence ( j 1 , . . . , j l ), our choice of indexing on the curves ensures f j l ... j 1 → g j l ... j 1 . Since the contraction algebra of both f and g is B con , part (2) of Theorem 4.12 shows the contraction algebra of both f j l ... j 1 and g j l ... j 1 is and hence the bijection preserves contraction algebras.

Example
A large class of examples can be obtained from complete local c A m−1 singularities. These can be written in the form where m is the order of the polynomial f (x, y) considered as a power series. We will only consider isolated singularities, which can be characterised using the irreducible factors of f (x, y); namely, if f factors into n irreducible power series f 1 , . . . , f n , then R is isolated precisely when ( f i ) = ( f j ) for all i = j. For these singularities, the maximal rigid objects in CM R have been completely determined.

Theorem 4.16
In particular, if we choose n = 3 , there are six maximal rigid objects, each with two summands and the mutation graph is shown in Fig. 1. Note that in this geometric setting, it is shown in [21, §1, §7] that fixing the ordering of the summands of one rigid object in CM R fixes an ordering on all the other rigid objects in mut(M), independent of the mutation sequences taken. Thus, each N ∈ mut(M) will appear precisely once as a vertex of mut(M).
Going back to the n = 3 example, by Theorem 4.11 there are six minimal models of any such Spec R, each with two curves in the exceptional locus. Choosing f 1 = x, f 2 = y and f 3 = x 2 + y 3 the contraction algebras associated to these minimal models are given in Fig. 2.
By Corollary 4.13, these are the only basic members of a derived equivalence class. Further, as in Example 3.15, we obtain a picture of this derived equivalence class, shown on the right hand side of Fig. 1, which controls the tilting complexes and hence the derived equivalences.

Remark 4.17
More generally, if A con is the contraction algebra of a minimal model of an isolated c A m−1 singularity given by uv − f 1 . . . f n , then the quiver of A con will be the double of the A n−1 Dynkin quiver, possibly with up to two loops at each vertex [20, 5.29].

Global setting
In this final subsection, we remove the restrictions in Setup 4.1 that the base of the flopping contraction needs to be complete local, or even affine.

Setup 4.18
Take f : X → X con to be a 3-fold flopping contraction between quasi-projective varieties where X has at worst Gorenstein terminal singularities.
In this more general setup, Donovan-Wemyss introduce a more general invariant given by a sheaf of algebras [14]. As with the construction of the contraction algebra, the construction involves a vector bundle V := O X ⊕ V 0 on X satisfying Although this bundle may not be tilting (as it is in the complete local case) there is a technical condition on V, detailed in [14, 2.3], which ensures that for any choice of affine open Spec R in X con , the bundle V| f −1 (Spec R) is a tilting bundle.
With this bundle V, they define the sheaf of contraction algebras to be D := f * End X (V)/I where I is the ideal sheaf of local sections that at each stalk at v ∈ X con factor through a finitely generated projective O X con ,v -module (see [14, 2.8] for details).
Writing Z for the locus of points on X con above which f is not an isomorphism, [14, 2.16] showed that the support of the sheaf D is precisely Z . In particular, in the setup of 4.18, the condition on X ensures that Z = {p 1 , . . . , p n } where each p i is an isolated singularity and thus where D p i is the O X con , p i -algebra given by the stalk of D at p i . Specifically, in the setup of 4.18, D is a finite dimensional algebra which splits into a direct sum of algebras, one for each point p i .
Alternatively, for each p i , it is possible to choose an affine neighbourhood R i of p i which contains no other p j . Localising if necessary, we can assume p i is the unique closed point of R i and setting U i := f −1 (Spec R i ), we can consider the map f i := f | U i . Further, we can complete this map to obtain a map This map now satisfies the conditions of the complete local setup in 4.1 and thus we get an associated contraction algebra A i := End R i (N i ) where N i is a rigid object in CM R i . [14, 2.24] The completion of D p i is Morita equivalent to A i .

Theorem 4.19
As D p i is a finite length module over O X con , p i , there is an isomorphism D p i ∼ = D p i of O X con , p i -algebras, where D p i denotes the completion of D p i . Combining this with our earlier results gives the following. where M i ∈ mut R i (N i ). In particular, there are only finitely many basic algebras in the derived equivalence class.
Proof Any algebra derived equivalent to D must be of the form where B i is derived equivalent to D p i . However, by Theorem 4.19 and the remark after, D p i is Morita equivalent to A i := End R i (N i ) and thus each B i must be derived equivalent to A i . However, for each i, Corollary 3.13 shows that the only basic algebras in the derived equivalence class of A i are End R i (M i ) for M i ∈ mut R (N i ) and thus B i must be Morita equivalent to one of these algebras.