NONCOMMUTATIVE DEFORMATIONS AND FLOPS

arXiv:1309.0698v4 [math.AG] 3 Oct 2015

WILL DONOVAN AND MICHAEL WEMYSS
Abstract. We prove that the functor of noncommutative deformations of every flipping or flopping irreducible rational curve in a 3-fold is representable, and hence associate to every such curve a noncommutative deformation algebra Acon . This newinvariant extends and unifies known invariants for flopping curves in 3-folds, such as
the width of Reid [R83], and the bidegree of the normal bundle. It also applies inthe settings of flips and singular schemes. We show that the noncommutative deformation algebra Acon is finite dimensional, and give a new way of obtaining thecommutative deformations of the curve, allowing us to make explicit calculations ofthese deformations for certain (−3, 1)-curves.
We then show how our new invariant Acon also controls the homological algebraof flops. For any flopping curve in a projective 3-fold with only Gorenstein terminal
singularities, we construct an autoequivalence of the derived category of the 3-foldby twisting around a universal family over the noncommutative deformation algebra Acon , and prove that this autoequivalence is an inverse of Bridgeland’s flop–flopfunctor. This demonstrates that it is strictly necessary to consider noncommutative
deformations of curves in order to understand the derived autoequivalences of a 3-fold,
and thus the Bridgeland stability manifold.

Contents
1. Introduction
2. Noncommutative Deformation Functors
3. Representability and Contraction Algebras
4. Strategy for Noncommutative Twists
5. Complete Local Mutation and Ext Vanishing
6. Zariski Local Twists
7. Noncommutative Twist Functors: Global Case
Appendix A. List of Notations
References

1
7
16
24
25
31
38
48
49

1. Introduction
To understand the birational geometry of algebraic varieties via the minimal model
program, it is necessary to understand the geometry of certain codimension two modifications known as flips and flops. Even for modifications of irreducible rational curves indimension three this geometry is extraordinarily rich, and has been discussed by many
authors, for example [P83, R83, CKM, KaM, KoM, K94]. A central problem is to classify
flips and flops in a satisfying manner, and to construct appropriate invariants.
In the first half of this paper we associate a new invariant to every flipping or flopping
curve in a 3-fold, using noncommutative deformation theory. This generalises and unifies
the classical invariants into one new object, the noncommutative deformation algebra Acon
associated to the curve. Our new invariant is a finite dimensional algebra, and can be2010 Mathematics Subject Classification. Primary 14D15; Secondary 14E30, 14F05, 16S38, 18E30.
The first author was supported by the EPSRC grant EP/G007632/1 and by the Erwin Schrödinger
Institute, Vienna, during the course of this work. The second author was supported by EPSRC
grant EP/K021400/1.
1

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WILL DONOVAN AND MICHAEL WEMYSS

associated to any contractible rational curve in any 3-fold, regardless of singularities. It
recovers classical invariants in natural ways. Moreover, unlike these classical invariants,
Acon is an algebra, and in the second half of the paper we exploit this to give the first
intrinsic description of a derived autoequivalence associated to a general flopping curve.
1.1. Background on 3-folds. The simplest example of a 3-fold flop is the well-known
Atiyah flop. In this case the flopping curve has normal bundle O(−1) ⊕ O(−1) and isrigid, and the flop may be factored as a blow-up of the curve followed by a blow-down.
However, for more general flops the curve is no longer rigid, and the factorization of the
birational map becomes more complicated. For a general flopping irreducible rational
curve C in a smooth 3-fold X, the following classical invariants are associated to C.
(1) Normal bundle. Denoting O(a) ⊕ O(b) as simply (a, b), the normal bundle mustbe (−1, −1), (−2, 0) or (−3, 1) [P83, Prop. 2].
(2) Width [R83, 5.3]. In the first two cases above, the width of C is defined to be"!
sup n | ∃ a scheme Cn ∼= C × Spec C[x]/xn with C ⊂ Cn ⊂ X .

More generally, in the case (−3, 1), it is not possible to define the width invariant
in this way.
(3) Dynkin type [KaM, K94]. By taking a generic hyperplane section, Katz–
Morrison and Kawamata assign to C the data of a Dynkin diagram with a marked
vertex. Only some possibilities arise, as shown in Table 1 and (3.B).
(4) Length [CKM, p95–96]. This is defined to be the multiplicity of the curve C in
the fundamental cycle of a generic hyperplane section.
X ′ factors into a(5) Normal bundle sequence [P83, Thm. 4]. The flop f : X
sequence of blow-ups in centres C1 , . . . , Cn , followed by blow-downs. The normal
bundles of these curves form the N -sequence.
The following table summarises the relations between the invariants above, and showsthat none of these invariants classify all analytic equivalence types of flopping curves.
(1)

(2)

(3)

(4)

(5)

NC|X

width

Dynkin type

lengthN -sequence(−1, −1)1A1

1(−2, 0)(−3, 1)n!
"#$(−2, 0) . . . (−2, 0) (−1, −1)

n>1

undefined(−1, −1)

D4

2

E6

3

E7

4

E8 (6)

6

E8 (5)

5(−3, 1) (−2, −1) (−1, −1)(−3, 1) (−3, 0) (−2, −1) (−1, −1)(−3, 1) (−3, 0) (−3, 0) (−2, −1) (−1, −1)

Table 1. Classical invariants of flopping rational curves in smooth 3-folds.
1.2. Noncommutative deformations and invariants. Our new invariant of floppingand flipping rational curves C in a 3-fold X is constructed by noncommutatively deforming the associated sheaf E := OC (−1). Classically, infinitesimal deformations of C arecontrolled by Ext1X (E, E), the dimension of which is determined by the normal bundle
NC|X . If dim C Ext1X (E, E) ≤ 1, the curve C deforms over an artinian base C[x]/xn .
However, when dim C Ext1X (E, E) ≥ 2, deformations of C are less well understood, andfor example in the setting of flops the width invariant of Reid is no longer suﬃcient tocharacterize them.

NONCOMMUTATIVE DEFORMATIONS AND FLOPS

3

In this paper, we solve this problem by using noncommutative deformation theory. We
give preliminary definitions here, leaving details to §2. Recall that the formal commutative
deformations of the simple sheaf E associated to C are described by a functor from
commutative local artinian C-algebras to sets, given by
cDefE : CArt → Sets
R *→ {flat R-families of coherent sheaves deforming E}/∼ .
In the above we take the point of view that a family is an R-module in coh X, as thisturns out to generalise. Following [L02, E07], we may then define a noncommutative deformation functor DefE : Art1 → Sets from 1-pointed artinian not-necessarily-commutative
C-algebras to sets, which extends cDefE (for details see 2.4). Our first main theorem isthe following.
Theorem 1.1 (=3.3(2)). For a flopping or flipping curve C in a 3-fold X, the noncommutative deformation functor DefE is representable.
As a consequence of 1.1, from the noncommutative deformation theory we obtain auniversal family E ∈ coh X. At the same time, cDefE is also representable, giving anotheruniversal family F ∈ coh X. The following are our new invariants.
(6) Noncommutative deformation algebra. This is defined Adef := EndX (E).
(7) Commutative deformation algebra. We define this to be EndX (F ), and provein 3.9 that it is isomorphic to the abelianization Aabdef , given by quotienting Adefby the ideal generated by commutators.
The above objects are algebras, and not just vector spaces. Nevertheless, taking
dimensions we obtain the following new numerical invariants.
(8) Noncommutative width. We take wid(C) := dim C Adef .
(9) Commutative width. We set cwid(C) := dim C Aabdef .
For flopping and flipping curves these numerical invariants are always finite by 1.2,
regardless of singularities. We explain in 5.2 that the noncommutative width wid(C) isreadily calculated, and is indeed easier to handle than the commutative width cwid(C).
From the viewpoint of the homological algebra, we will see later in §1.7 that the noncommutative width is the more natural invariant.
For floppable curves of Type A, when Reid’s width invariant is defined [R83], weprove in 3.15 that all three width invariants agree. For all other contractible curves,
including flips and singular flops, our invariants are new. We will see in 1.3 below that thenoncommutative width may be strictly larger than the commutative width. Remark 3.17
also gives lower bounds for the width according to Dynkin type. This makes it clear thatthe algebras Adef can be quite complex: for instance in type E8 we find that wid(C) ≥ 40.
1.3. Contraction algebras. In addition to the deformation-theoretic viewpoint given
above, we explain in §3 how Adef arises also as a contraction algebra associated to an
algebra obtained by tilting. It is this alternative description of Adef that allows us tocalculate it, and also control it homologically.
Briefly, for any contractible curve C in a 3-fold X, by passing to an open neighbour# is# → Spec R,# it is well known [V04] that U
hood U of C, and then to the formal fibre U##
derived equivalent to an algebra A := EndR% (R ⊕ N1 ). See §2.4 for more details. Let [R]# ⊕ N1 → R# ⊕ N1 that factor
be the two-sided ideal of A consisting of homomorphisms R##
through add R. We write Acon := A/[R] and call Acon the contraction algebra.
Theorem 1.2 (=3.9, 2.13). For a flopping or flipping curve in a 3-fold, Adef ∼= Acon ,and furthermore this is a finite dimensional algebra.

We refer the reader to 3.10 for the calculation of Acon in the case of the Pagoda flop,in which case Acon ∼= C[x]/xn , as expected. More generally, our approach allows us tomake calculations beyond Dynkin type A. We illustrate this in the following example.

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WILL DONOVAN AND MICHAEL WEMYSS

Example 1.3. The simplest non-type-A flopping contraction is obtained by settingC[[u, v, x, y]]# :=,R
u2 + v 2 y = x(x2 + y 3 )# obtained by blowing up a generator of the classand considering the morphism X → Spec R
group. There is a single floppable (−3, 1)-curve above the origin, and this is a length two# ⊕ N1 )D4 flop, known as the Laufer flop [R83, AM]. In this case, by [AM], A := EndR% (Rcan be presented as the completion of the quiver with relations
x

%
R

a

N1

b

y

ay 2 = −abay 2 b = −babxy = −yxx2 + yba + bay = y 3 .

It follows immediately thatC⟨⟨x, y⟩⟩C⟨x, y⟩=.
23xy = −yx, x = y
xy = −yx, x2 = y 3
We call this algebra a quantum cusp. From this presentation, it can be seen that wid(C) =9 (for details, see 3.14). Furthermore, factoring by commutators it follows that the commutative deformation algebra is given by∼ C[x, y]Aabcon =xy, x2 = y 3
Acon ∼=and so cwid(C) = dimC Aabcon = 5.
The deformations are sketched in Figure 1. In the commutative case on the left, theoriginal curve moves infinitesimally, fibred over Spec Aabcon , with the dots in the base representing the vector space basis {1, x, x2 , y, y 2 } of Aab.
con The noncommutative deformations
on the right are harder to draw, but roughly speaking they thicken the universal sheaf F
to a sheaf E ∈ coh X over Acon , where the nine dots represent a basis of Acon .
X

y

Aab
con

x

y
x

Acon

Figure 1. Comparing commutative and noncommutative deformationsof the Laufer D4 floppable curve in 1.3.
Using this example, and its generalization 3.14, we are able to extend the previous
known invariants in Table 1. This is summarised in Table 2. We feel that the further
study of the structure of these newly-arising algebras Acon , and of their representations, isof interest in itself. Furthermore, we conjecture that the contraction algebra distinguishes
the analytic type of the flop:
Conjecture 1.4. Suppose that Y → Ycon and Z → Zcon are flopping contractions ofan irreducible rational curve in smooth projective 3-folds, to points p and q respectively.
To these, associate the contraction algebras Acon and Bcon . Then the completions of thestalks at p and q are isomorphic if and only if Acon ∼= Bcon .

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NONCOMMUTATIVE DEFORMATIONS AND FLOPS

In the Type A and D cases for which we know the algebra Acon explicitly, the conjecture is true. We will return to the general case in future work.

ExampleAtiyah

(3)

(6)

(7)

(8)

(9)

Dynkin type

Acon

Aab
con

wid(C)cwid(C)

C

C

1

1

C[x]
xn
C⟨x,y⟩
xy=−yx, x2 =y 2n+1
quantum cusp

C[x]xn
C[x,y]
xy, x2 =y 2n+1
cusp

n

n

3(2n + 1)

2n + 3

A1

Pagoda (3.10)
Laufer (3.14)

D4

Table 2. Comparison of new invariants in known examples.

1.4. Homological algebra and derived autoequivalences. We now restrict our attention to flopping contractions, and use the contraction algebra Acon to give a unified
construction of spherical-type twists related to flops. This works even in the mildly singular setting.
Given X, a projective 3-fold with at worst terminal Gorenstein singularities, and aflopping contraction X → Xcon with flop X ′ → Xcon , then there is an autoequivalence ofDb (coh X) given as the composition of the flop equivalencesFF′

Db (coh X) −→ Db (coh X ′ ) −→ Db (coh X)
where the Fourier–Mukai kernel OX×Xcon X ′ is used for both F and F′ [B02, C02]. We
denote this composition by FF.
Without assuming knowledge of the flop, we construct an autoequivalence of Db (coh X)
which we call the noncommutative twist functor, using the noncommutative deformation
theory from the previous section. Then, using that FF is an autoequivalence, we showthat it is an inverse of the noncommutative twist. This gives an intrinsic description of
FF in terms of noncommutative deformation theory.
1.5. Known autoequivalence results. When X is smooth and the flopping contraction
is of Type A, the functor FF has previously been described intrinsically as some form ofspherical twist, which we now briefly review.
Spherical twists about spherical objects, in the sense of [ST01], are the simplest typeof non-trivial derived autoequivalence. If C is a (−1, −1)-curve in a smooth 3-fold, thesheaf E := OP1 (−1) ∈ coh X is a spherical object, and it was very well known to expertsthat the resulting twist autoequivalence was an inverse of FF.
For all other types of floppable curves, the structure sheaf E is not spherical, so thetwist of [ST01] is not an autoequivalence. Toda proposed in [T07] a conceptual way to
overcome this. The commutative deformation functor cDefE defined above gives us auniversal sheaf F ∈ coh X. Using this sheaf, Toda constructed a Fourier–Mukai functor
TF : Db (coh X) → Db (coh X).
Under the assumption that X is smooth, using explicit knowledge of the deformation base,Toda proved the following.
• There is a functorial triangle RHomX (F , −) ⊗L
S F → Id → TF →.
• For flopping contractions of Type A, TF is an autoequivalence.
• For flopping contractions of Type A, TF is an inverse of FF.
The techniques in [T07] have been very influential, and may be considered in themuch more general framework of ‘spherical functors’, see [AL10, AL13]. However, it hasremained an open question as to whether the above results hold for all flopping curves,

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WILL DONOVAN AND MICHAEL WEMYSS

and whether they extend to mildly singular schemes. The commutative deformation baseis extremely hard to calculate, and so it is diﬃcult to determine whether the techniquesin [T07] can be extended.
1.6. New autoequivalence results. We now consider the general setting where X is a
projective 3-fold with at worst terminal Gorenstein singularities, and there is a flopping
contraction X → Xcon . By exploiting the universal sheaf E provided by the noncommutative deformation theory in §1.2, in 7.13 we construct a Fourier–Mukai functor
TE : Db (coh X) → Db (coh X).
Toda’s functor TF was constructed using Spec C[x]/xn in a crucial way, but in this general setting we cannot take Spec of Acon since it may not be commutative, and so ourconstruction of TE is necessarily more abstract.
The following is the main result of this paper.
Theorem 1.5 (=7.14, 7.18). For any flopping contraction of an irreducible rational curve
C in X, a projective 3-fold with only terminal Gorenstein singularities,• There is a functorial triangle RHomX (E, −) ⊗L
Acon E → Id → TE →.
• The functor TE : Db (coh X) → Db (coh X) is an autoequivalence.
• The functor TE is an inverse of FF.
Thus in addition to extending the previous work to cover (−3, 1)-curves, our noncommutative deformation technique also extends to the setting of mildly singular schemes. In
the course of the proof, we establish the following.
Proposition 1.6 (=5.7, 7.1). With the assumptions as in 1.5, and writing as above
E := OP1 (−1) ∈ coh X,• The contraction algebra Acon is a self-injective finite dimensional algebra.
• The universal object E is a perfect complex (although E need not be).
• We have$
C if t = 0, 3
ExttX (E, E) =0 else.
Remark 1.7. We will not discuss here whether a suitable notion of spherical functors
could be applied in our noncommutative setting, but the above suggests that it is possible
to use this axiomatic approach, potentially giving noncommutative twists in other triangulated and DG categories. However, we remark that the results in the proposition are
not formal consequences of deformation theory, and so proving results in any axiomatic
framework will necessarily need to use results obtained in this paper, or find anotherapproach to show 1.6.
The noncommutative deformation base Acon is artinian, and thus the noncommutative deformations of flopping curves are infinitesimal and may be analysed on the formal
fibre. Consequently, to establish the Ext vanishing and other properties we pass to thisformal fibre. There, we adapt the techniques of [IW10, §6] and prove 1.6 without an
explicit presentation of Acon , or equivalently a filtration structure on E.
In the course of proving that TE is an autoequivalence, once we have shown that
TE is fully faithful, the usual Serre functor trick [BKR, 2.4] to establish equivalence does
not work in our possibly singular setting. However we are able to bypass this by instead
giving an elementary argument based on the ‘diﬃcult to check’ criterion in [H06, 1.50].
This may be of independent interest.
Lemma 1.8 (=7.11). Let C be a triangulated category, and F : C → C an exact fully
faithful functor with right adjoint F RA . Suppose that there exists an object c ∈ C suchthat F (c) ∼= c[i] for some i, and further F (x) ∼= x for all x ∈ c⊥ . Then F is an equivalence.

NONCOMMUTATIVE DEFORMATIONS AND FLOPS

7

1.7. Necessity of noncommutative deformations. Restricting to the case when X issmooth, by studying Acon we can also prove that Toda’s functor TF defined using commutative deformations is not an autoequivalence for (−3, 1)-curves. This shows that ournoncommutative deformation approach is strictly necessary. To achieve this, we investigate the conditions under which Acon is commutative.
Theorem 1.9 (=3.15). Let X → Xcon be the flopping contraction of an irreducible rational curve C, where X is smooth. Then the contraction algebra Acon is commutative if
and only if C is a (−1, −1)-curve or a (−2, 0)-curve.
The main content in the above theorem is the ‘only if’ direction, and we prove it
without any explicit case-by-case analysis. When Acon is not commutative, we prove in
3.19 that Toda’s functor TF is not an equivalence. Combining with 1.9, we obtain thefollowing corollary.
Corollary 1.10 (=3.19). With the setup as in 1.9, suppose that C is a (−3, 1)-curve.
Then Toda’s functor TF is not an autoequivalence of Db (coh X).
1.8. Conventions. Unqualified uses of the word ‘module’ refer to right modules, andmod A denotes the category of finitely generated right A-modules. Sometimes left modules
will appear, in which case we will either emphasize the fact that they are left modules,
or consider them as objects of mod Aop . If M ∈ mod A, we let add M denote all possible
summands of finite sums of M . We say that M is a generator if R ∈ add M .
We use the functional convention for composing arrows, so f ·g means g then f . With
this convention, M is a EndR (M )op -module. Furthermore, HomR (M, N ) is a EndR (M )module and a EndR (N )op -module, in fact a bimodule. Note also that HomR (S MR , T MR )is an S–T bimodule and HomRop (R MS , R MT ) is a T –S bimodule.
Throughout, we work over the field of complex numbers C, but many of our arguments
work without this assumption.
1.9. Notations. In Appendix A we list our main notations, along with cross-references
to their definitions.
Acknowledgements. It is a pleasure to thank Alessio Corti for conversations concerning[P83], Colin Ingalls for bringing to our attention [L02], Osamu Iyama for explaining aspectsof [IR08], Tim Logvinenko for explaining both the technical and philosophical aspects of[AL10] and [AL13], Ed Segal for answering our deformation questions related to his thesis[S08], Miles Reid for discussions regarding [R83], and the referees for their careful reading
and simplifying suggestions. The first author is grateful for the support of the Erwin
Schrödinger Institute, Vienna, and for the hospitality of Matthew Ballard, David Favero,
and Ludmil Katzarkov. The second author also wishes to thank the Mathematical Sciences
Research Institute, Berkeley, for its hospitality whilst this manuscript was being prepared.
2. Noncommutative Deformation Functors
Noncommutative deformations of modules over algebras were first introduced andstudied by Laudal [L02]. Later this was extended to the setting of quasi-coherent sheaves
over schemes by Eriksen [E07], and also studied by Segal [S08], Efimov–Lunts–Orlov[ELO1, ELO2, ELO3] and others. These works all establish prorepresentability of certain
functors, whereas in the situation of flips and flops we will require the much stronger
property of representability. The above approaches are not well suited to this problem,so in this section we will adopt a diﬀerent strategy, based on tilting.
Classically, formal deformation functors are certain covariant functors CArt → Sets
from local commutative artinian C-algebras to sets. The idea behind noncommutative
deformation theory is that the source category should be enlarged to include noncommutative algebras. It is well known that contractible irreducible rational curves have only
infinitesimal commutative deformations, and that this can be detected by passing to a

8

WILL DONOVAN AND MICHAEL WEMYSS

suitable formal neighbourhood. In this section we will establish that this is also the casefor noncommutative deformations, by relating deformations on the 3-fold X to deformations over a certain tilting algebra associated to a formal neighbourhood of the curve C.
We prove this in a sequence of reduction steps, first to a Zariski local neighbourhood U
in 2.6, then to a C-algebra Λ in 2.8, then finally to the formal fibre in 2.19. From this,representability will be established in §3.
2.1. Algebras and functors. A key feature of the theory of noncommutative deformations, following [L02, E07, S08], is to consider simultaneous deformations of discrete
families of n objects, which may be parametrised by the n-pointed algebras defined below.
For the purposes of this paper, only the case n = 1 will be used, however we give thegeneral definition as it highlights features that are hidden in the n = 1 case.
Definition 2.1. An n-pointed C-algebra Γ is an associative C-algebra, together with
C-algebra morphisms p : Γ → Cn and i : Cn → Γ such that p ◦ i = Id. A morphismof n-pointed C-algebras ψ : (Γ, p, i) → (Γ′ , p′ , i′ ) is an algebra homomorphism ψ : Γ → Γ′such that
Γ p
i
Cn

Cn

ψ
i′

Γ′

p′

commutes. We denote the category of n-pointed C-algebras by Algn . We denote the fullsubcategory consisting of those objects that are commutative rings by CAlgn .
We write Artn for the full subcategory of Algn consisting of objects (Γ, p, i) for which
dim C Γ < ∞ and the augmentation ideal n := Ker(p) is nilpotent. We write CArtn for thefull subcategory of Artn consisting of those objects that are commutative rings.
Remark 2.2. In the literature, an n-pointed C-algebra is sometimes called an augmented
Cn -algebra. We also remark that when n = 1 the morphism i is the structure morphism, so
can be omitted. Furthermore, CArt1 is simply the category of commutative artinian local
C-algebras, with maximal ideal n. In our setting below, Art1 will be the source category
for our noncommutative deformation functors, since we will restrict to contractions ofcurves with 1 irreducible component. The category Artn is required for contractions ofcurves with n irreducible components [DW2].
Throughout we will only consider abelian categories A = Mod Λ or A = Qcoh Y
where Λ is a C-algebra and Y is a quasi-projective C-scheme, since considering a general
C-linear abelian category requires significantly more technology [LV, S08, ELO1, DW2].
With this restriction, it is possible to pick an object of A and study its noncommutative
deformation functor in a naive way suﬃcient for our purposes.
Definition 2.3. For A an abelian category as above, and an associative C-algebra Γ,
the category Pairs(A, Γ) has as objects pairs (b, φ) where b ∈ A and φ is a C-algebra
homomorphism φ : Γ → EndA (b). Morphisms in Pairs(A, Γ) are defined to intertwine
the action maps φ: explicitly, a morphism f : (b, φ) → (b′ , φ′ ) is given by a morphismf : b → b′ in A such thatbfb′φ′ (r)φ(r)bcommutes for all r ∈ Γ.

f

b′

9

NONCOMMUTATIVE DEFORMATIONS AND FLOPSDefinition 2.4. Let A be the abelian category A = Mod Λ or A = Qcoh Y where Λ is a
C-algebra and Y is a quasi-projective C-scheme. Pick a ∈ A, then the noncommutative
deformation functor is defined
DefaA : Art1 → Setsby sending⎧(⎫,( (b, φ) ∈ Pairs(A, Γ)⎨⎬((Γ, n) *→ ((b, φ), δ) (( − ⊗Γ b : mod Γ → A is exact∼⎩⎭∼( δ : (Γ/n) ⊗Γ b −→awhere ((b, φ), δ) ∼ ((b′ , φ′ ), δ ′ ) if there exists an isomorphism τ : (b, φ) → (b′ , φ′ ) in
Pairs(A, Γ) such that1⊗τ(Γ/n) ⊗Γ b(Γ/n) ⊗Γ b′δ′δacommutes. The commutative deformation functor is defined to be the restriction of theabove functor to the category CArt1 , and is denoted by
cDefaA : CArt1 → Sets.
Remark 2.5. When A = Mod Λ, the functor − ⊗Γ b in 2.4 is the standard tensor functor
of modules. When A = Qcoh Y , the functor − ⊗Γ b is defined in exactly the same
way as in the case of tilting equivalences, and can be thought of as tensoring over thenoncommutative sheaf of algebras OY ⊗C Γ, where C and Γ are the constant sheaves on
Y associated to C and Γ (see e.g. §6.4).
The fact that DefaA is a functor is routine, and we refer the reader to [L02, E07].
2.2. Contractions and deformation functors. Our setup is the contraction of anirreducible rational curve C (C red ∼= P1 ) in a quasi-projective normal 3-fold X with only
Cohen–Macaulay canonical singularities. We remark that throughout the paper, there
will be no extra assumptions on the singularities of X unless stated. By a contraction, wemean a projective birational morphism f : X → Xcon contracting C to a point p, satisfying
Rf∗ OX = OXcon , such that X\C ∼= Xcon \p. This incorporates both flips and flops.
We are interested in deformations of the sheaf OP1 (−1) viewed as a sheaf on X, so wechoose an aﬃne open neighbourhood Ucon ∼= Spec R around p. Putting U := f −1 (Ucon ),we have a commutative diagram
e

C

i

U
f |U

p

Ucon

f

Xcon

(2.A)

∼
=

=
m

∈

X

∈ Spec R

where C red ∼= P1 , e is a closed embedding, and i is an open embedding. Since Xcon
is separated and Ucon is aﬃne, the morphism Ucon %→ Xcon is aﬃne, and since aﬃne
morphisms are preserved under pullback it follows that i is an aﬃne morphism. This,
together with the fact that i is an open embedding, implies that there is a fully faithfulembedding of derived categories
Ri∗ = i∗ : D(Qcoh U ) %→ D(Qcoh X)(2.B)

with adjoints i! and i∗ .
Throughout we write E := e∗ OP1 (−1) ∈ coh U , so that i∗ E ∈ coh X is just thesheaf OP1 (−1) viewed as a sheaf on X. From 2.4, we thus have two deformation functors

10

WILL DONOVAN AND MICHAEL WEMYSSXDefEQcoh U and DefiQcoh, both of which take Art1 → Sets. For brevity, we denote them by∗E
UX
DefE and Defi∗ E respectively. The isomorphism of these deformation functors below is
not surprising: it says that noncommutative deformations can be detected Zariski locally.
Although elementary, we give the proof in full, since we will need variations of it later.
∼

Proposition 2.6. There is a natural isomorphism DefEU −→ DefiX∗ E induced by i∗ , with∼∗inverse induced by i . By restriction, there is also a natural isomorphism cDefEU −→X
cDefi∗ E , with a corresponding inverse.
φi∗Proof. Given ((F , φ), δ) ∈ DefEU (Γ), composing Γ −→ EndU (F ) −→EndX (i∗ F ) gives apair (i∗ F , i∗ φ) ∈ Pairs(Qcoh X, Γ). We check that this belongs to DefiX∗ E (Γ). Firstly, by
the projection formula and (2.B), the functor − ⊗Γ i∗ F : mod Γ → Qcoh X factorizes as

−⊗ F

i

∗
− ⊗Γ i∗ F : mod Γ −−−Γ−→ Qcoh U −→Qcoh X.

(2.C)

The first functor is exact by definition, and the second functor is exact by (2.B), so thecomposition − ⊗Γ i∗ F is exact. Next, we have thati∗ (δ)∼

Γ/n ⊗Γ i∗ F −→ i∗ (Γ/n ⊗Γ F ) −−−→ i∗ E
is an isomorphism, so denoting the first isomorphism by α, it follows that.
(i∗ F , i∗ φ), i∗ (δ) · α ∈ DefiX∗ E (Γ).

It is easy to check that this preserves the equivalence relation ∼ and is functorial, so givesa natural transformationi∗ : DefEU → DefiX∗ E .
i∗ψConversely, given ((G, ψ), γ) ∈ DefiX∗ E (Γ), composing Γ −→ EndX (G) −→ EndU (i∗ G)
gives a pair (i∗ G, i∗ ψ) ∈ Pairs(Qcoh U, Γ). The functor − ⊗Γ i∗ G : mod Γ → Qcoh U
factorizes as
i∗

−⊗ G

Γ
− ⊗Γ i∗ G : mod Γ −−−−
→ Qcoh X −→ Qcoh U,(2.D)

so as a composition of two exact functors it is exact. Further we have thati∗ (γ)∼∼

Γ/n ⊗Γ i∗ G −→ i∗ (Γ/n ⊗Γ G) −−−→ i∗ i∗ E −→ E
is an isomorphism, so denoting the first isomorphism by β, it follows that.
(i∗ F , i∗ φ), εE · i∗ (γ) · β ∈ DefiX∗ E (Γ)

where εE is the counit morphism. Again it is easy to check that this preserves theequivalence relation ∼ and is functorial, so gives a natural transformationi∗ : DefiX∗ E → DefEU .
To show that these natural transformations are isomorphisms, we need to show that.
((F , φ), δ) ∼ (i∗ i∗ F , i∗ i∗ φ), εE · i∗ (i∗ (δ) · α) · β.
((G, ψ), γ) ∼ (i∗ i∗ G, i∗ i∗ ψ), i∗ (εE · i∗ (γ) · β) · α .

We explain the second, the first being slightly easier. Since G is filtered by the sheaf i∗ E,
its support is contained in U and so the unit map ηG : G → i∗ i∗ G is an isomorphism. By
functoriality of the adjunction it is clear thatG

ηGi∗ i∗ Gi∗ i∗ ψ(r)

ψ(r)

G

ηG

i∗ i∗ G

11

NONCOMMUTATIVE DEFORMATIONS AND FLOPS

commutes for all r ∈ Γ, and so (G, ψ) ∼= (i∗ i∗ G, i∗ i∗ ψ) in Pairs(Qcoh X, Γ). Finally, we
claim that(Γ/n) ⊗Γ G

1⊗ηG(Γ/n) ⊗Γ i∗ i∗ G
(i∗ εE )·(i∗ i∗ (γ))·(i∗ β)·αγi∗ Ecommutes. We rewrite this diagram, adding in a unit morphism as the dotted arrow:1⊗ηG(Γ/n) ⊗Γ G(Γ/n) ⊗Γ i∗ i∗ Gαη(Γ/n⊗G)
γi∗ ((Γ/n) ⊗Γ i∗ G)i∗ βi∗ i∗ ((Γ/n) ⊗Γ G)i∗ i∗ (γ)
i∗ εE

i∗ E

i∗ i∗ i∗ E

The bottom half commutes by the functoriality of the unit, since i∗ εE = (ηi∗ E )−1 by thetriangular identity. It remains to show that the top half commutes. This just follows byinspection, since locally the maps send1 ⊗ i∗ i∗ a1⊗ai∗ (1 ⊗ i∗ a)i∗ i∗ (1 ⊗ a).

!

2.3. From geometry to algebra. In 2.6 we reduced the problem of deforming i∗ E in X
to deforming E in U . Now we further reduce the problem to deforming a simple module
over a certain noncommutative algebra, using a derived equivalence of Van den Bergh[V04]. With respect to our applications later, this step is crucial, since it yields a diﬀerent
description of the universal object, one which is homologically much easier to control.
We keep the setup as in (2.A), so we have a projective birational map f : U → Spec R
of 3-folds, with at most one-dimensional fibres, such that Rf∗ OU = OR , where R is a
Cohen–Macaulay C-algebra. Since the fibre is at most one-dimensional, in this setup it is
known [V04, 3.2.8] that there is a bundle V := OU ⊕ N inducing a derived equivalence
Db (coh U )RHomU (V,−)∼

Db (mod EndU (V))
(2.E)

0

Per U

∼

mod EndU (V)Here 0 Per U denotes the category of perverse sheaves, defined to be⎧(⎫(
H i (a) = 0 if i ̸= 0, −1⎨⎬(
0
Per U := a ∈ Db (coh U ) (( f∗ H −1 (a) = 0, R1 f∗ H 0 (a) = 0⎩( Hom(c, H −1 (a)) = 0 for all c ∈ C ⎭where

C := {c ∈ coh U | Rf∗ c = 0}.

12WILL DONOVAN AND MICHAEL WEMYSS

By birationality [V04, 4.2.1], EndU (V) ∼= EndR (f∗ V), hence we write N := f∗ N andthroughout this section we setΛ := EndU (V) = EndU (OU ⊕ N ) ∼= EndR (R ⊕ N ).
0

Since it is clear that E = e∗ OP1 (−1), viewed as a complex in degree zero, belongs to
Per U , sending it across the derived equivalence (2.E) yields a module in degree zero.

Definition 2.7. We write T := HomU (V, E) ∼= RHomU (V, E).
By 2.4, there is a deformation functor DefTMod Λ : Art1 → Sets, which to ease notation
we denote by DefTΛ . The following is an analogue of 2.6, and is proved in a very similarway.
Proposition 2.8. There is a natural isomorphism DefEU ∼= DefTΛ induced by HomU (V, −)and − ⊗Λ V, and consequently by restriction a natural isomorphism cDefEU ∼= cDefTΛ .
Proof. The functors HomU (V, −) : Qcoh U → Mod Λ and − ⊗Λ V : Mod Λ → Qcoh U are
adjoint. Since T is a module corresponding under the derived equivalence to a sheaf E,
we know that T ∼= HomU (V, T ⊗Λ V) and E ∼= HomU (V, E) ⊗Λ V via the unit and counit
morphisms.
The proof is now identical to 2.6, where in the analogue of (2.C) and (2.D), the second
functors are no longer exact, but they are exact out of the image of the first functor (using
RHomU (V, E) = HomU (V, E) and T ⊗L
Λ V = T ⊗Λ V respectively), which is suﬃcient for
the remainder of the proof.
!
Definition 2.9. We define [R] to be the two-sided ideal of Λ = EndR (R ⊕ N ) consistingof those morphisms R ⊕ N → R ⊕ N which factor
R⊕N

R⊕N
P

through an object P ∈ add R. We set Icon := [R] and define the contraction algebra
associated to Λ to be Λcon := Λ/Icon .
Remark 2.10. The contraction algebra is the fundamental object in our paper, but weremark that the algebra Λcon defined above in 2.9 depends on Λ, which in turn depends
on the choice of derived equivalence in (2.E); diﬀerent equivalences may be obtained dueto the lack of Krull–Schmidt decompositions of tilting sheaves, see [V04, 3.2.7]. In 2.12
we will define a contraction algebra associated to C which is intrinsic to the geometryof X, and does not involve choices, as shown in 3.9. It will turn out in 5.4 that Λconwill be morita equivalent, though not necessarily isomorphic, to the contraction algebra
associated to C.
2.4. Complete local geometric setting. Having in 2.6 and 2.8 reduced the problem
of deforming i∗ E in X to the purely algebraic problem of deforming T in mod Λ, we
next search for a candidate representing object. This requires passing through a morita
equivalence; indeed we would like to say that Λcon belongs to Art1 , since then it is anatural candidate for the representing object. However, the algebras in Art1 have only
one simple module, and it is one-dimensional: this is not in general true for Λcon for thereasons in 2.10. Thus to obtain a candidate representing object in Art1 , we need to passto the basic algebra associated to Λcon . The purpose of this subsection is to obtain thisfrom the formal fibre, and construct the contraction algebra Acon . We will show that Acon
is indeed the representing object in §3.
For X ∈ Mod Λ and n ∈ Max R, we write Xn := X ⊗R Rn , and for Y ∈ Mod Λ we/n . We require a better description of Λ.
# Completing the baseoften write Y# := Yn ⊗Rn R

NONCOMMUTATIVE DEFORMATIONS AND FLOPS

13

with respect to m in (2.A), since endomorphism rings of finitely generated modules behave
well under completion [E95, 2.10], it is standard that
# := Λ ⊗R R
# = EndR (R ⊕ N ) ⊗R R
#∼
#⊕N
# ).
Λ
= End % (RR

# and the formal fibre f : U# → Spec R,# it is also
On the other hand, considering Spec R#
standard that the tilting bundle V = OU ⊕ N on U restricts to a tilting bundle OU% ⊕ N#
on U . Then by birationality [V04, 3.2.10]∼ End % (f∗ (O % ⊕ N
#) =
# )),
End % (O % ⊕ N
U

U

R

U

#⊕N
# ). Hence passing to the formal fibre induces a derived
and this is clearly just EndR% (Requivalence
%
RHomU
% (OU
% ⊕N ,−)#)Db (coh U#
Db (mod Λ).

∼# , constructed in [V04], whose associated
There is a more explicit tiling bundle on U
# For
tilting algebra, defined in 2.12 below, is the basic algebra morita equivalent to Λ.
−1#
this, let C = π (m) where m is the unique closed point of Spec R, then giving C the# suchreduced scheme structure, we have C red ∼= P1 . Let L denote the line bundle on U∗
that L · C = 1. If the multiplicity of C is equal to one, set M := L and N1 := M [V04,
3.5.4], else define M to be given by the maximal extension⊕(r−1)

0 → OU%→M→L→0# , L∗ ) and set N1 := M∗ . Then
associated to a minimal set of r − 1 generators of H 1 (U
# [V04, 3.5.5]. Again by birationality
OU% ⊕ N1 is a tilting bundle on U
End % (O % ⊕ N1 ) ∼= End % (f∗ (O % ⊕ N1 )),
U

U

R

U

# ⊕ N1 where N1 := f∗ N1 . We remark that rank % N1 isand we have f∗ (OU% ⊕ N1 ) = R
Requal to the scheme-theoretic multiplicity of the curve C [V04, 3.5.4].
#⊕N#∼#⊕a0 ⊕ N ⊕a1 for some a0 , a1 ∈ N.
Lemma 2.11. We can write R=R1# ∈ 0 Per U# is aProof. Since OU ⊕ N ∈ 0 Per U is a progenerator [V04, 3.2.7], OU% ⊕ Nprogenerator, so by [V04, 3.5.5], there exist positive integers such that#∼
O% ⊕ N= O⊕a0 ⊕ N ⊕a1 .
U

1

%
U

#⊕N
#∼
# ⊕a0 ⊕ N ⊕a1 .
Pushing down, it follows that R
=R
1

!

#
To ease notation with hats, we put R := R.

Definition 2.12. We define A := EndR (R ⊕ N1 ), the basic algebra morita equivalent to# and define the contraction algebra associated to C to be Acon := EndR (R ⊕ N1 )/[R],Λ,
where [R] is defined similarly as in 2.9.

We show later in 3.9 that the contraction algebra Acon is intrinsic to the geometryof X, and in particular does not depend on the choice of derived equivalence in (2.E).
It is not hard to see, and we will prove it later in 5.4, that Acon is morita equivalent to# con . However, the morita equivalence between A and Λ# is easy to describe, and we doΛthis now. We write
Y := R ⊕ N1 ,Z := R⊕a0 ⊕ N1⊕a1 ,# = EndR (Z). Then we put
so that A = EndR (Y ) and Λ
P := HomR (Y, Z),

Q := HomR (Z, Y ).

These have the structure of bimodules, namely Λ% PA and A QΛ% . It is clear that P is aprogenerator, and we have a morita context# P, Q).
(A, Λ,

14

WILL DONOVAN AND MICHAEL WEMYSS

In particular this implies that
F:=HomA (P,−)=−⊗A Q

mod A

HomΛ
% (Q,−)=−⊗Λ
%P

#
mod Λ

are equivalences.
We now show that, in our situation of flips and flops, the algebras Λcon and Acon are
finite dimensional. This will be used later to obtain numerical invariants associated to thecurve C.
Proposition 2.13. With the setup as in §2.2, and with notation as above,(1) Λcon and Acon are finite dimensional algebras, which as R-modules are supported
only at m.
(2) As an R-module, FAcon is supported only at m.
(3) T is a simple Λ-module, which as an R-module is supported only at m.
Proof. (1) The contraction U → Spec R is an isomorphism away from a single pointm ∈ Max R. By base change, it is clear that add Np = add Rp for all p ̸= m, so Λcon p = 0
for all p ̸= m. It follows that Λcon is supported only on m, and so in particular is finite# → Spec R# is andimensional. The proof for Acon is identical, using the fact that U
isomorphism away from the closed point.
(2) follows immediately from (1).
# is [V04, 3.5.8]. Thus T# is a simple Λ-module,#(3) The fact that E is simple in 0 Per U
since
it corresponds to a simple across a morita equivalence. Now let e denote the idempotent
in Λ corresponding to R, thenDb (coh U )Rf∗

Db (coh Spec R)

RHomU (O⊕M,−)∼

Db (mod Λ)e(−)

Db (mod R)

commutes. Since Rf∗ E = 0, it follows that eT = 0, and so T is a finitely generated
Λcon -module. In particular, by (1) T is supported only at m, so T ∼= T#. It follows that T
is a simple Λ-module.
!
Definition 2.14. We denote by S the simple A-module corresponding across the morita
equivalence to T#.

Note that since A is basic, dimC S = 1. It is also clear that S can be viewed as asimple Acon -module, and it is the unique simple Acon -module. Under the running flips
and flops setting, Acon is further finite dimensional by 2.13(1). Thus Acon ∈ Art1 , making
it the candidate for the representing object.

2.5. Reduction to complete local setting. The proof that Acon represents the deformation functors in 2.6 and 2.8 requires one more reduction step, namely we must relatethem to a similar deformation functor on the formal fibre. This step is largely routine,
although in this subsection we do introduce notation and known results that will be usedlater.
The following is well known, and will be used throughout.
Lemma 2.15. Let X ∈ Db (mod Λ), Y ∈ D(Mod Λ), then
(1) RHomΛ (X, Y ) ⊗R Rm ∼= RHomΛm (Xm , Ym ).
#m ∼# Y# ).
(2) RHomΛ (X, Y )m ⊗Rm R= RHomΛ% (X,00#(3) If X ∈ mod Λ with dimC X < ∞, then X ∼= m∈SuppR X X= m∈SuppR X Xm ∼as Λ-modules.

NONCOMMUTATIVE DEFORMATIONS AND FLOPS

15

Proof. (1) and (2) are [IR08, p1100].
0(3) The natural map ψ : X →m∈SuppR X Xm is clearly a Λ-module homomorphism.
Viewing ψ as an R-module homomorphism, since X has finite length, ψ is bijective by[E95, 2.13b]. It follows that ψ is a Λ-module isomorphism. The last isomorphism follows# as Λm -modules.
since if Xm has finite length, then Xm ∼
!
=X

∼
# =
Now Λ = EndR (R ⊕ N ) and Λcon = Λ/[R]. Completing with respect to m, Λ###
EndR% (R ⊕ N ) and we already know from §2.4 that N may decompose into many moresummands than N does. However, the following states that the localization (resp. completion) of the contraction algebra associated to Λ is the contraction algebra associated# This will allow us to reduce many of ourto the localization Λm (resp. completion Λ).
problems to the formal fibre.
Lemma 2.16. With notation as above,(1) [R] ⊗R Rm ∼= [Rm ]. We denote this by Im con .
## We denote this by I#con .
(2) [Rm ] ⊗R R ∼= [R].
∼#∼# R].
#(3) Λcon ⊗R Rm = Λm /[Rm ] and Λcon ⊗R R
= Λ/[

Proof. Since localization and completion are exact, (3) follows from the first two. We
prove (2), with (1) being similar. To ease notation, we temporarily write M := R ⊕ N .
# ⊆ [R],# so let ψ ∈ [R].
# Then ψ : M/→M/ factors through
It is clear that [Rm ] ⊗R R
# so by definition ψ factors through a summand of R
#a for some a ∈ N. This implies
add R,#a , so there is a commutative diagramthat ψ factors through R
ψ

/
M

α

β

#a
R

/
M

a
#∼
/, R
# a ) etc, so we can find Λm -module
As in 2.15(2) HomΛm (Mm , Rm
) ⊗ Rm R= HomΛ% (Mhomomorphisms
f

Mm
g

a
Rm

Mm
h

(2.F)

such that f# = ψ, #g = α and #h = β. It remains to show that (2.F) commutes, since thenf ∈ [Rm ]. But since the completion is exact#∼
Im(f − hg) ⊗ R− hg = 0,= Im f!

so the completion of Im(f −hg) is zero, which implies that Im(f −hg) = 0, i.e. f = hg. !
# con for theBelow, we write Λmcon for the contraction algebra associated to Λm , and Λ!
∼##
contraction algebra associated to Λ. By the above, Λm con = (Λcon )m and Λcon ∼= (Λcon ),
so there is no ambiguity.
# con both as Λ-modules and as algebras.
Lemma 2.17. Λcon ∼=ΛProof. Since Λcon is supported only at m by 2.13, it follows from 2.16 and 2.15(3) that the# con is bijective. Since it is both a Λ-module homomorphism and an
natural map Λcon → Λ# con both as Λ-modules, and as algebras.
algebra homomorphism, we have Λcon ∼!
=Λ

Warning 2.18. Often when studying the birational geometry of global 3-folds, we areforced to flip or flop together several irreducible curves that don’t intersect. When this0
# con , instead Λcon ∼# con .
occurs, Λcon " Λ= m∈Supp Λcon Λ

16

WILL DONOVAN AND MICHAEL WEMYSS

Hence we have the following functors⊗ R Rmb

D (Mod Λ)
T
Λcon
FAcon

res

%
⊗R m Rb

D (Mod Λm )Tm(Λm )con
FAcon

res

#
Db (Mod Λ)
T#
#
Λcon
FAcon

F

Db (Mod A)
S
Acon

where res denotes restriction of scalars along the natural ring homomorphisms. The linefor Λcon follows by 2.16. Note that the first two functors are not equivalences, however,since T , Λcon and Acon are supported only at m we have T ∼= res(Tm ) etc, so the arrows in
the bottom three lines in the above diagram can also be drawn in the opposite direction.
Given this, the following is not surprising, and is a standard application of the proofs
and techniques developed so far.
Proposition 2.19. There are natural isomorphisms% ∼A
DefTΛ ∼
= DefTΛmm ∼= DefTΛ% = DefS ,and consequently by restriction natural isomorphisms%

A∼
cDefTΛ ∼= cDefTΛmm ∼= cDefTΛ
% = cDefS .

Proof. The functors −⊗R Rm : Mod Λ → Mod Λm and res : Mod Λm → Mod Λ are adjoint,
and are both exact. Since, by 2.13, T is a finite length module supported only at m,
T ∼= res(T ⊗R Rm ) and Tm ∼= res(Tm ) ⊗R Rm via the unit and counit morphisms.
The proof is now identical to 2.6, where in the analogue of (2.C) and (2.D), thesecond functors are replaced by − ⊗R Rm and res, which are still exact. This establishes
the first isomorphism.
For the second, again restriction and extension of scalars give an exact adjoint pair,and since Tm is a finite length module, the necessary unit and counit maps are isomorphisms, since restriction and extension of scalars are an equivalence on finite length modules. The proof is then identical to the above, establishing the second isomorphism. The
third isomorphism also follows in an identical manner.
!
Thus, in conclusion, combining 2.6, 2.8 and 2.19 gives a natural isomorphism
DefiX∗ E ∼= DefSA : Art1 → Sets,(2.G)

and so to prove representability, we have reduced the problem to showing that DefSA isrepresentable. Since A is basic (unlike Λ), by 2.13 Acon ∈ Art1 and so this is the natural
candidate for the representing object.
3. Representability and Contraction Algebras
We keep the setup in §2.2. In the previous section we established natural isomorphisms of functors
DefiX∗ E ∼= DefEU ∼= DefSA : Art1 → Sets

(3.A)

and showed that Acon ∈ Art1 . In this section we will show that Acon is a representing object
for these functors, and describe the associated universal families. This then allows us todefine numerical invariants associated to the curve C, and show that Toda’s commutative
deformation functor is not an equivalence in general.

NONCOMMUTATIVE DEFORMATIONS AND FLOPS

17

3.1. Representability on the basic algebra A. In this subsection we show that DefSA
and cDefSA are representable. This statement turns out to be an elementary consequenceof the definitions, and is implicit in [E03, E07, S08], albeit in a slightly modified setting.
Proposition 3.1. DefSA ∼= HomArt1 (Acon , −) : Art1 → Sets.
Proof. Since S = C is an A-module, this induces a homomorphism A → C which wedenote by q. Since objects in Pairs(Mod A, Γ) are bimodules Γ MA ,($12
( − ⊗Γ M : mod Γ → mod A is exact
DefSA (Γ) = (Γ MA , δ) ((∼∼δ : (Γ/n) ⊗Γ M −→ S as A-modules⎫,⎧⎨ • A right A-module structure on Γ ⊗C S such that ⎬Γ ⊗C S becomes a Γ-A bimodule.
∼
=
⎭
⎩
∼
• δ : (Γ/n) ⊗Γ (Γ ⊗C S) −→ S as A-modules$
12• A C-algebra homomorphism A → Γ such that=∼the composition A → Γ → Γ/n = C is q
which is just HomAlg1 (Acon , Γ) since q factors through Acon . The result then follows sinceAcon ∈ Art1 which is by definition a full subcategory of Alg1 .
!
Corollary 3.2. cDefSA ∼= HomCArt1 (Aabcon , −) : CArt1 → Sets.
Proof. Since any homomorphism from Acon to a commutative ring Γ must kill the twosided ideal in Acon generated by the commutators, the result follows since
3.1
ab
cDefSA (Γ) = HomAlg1 (Acon , Γ) ∼= HomAlg1 (Aabcon , Γ) = HomCArt1 (Acon , Γ).

!

3.2. Chasing through the representing couple. It follows from 3.1 that all the functors in (3.A) are representable. In this subsection we chase the representing couples
through the relevant isomorphisms, as we need the universal sheaves in geometric terms
for our applications later.
If F : Art1 → Sets is a deformation functor, recall that (Γ, ξ) is a couple for F if
Γ ∈ Art1 and ξ ∈ F (Γ). A couple (Γ, ξ) induces a natural transformationαξ : HomArt1 (Γ, −) → F
which when applied to Γ2 ∈ Art1 simply takes φ ∈ HomArt1 (Γ, Γ2 ) to the element F (φ)(ξ)
of F (Γ2 ). We say that a couple (Γ, ξ) represents F if αξ is a natural isomorphism offunctors on Art1 . If a couple (Γ, ξ) represents F , then it is unique up to unique isomorphismof couples.
∼
Now given the natural isomorphism β : HomArt1 (Acon , −) → DefSA in 3.1, the couple
A(Acon , β(IdAcon )) = (Acon , Acon (Acon )A ) represents DefS . Simply tracking the element
A
Acon (Acon )A ∈ DefS (Acon ) through the isomorphisms
F

% res

−⊗ V

i

∗
DefSA −
→ DefTΛ−→ DefTΛ −−−Λ−→ DefEU −→DefiX∗ E% −we obtain the following:
Corollary 3.3. In the setup of §2.2,(1) The couple (Acon , FAcon ⊗Λ V) represents DefEU .
(2) The couple (Acon , i∗ (FAcon ⊗Λ V)) represents DefiX∗ E .
We now fix notation.
Definition 3.4. We define EU := FAcon ⊗Λ V ∈ coh U , and E := i∗ (EU ).
Thus (Acon , E) is the representing couple for DefiX∗ E . Similarly to the above, we candeduce from 3.2 that all the commutative deformation functors have representing couples,with the left-hand term Aabcon . Chasing through, it is obvious that the representing couple
for cDefEU is given by a coherent sheaf. We again fix notation.

18

WILL DONOVAN AND MICHAEL WEMYSSDefinition 3.5. We write FU ∈ coh U for the coherent sheaf such that (Aabcon , FU ) repreXsents cDefEU , and write F := i∗ FU so that (Aab,F)
represents
cDef
.
con
i∗ E
Since i∗ need not preserve coherence, we must prove that E and F are coherent.
Lemma 3.6. With assumptions and notation as above,(1) FU and EU are filtered by the sheaf E.
(2) Ri∗ EU = i∗ EU = E and Ri∗ FU = i∗ FU = F .
(3) F and E are filtered by the sheaf i∗ E, hence are coherent.
Proof. Denote the radical of Acon by J.
(1) Since EU ∈ DefEU (Acon ), the functor − ⊗Acon EU : mod Acon → Qcoh U is exact, and(Acon /J) ⊗Acon EU ∼= E. Since Acon is a finite dimensional algebra, filtering Acon by its
simple module (Acon /J) and applying − ⊗Acon EU yields the result. The case of FU is
identical.
!
(2) follows from (2.B), and (3) follows by combining (1) and (2).
In a similar vein, using the fact HomU (V, E) = RHomU (V, E) in 2.7 together with afiltration argument gives the following, which will be used later.
Lemma 3.7. With assumptions and notation as above,∼(1) RHomU (V, EU ) ∼= FAcon and FAcon ⊗LΛ V = EU .
ab
ab
L
∼
(2) RHomU (V, FU ) = FAcon and FAcon ⊗Λ V ∼= FU .
3.3. The contraction algebra of C and wid(C).
Definition 3.8. With the setup and assumptions as in §2.2, suppose that X → Xcon
is the contraction of an irreducible rational curve C. We define the noncommutative
deformation algebra of C to be EndX (E).
The noncommutative deformation algebra is the fundamental object in our paper.
The above definition is intrinsic to the geometry of X, since E arises from the representing
couple of the noncommutative deformations of the sheaf OP1 (−1) in X. However, it ishard to calculate in the above form. It is the following description of EndX (E) as the# ⊕ N1 )/[R]# that both makes it possible to calculate
contraction algebra Acon = EndR% (REndX (E), and also to control it homologically.
Lemma 3.9. There are isomorphisms∼ EndU (EU ) =∼ EndΛ (FAcon ) ∼EndX (E) == End % (FAcon ) ∼= EndA (Acon ) ∼= Acon .
ΛFurthermore, EndX (F ) ∼= EndX (E)ab .
Proof. The first isomorphism uses the fully faithful embedding of derived categories Ri∗ ,together with the fact that Ri∗ EU ∼= E by (2.B). The second isomorphism follows fromthe equivalence (2.E) together with 3.7. The third isomorphism follows since FAcon is
supported only at m (by 2.13), and the fact that restriction and extension of scalars arean equivalence on these finite length subcategories. The fourth isomorphism is just chasing
across a morita equivalence, and the last is simply
EndA (Acon ) = EndA (A/AeA) ∼= A/AeA = Acon#
where e is the idempotent in A corresponding to R.
∼
For the final statement that EndX (F ) = EndX (E)ab , observe that
3.7

3.6

ab
ab
ab
∼
∼
∼
∼
∼
Aab
con = HomA (A, Acon ) = HomA (Acon , Acon ) = EndA (Acon ) = EndU (FU ) = EndX (F )and chasing through this establishes that Aab ∼!
= EndX (F ) as rings.
conThus throughout the remainder of the paper, we use the terms ‘contraction algebra’and ‘noncommutative deformation algebra’ interchangeably.

19

NONCOMMUTATIVE DEFORMATIONS AND FLOPS

Example 3.10. (Pagoda [R83]) Consider
C[[u, v, x, y]]uv = (x − y n )(x + y n )for some n ≥ 1. It is well known that N1 ∼= (u, x + y n ).
# ⊕ N1 ) as the completion of the quiver with relations
We can present A := End % (R
# :=
R

R

x+y
u

y

%
R

n

inc

N1

a1
a2

y

y1

y2

b1
b2

x−yn
u

y 1 a1 = a1 y 2
y 1 a2 = a2 y 2
y2 b1 = b1 y1
y2 b2 = b2 y1
2y1n = a1 b1 − a2 b2n2y2 = b1 a1 − b2 a2 .

# ⊕ N1 )/[R]# we factor by all arrows that factor through the
To obtain Acon = EndR% (R# and so Acon ∼vertex R,= C[[y2 ]]/2y2n ∼= C[y]/y n .

Definition 3.11. Let X → Xcon be a contraction of an irreducible rational curve C andlet E ∈ coh X be as defined in 3.4. We define(1) the width of C to be wid(C) := dimC EndX (E).
(2) the commutative width of C to be cwid(C) := dimC EndX (E)ab .

We remark that the width of a (−3, 1)-curve is not defined [R83], and so in this case(as well as in the singular setting, and the flipping setting) the invariants in 3.11 are new.
In contrast to determining an explicit presentation for EndX (E), which sometimes can be
hard, to calculate wid(C) is much simpler, and can be achieved by using just commutative# We refer the reader to 5.2 later.
algebra on the base singularity R.
Example 3.12. (Pagoda) In 3.10, Acon ∼= C[y]/y n which is commutative, sowid(C) = cwid(C) = n.
Example 3.13. (Francia flip) Consider the ring R := C[[a, b, c1 , c2 , d]]/I where I isgenerated by the 2 × 2 minors of the following matrix:
3
4
a b c1b c2 dThere is a flip
X nef

X

g

f

Spec R
contracting a curve C in X and producing a nef curve C nef in X nef . The scheme X is
obtained by blowing up the ideal S1 := (c2 , d2 ) whereas X nef is obtained by blowing up
the ideal S2 := (c1 , d). In fact X is derived equivalent to A := EndR (R ⊕ S1 ), which canbe presented as
c2

c1

c1

d2

A∼
=

R
d

c2
a
1
c2 = d 2c1bc2 = dS1
dinc

At the vertex S1 , the loops c1 and d commute since they belong to EndR (S1 ) ∼= R, whichis commutative. Also, at the vertex S1 , the path c21 factors through the vertex R as thec2arrow d12 followed by the arrow d2 . Similarly c1 d and d2 factor through the vertex R, so
C[[c1 , d]] ∼ C[x, y]
Acon ∼.
= 2= 2
c1 , c1 d, d2
x , xy, y 2

20WILL DONOVAN AND MICHAEL WEMYSS

In particular Acon is commutative, thuswid(C) = cwid(C) = 3.
On the other hand, X nef is derived equivalent to B := EndR (R ⊕ S2 ) which can bepresented as
c1
d

B∼
=

R

a
b
c1 = dc2bc1 = dS2

inc

Thus we see Bcon = C, which is commutative, and wid(C nef ) = cwid(C nef ) = 1.
Example 3.14. (The Laufer D4 flop) We consider the more general case than in the
introduction, namely the family of D4 flops given by
Rn :=C[[u, v, x, y]]u2 + v 2 y = x(x2 + y 2n+1 )where n ≥ 1. In this case A := EndR (R ⊕ N1 ) where N1 is a rank 2 Cohen–Macaulay
module, and by [AM], A can be written abstractly as the completion of the quiver withrelationsxax

Ray 2 = −xa
y 2 b = −bx
ab = −xnxy = −yxx2 + yba + bay = (−1)n+1 y 2n+1 .

N1

b

y

Thus we see immediately that
Acon ∼=

C⟨⟨x, y⟩⟩C⟨⟨x, y⟩⟩
∼
.
=
2
n+1
2n+1
xy = −yx, x = (−1)
yxy = −yx, x2 = y 2n+1

The contraction algebra Acon is finite dimensional by 2.13. To see this explicitly, postand premultiplying the second equation by x we obtain x3 = y 2n+1 x = xy 2n+1 . But using
the first equation repeatedly gives xy 2n+1 = −y 2n+1 x, thus x3 = 0. Consequently thealgebra Acon is spanned by the monomials
1
x
x2

y
xy
x2 y

y2
xy 2
x2 y 2...
y 2n. . . xy 2n. . . x2 y 2n

In fact this is a basis (which for example can be checked using the Diamond Lemma) andso wid(C) = dimC Acon = 3(2n + 1). Furthermore
∼
Aab
con =

C[[x, y]]C[x, y]∼=xy = 0, x2 = y 2n+1
xy = 0, x2 = y 2n+1

and so cwid(C) = dimC Aabcon = 2n + 3.
3.4. Commutativity of Acon and flops. When C is an irreducible rational flopping
curve and U is smooth, necessarily R is Gorenstein. Below in 3.15 we show that in thisflopping setup, wid(C) = cwid(C) if and only if C has normal bundle (−1, −1) or (−2, 0),and furthermore in this case this invariant recovers Reid’s notion of width from [R83].
Theorem 3.15. Let U → Spec R be the flopping contraction of an irreducible rational
curve C, where U is smooth. Then EndU (EU ) is commutative if and only if C is a (−1, −1)or (−2, 0)-curve.

NONCOMMUTATIVE DEFORMATIONS AND FLOPS

21

Proof. Since EndU (EU ) ∼= Acon by 3.9, we assume throughout that R is complete local.
We check commutativity on Acon .
(⇐) Suppose that C is not a (−3, 1)-curve, then it is well known thatR∼=

C[[u, v, x, y]]uv = (x − y n )(x + y n )[R83] for some n ≥ 1, and that A is isomorphic to the completion of the quiver withrelations in 3.10. As shown in 3.10, Acon ∼= C[y]/y n , which is evidently commutative andhas dimension n, which is the same as Reid’s width.
(⇒) By contraposition. Suppose that C is a (−3, 1)-curve, then since U is smooth,dim C Ext1U (E, E) = dim C Ext2U (E, E) = 2
by CY duality. Hence by the standard deformation theory argument [L02, 1.1]
C⟨⟨x, y⟩⟩Acon ∼=(f1 , f2 )where (f1 , f2 ) is the closure of the ideal generated by f1 and f2 , and further when we
write both f1 and f2 as a sum of words, each word has degree two or higher. We know
Acon is finite dimensional by 2.13, hence it follows from [S13, 1.2] that Acon ∼= EndU (EU )cannot be commutative.
!
The above is somewhat remarkable, since it does not use the classification of flops,
and in fact it turns out later (3.19) that the above is the key step in showing that Toda’scommutative deformation twist functor is not an equivalence. However, below we give asecond proof of (⇒) in 3.15 that does use the classification of flops, since this proof gives
us extra information regarding lower bounds of the possible widths of curves. We do not
use this second proof (and thus the classification) anywhere else in this paper.
Proof. As in the previous proof, we can assume that R is complete local. Since R is acompound Du Val singularity and C is a (−3, 1)-curve, a generic hyperplane section g ∈ R
gives a Du Val surface singularity R/gR of type D4 , E6 , E7 or E8 . In fact, taking thepullback
X
U
Spec(R/gR)

Spec Rthen X is a partial resolution of the Du Val singularity where there is only one curve
above the origin, and that curve must correspond to a marked curve in one of the following
diagrams

D4

E6

E7

(3.B)

E8 (5)

E8 (6)[KaM, K94]. Now set Y := R ⊕ N1 as in §2.4, so A = EndR (Y ), and let e be theidempotent corresponding to R, so Acon = A/AeA. We denote the category of maximal
Cohen–Macaulay R-modules by CM R. Since flopping contractions are crepant, A ∈ CM R
[V04, 3.2.10], so since R is an isolated singularity, by the depth lemma Ext1R (Y, Y ) = 0.
Also, since Y is a generator, necessarily Y ∈ CM R.
Thus applying HomR (Y, −) tog0→Y −→ Y → Y /gY → 0

22

WILL DONOVAN AND MICHAEL WEMYSS

gives
g·

0 → HomR (Y, Y ) −→ HomR (Y, Y ) → HomR (Y, Y /gY ) → 0,
which implies that
A/gA ∼= HomR (Y, Y /gY ) ∼= EndR/gR (Y /gY )where the last isomorphisms arises from the extension–restriction of scalars adjunction
for the ring homomorphism R → R/gR. Since EndR/gR (Y /gY ) ∈ CM R/gR with
rankR N1 = rankR/gR (N1 /gN1 ), in the McKay correspondence N1 /gN1 is the CM module
corresponding to one of the marked curves in (3.B).
Now let e′ be the idempotent in ∆ := EndR/gR (Y /gY ) corresponding to R/gR, so∆con = ∆/∆e′ ∆. We claim that ∆con is a factor of Acon . This follows since setting
I = AeA we have
A
A/gA
A/I
Acon ∼
∼
∼
(3.C)
=
=
=
gAcon
(gA + I)/I
gA + I(gA + I)/gA
and it is easy to show that as a ∆-ideal, (gA + I)/gA is equal to ∆e′ ∆. Hence (3.C) is
isomorphic to ∆con , and so indeed ∆con is a factor of Acon .
It follows that if we can show ∆con is not commutative, then Acon is not commutative.
But ∆con ∼= HomR/gR (N1 /gN1 , N1 /gN1 ), and it is well known that we can calculate thisusing the AR quiver of CM R/gR (see e.g. [IW08, Thm 4.5, Example 4.6]). Below in 3.16
we show by a case-by-case analysis of the five possible N1 appearing in (3.B) that each∆con is not commutative, and so each Acon is not commutative.
!
Lemma 3.16. With notation as in the proof of 3.15, the ∆con corresponding to partial
resolutions of the Du Val surfaces involving only the marked curves in (3.B) are nevercommutative.
Proof. We give details for the E7 flop, the remaining cases being very similar. We wish to
calculate HomR/gR (N1 /gN1 , N1 /gN1 ), which is graded by path length in the AR quiverof CM R/gR. To calculate the dimension of each graded piece, we begin by placing a 1(corresponding to the identity) in the place of N1 /gN1 , and proceed by knitting:
0

1
1
1
2
1
1
1
0
1
2
2
3
3
2
2
1
0
1 1 2 1 3 2 4 2 4 2 4 2 3 1 2 1 1 0
1
2
3
3
3
3
2
1
0
0
1
2
2
2
2
2
1
0
0
1
1
1
1
1
1
0
0
For details, see for example [IW08, §4]. The above shows that the degree zero morphism
set is one-dimensional (spanned by the identity), the degree two morphism set is twodimensional, the degree four morphism set is three-dimensional, etc. Summing up, we seethat dimC ∆con = 1 + 2 + 3 + 4 + 4 + 4 + 3 + 2 + 1 = 24. Further, if we set
0
1
0

1
1 x
1y 2
1
1
0

2
1
2
1

then both x and y have degree two, and by the mesh relations y 2 = 0. Since the degree
four morphism set is three-dimensional, necessarily xy ̸= yx.
!
Remark 3.17. Although we do not know explicitly all the contraction algebras EndX (E) ∼=Acon for all flopping contractions (except for some of type A and D4 ), a more detailed

NONCOMMUTATIVE DEFORMATIONS AND FLOPS

23

analysis of the proof of 3.16 gives a precise value for dimC (∆con ), and thus gives a lower
bound for the possible wid(C):Dynkin typewid(C)AD4
E6E7
E8 (5)
E8 (6)

≥1
≥4
≥ 12
≥ 24
≥ 40
≥ 60

3.5. On Toda’s commutative deformation functor. In this subsection we show thatToda’s functor defined in [T07] using commutative deformations is never an equivalence for
floppable (−3, 1)-curves. In fact, we work much more generally, without any assumptions
on the singularities of X, then specialize down to establish the result. We first establish
the results locally on U , then lift to X.
Theorem 3.18. Let U → Spec R be a contraction of an irreducible rational curve C.
Then(1) If EndU (EU ) is not commutative, then Ext1U (FU , E) ̸= 0.
(2) If U is smooth and C is a flopping (−3, 1)-curve, then Toda’s functor does notgive an autoequivalence of Db (coh U ).
Proof. (1) Since the Ext group is supported only on m,
Ext1U (FU , E) ∼= Ext1A (Aabcon , S).
# ⊕ N1 ) as the (completion of) a quiver with relations, where
We can present A = EndR% (Rthere are two vertices 0 and 1, corresponding to projectives as follows:# ⊕ N1 , R),#
P0 := HomR% (R# ⊕ N1 , N1 ).
P1 := HomR% (RWe remark that P1 is the projective cover of S, and that S is the vertex simple at 1.
We know that EndU (EU ) ∼= Acon by 3.9. Since Acon is not commutative, Acon ̸= Aabcon .
Hence in the minimal projective resolution of AabthekernelKofthenaturalmap
con
P1 → Aabcon contains some non-zero element x which is a composition of cycles at vertex 1,such that x is zero in Aabcon but not in Acon . Since this element x does not factor throughother vertices, there cannot be a surjective map P0a → K, as we need some P1 → K inorder to surject onto the element x. Consequently the projective cover of K must contain
P1 , and hence Ext1A (Aabcon , S) ̸= 0.
(2) By 3.15, EndU (EU ) is not commutative, so this follows by (1) [AL10, Theorem 3.1]. !
Corollary 3.19. Suppose that X → Xcon is a contraction of an irreducible rational curveC.
(1) If EndX (E) is not commutative, then Ext1X (F , i∗ E) ̸= 0.
(2) If X is smooth and C is a flopping (−3, 1)-curve, Toda’s functor does not give an
autoequivalence of Db (coh X).
∼ EndU (EU ) is not commutative, we know that Ext1 (FU , E) ̸=
Proof. (1) Since EndX (E) =U
0 by 3.18. The result then follows since(2.B)

0 ̸= Ext1U (FU , E) ∼= Ext1X (F , i∗ E).
(2) By 3.15, EndX (E) ∼= EndU (EU ) is not commutative, so this follows by (1).

!

24

WILL DONOVAN AND MICHAEL WEMYSS

4. Strategy for Noncommutative Twists
In the remainder of the paper we restrict to the setting of a flopping contraction X →Xcon of an irreducible rational curve C, where X is projective and has only Gorensteinterminal singularities. Our aim is to overcome 3.19 and use the universal sheaf E toproduce a noncommutative twist functor, then prove that it is an autoequivalence whichfurthermore gives an intrinsic description of the flop–flop functor of Bridgeland and Chen.
4.1. Notation. As before, and so as to fix notation for the remainder of the paper, oursetup is the contraction of a irreducible rational floppable curve C in a projective normal
3-fold X with at worst Gorenstein terminal singularities (e.g. X is smooth). We denotethe contraction map f : X → Xcon and remark that necessarily Rf∗ OX = OXcon and Xcon
has only Gorenstein terminal singularities. As before, there is a commutative diagram
e

C

i

U

X

f |U

p

∈

Xcon

(4.A)

∼
=

=
m

Ucon

f

∈ Spec R

where C red ∼= P1 , e is a closed embedding and i is an open embedding. As at the
beginning of §3, U is derived equivalent to an algebra Λ ∼= EndR (R ⊕ N ), and we set
Λcon := Λ/Icon where Icon = [R]. Since Xcon has only terminal Gorenstein singularities,
R has only isolated hypersurface singularities. Furthermore, since a single irreducible
curve has been contracted to a point m, as an R-module Λcon is supported only on m and#∼# con . To ease notation, as before we set R := R.
#Λcon ∼= Λcon ⊗R R
=Λ

4.2. Mutation. Our strategy is to first work on the formal fibre R, and here our main new
tool is that of mutation, an algebraic operation developed in [IW10, §6] that extended othertheories of mutation to the setting of quivers with loops, 2-cycles, and no superpotential,
all of which occur for general flops.
One of the key properties of mutation exploited below is that it always gives rise
to derived equivalences as follows (see §5.3 for more details). As input, we consider amodifying R-module Y , chosen as in §2.3 so that EndR (Y ) is derived equivalent to theformal fibre. The output is another R-module, the mutation νY , together with a tilting
bimodule inducing an equivalenceΦ : Db (mod EndR (Y )) → Db (mod EndR (νY )).
It turns out that Φ is isomorphic to the inverse of the flop functor on the formal fibre [W,4.2], but we will not need this fact. Instead, we compose with a similar functor in the opposition direction to obtain a mutation–mutation autoequivalence of Db (mod EndR (Y )),denoted by Φ ◦ Φ. This will be easier to study, and forms our algebraic model for the
inverse of the (analytic) flop–flop functor. Writing A = EndR (Y ),(1) For the A-bimodule IA defined by the natural exact sequence
IA → A → Acon ,(4.B)Φ ◦ Φ = RHomA (IA , −).

(4.C)

we will show in 5.10(1) that
Although we do not give the details here, the functor RHomA (IA , −) may be interpreted as an (inverse) twist around the noncommutative deformation family of the simple
A-module S, and so (4.C) gives a twist description of Φ ◦ Φ. In the process of proving (1),
we further obtain:(2) A minimal projective resolution of Acon by A-modules in 5.6(2).

NONCOMMUTATIVE DEFORMATIONS AND FLOPS

25(3) A description of the action of Φ ◦ Φ on Acon in 5.9(2).
Point (3) is needed to produce a global spanning class later, and point (2) will establishthat E is a perfect complex such that$
C if t = 0, 3t
ExtX (E, i∗ E) =(4.D)
0 else,generalising [T07].

4.3. Strategy. The remainder of the paper involves lifting the above formal fibre results
to algebraic flops. From the complete local algebraic model in §4.2, a twist autoequivalence
around the universal sheaf E on X is produced using the following strategy:
Tilting algebra Λ: Using an algebraic idempotent trick, together with a morita
equivalence, we are able to lift the tilting bimodule IA giving rise to the complete
local equivalence Φ ◦ Φ to a Λ-bimodule Icon . We prove in 6.2 that this bimodule
gives rise to a derived autoequivalence of Λ, and we thus obtain an algebraic local
model RHomΛ (Icon , −) for our twist functor.
Local geometry U : Since U is derived equivalent to Λ, the above algebraic Λbimodule Icon induces a Fourier–Mukai kernel on U × U . Since Icon is a tilting
module, this kernel gives a twist autoequivalence on U , in §6.4.
Global geometry X: A gluing construction then produces a twist functor TE on X
in §7. It remains to show that TE is an autoequivalence. We are able to conclude
that E is perfect from the minimal projective resolution of Acon in point (2) above,
then, by inferring the action of TE on the universal sheaf E from the knowledge ofΦ◦Φ in (3), we are able to construct (in §7.4) a spanning class on which the actionof TE is known. A spanning class argument establishes that TE is fully faithful,and from there a straightforward lemma, 7.11, allows us to conclude that TE is an
autoequivalence in §7.5.
Furthermore, in 7.14(2) we show that TE fits into a functorial triangle
RHomX (E, −) ⊗L
Acon E → Id → TE →as a consequence of the short exact sequence (4.B) above. We finally prove in 7.18 that
TE is an inverse of the flop–flop functor FF, giving the global geometric version of (4.C).
5. Complete Local Mutation and Ext Vanishing
We keep the notation and assumptions as in §4. There, U → Spec R is a crepantmorphism, which in fact is equivalent to the condition Λ = EndR (R ⊕ N ) ∈ CM R [IW11],
where recall CM R denotes the category of maximal Cohen–Macaulay R-modules. Since
N ∼= HomR (R, N ) is a summand of Λ, this in turn forces N ∈ CM R. Writing ref R forthe category of reflexive R-modules, modules Y ∈ ref R satisfying EndR (Y ) ∈ CM R arethus of interest, and are called modifying R-modules [IW10].
5.1. Mutation. Now Λ = EndR (R ⊕ N ) and completing with respect to the maximal
ideal m in (4.A)
#∼
#⊕N
# ).
Λ
= EndR% (RAs in 2.11, we decompose#⊕N#∼#⊕a0 ⊕ N ⊕a1 .
R
=R
1
# and
Throughout the section, as in §2.4, to ease notation with hats we write R := R,
Y := R ⊕ N1 ,Z := R⊕a0 ⊕ N1⊕a1 ,# = EndR (Z). We write (−)∗ for the duality functor
so that A = EndR (Y ) and Λ
HomR (−, R) : ref R → ref R.

Definition 5.1. With the setup as above,

26

WILL DONOVAN AND MICHAEL WEMYSS(1) We take a minimal right (add R)-approximation
a

R0 −
→ N1
of N1 , which by definition means that(a) R0 ∈ add R and (a·) : HomR (R, R0 ) → HomR (R, N1 ) is surjective,(b) if g ∈ EndR (R0 ) satisfies a = ag, then g is an automorphism.
In other words, a is a projective cover of N1 . Necessarily a is surjective. Since R iscomplete, such an a exists and is unique up to isomorphism. We put K0 := Ker a,so we have an exact sequenceca

0 → K0 −→ R0 −→ N1 → 0.

(5.A)

Triviallyc·a·0 → HomR (R, K0 ) −→ HomR (R, R0 ) −→ HomR (R, N1 ) → 0
is exact, thus since by definition Acon = EndR (Y )/[R] = EndR (N1 )/[R], applying
HomR (Y, −) to (5.A) yields an exact sequencec·a·

0 → HomR (Y, K0 ) −→ HomR (Y, R0 ) −→ HomR (Y, N1 ) → Acon → 0(5.B)

of A-modules.
(2) We define the right mutation of Y asµY := R ⊕ K0 ,that is we remove the summand N1 and replace it with K0 .
(3) Dually, we consider a minimal right (add R∗ )-approximation
b

R∗1 −
→ N1∗
of N1∗ , and we put K1 := Ker b. Thus again we have an exact sequencedb

0 → K1 −→ R∗1 −
→ N1∗ → 0.

(5.C)

(4) We define the left mutation of Y as
νY := R ⊕ K1∗ .
In fact νY = (µY ∗ )∗ , so really we only need to define right mutation.
Remark 5.2. Applying HomR (Y, −) to (5.A) and observing the resulting long exact
sequence in Ext givesc·a·

0 → HomR (Y, K0 ) −→ HomR (Y, R0 ) −→ HomR (Y, N1 ) → Ext1R (Y, K0 ) → 0.
Combining this with (5.B) shows that, as R-modules
Acon ∼= Ext1 (Y, K0 ) ∼= Ext1 (N1 , ΩN1 ) ∼= D Ext1 (ΩN1 , N1 ) ∼= D Ext2 (N1 , N1 )
R

R

R

R

where the third isomorphism is just AR duality, since R is an isolated singularity. Taking
dimensions of both sides shows thatwid(C) = dimC Ext2R% (N1 , N1 ).
# and gives a very easy
This may be calculated by using computer algebra on the base R,
way to calculate wid(C). In contrast, computing the algebra structure on Acon is always
a little harder, and this is needed to calculate Aabcon and hence cwid(C). We remark that
there does not seem to be any easy description of cwid(C) in terms of Ext groups.
Remark 5.3. Mutation is defined in [IW10, §6] for any modifying module, so in particularalso for Z = R⊕a0 ⊕ N1⊕a1 . Summing the exact sequence (5.A) gives an exact sequence
c⊕a1

a⊕a1

1
0 → K0⊕a1 −−−→ R⊕a−−−→ N1⊕a1 → 0
0where a⊕a1

is a minimal (add R)-approximation of N1⊕a1 . Thus0µZ = R⊕a⊕ K0⊕a1 .
0

(5.D)

27

NONCOMMUTATIVE DEFORMATIONS AND FLOPS

Similarly, summing (5.C) gives a minimal (add R∗ )-approximation of (N1⊕a1 )∗ and so
νZ = R0⊕a0 ⊕ (K1∗ )⊕a1 .
From this, it is clear that EndR (νY ) is morita equivalent to EndR (νZ).
# is the morita equivalence from §2.4,
Recall that F = HomA (P, −) : mod A → mod Λ
where P = HomR (Y, Z).
# con ∼#
Lemma 5.4. Λ
In particular= (FAcon )⊕a1 as Λ-modules.
# con ∼
Λcon ∼=Λ= Ma1 (Acon ),# con
where Ma1 (Acon ) denotes the ring of a1 ×a1 matrices over Acon . Consequently Λcon ∼=Λ
and Acon are morita equivalent.
Proof. Applying HomR (Z, −) to (5.D) yields the exact sequence(a·)⊕a11# con → 0
HomR (Z, R⊕a) −−−−−→ HomR (Z, N1⊕a1 ) → Λ
0#of Λ-modules.
On the other hand applying F to (5.B) shows that the top sequence in thefollowing commutative diagram is exact

HomR (Z, R0 )F(Acon )∼HomA (HomR (Y, Z), HomR (Y, N1 ))∼HomA (HomR (Y, Z), HomR (Y, R0 ))a·

HomR (Z, N1 )where the vertical isomorphisms are just reflexive equivalence (see e.g. [IW10, 2.5]). It
follows that FAcon ∼= Cok(a·) and thus
⊕a1 ∼# con .
FAcon=Λ
= Cok(a⊕a1 ·) ∼The last statement follows by 2.17 since

⊕a1 ∼
1
∼
∼
# con ∼# con ) ∼Λ
= EndΛ% (Λ= EndΛ% (FA⊕acon ) = EndA (Acon ) = Ma1 (EndA (Acon )) = Ma1 (Acon ).

!

The following is important, and is a consequence of the fact that R is a hypersurface
singularity.
Proposition 5.5. With assumptions and notation as above,(1) ν(νY ) ∼= Y.
(2) K1∗ ∼= K0 , so that νY ∼= µY .
Proof. (1) By definition of mutation, ν = Ω−1 , where Ω−1 denotes the cosyzygy functor
on CM R. But N1 ∈ CM R since A ∈ CM R, and it is well known that on hypersurfacesΩ−2 (Z) ∼= Z for all Z ∈ CM R [E80].
∗(2) To calculate ν(νY ), we take a minimal right (add( νYK1∗ ) )-approximation, equivalently∗∗ ∗ ∼a minimal right (add R )-approximation, of (K1 ) = K1 and by doing so obtain an exact
sequence
0 → K3 → R∗2 → K1 → 0
with R∗2 ∈ add R∗ . Since we have ν(νY ) := R ⊕ K3∗ , by (1) and Krull–Schmidt it follows
that K3∗ ∼= N1 . Now dualizing the above sequence gives an exact sequence
0 → K1∗ → R2 → N1 → 0
where the last map is a minimal right (add R)-approximation. By uniqueness of minimal
approximations, this implies that R2 ∼= R0 , K1∗ ∼= K0 and further νY = R⊕K1∗ ∼= µY . !

0

28

WILL DONOVAN AND MICHAEL WEMYSS

Since all modules R, N1 , K0 , K1 ∈ CM R and (−)∗ is an exact duality on CM R, we
can dualize (5.C) and obtain an exact sequenceb∗d∗

0 → N1 → R1 → K1∗ → 0.

(5.E)

5.2. Ext vanishing. With regards to our applications, the point of this subsection is toprove 5.7. The following is an application of [IW10, §6].
Proposition 5.6. With assumptions and notation as above,(1) Applying HomR (Y, −) to the sequence (5.E) gives an exact sequence
b∗ ·

d∗ ·

0 → HomR (Y, N1 ) −−→ HomR (Y, R1 ) −−→ HomR (Y, K1∗ ) → 0.

(5.F)

(2) The minimal projective resolution of Acon as an A-module has the form
0 → P → Q1 → Q0 → P → Acon → 0
where P := HomR (Y, N1 ), and Qi ∈ add Q for Q := HomR (Y, R).
# con = 3 and pd % I#con = 2.
(3) pdΛ% ΛΛProof. (1) Since Acon is finite dimensional by 2.13, this follows from the argument exactly
as in [IW10, (6.Q)].
(2) Since K1∗ ∼= K0 by 5.5, splicing (5.F) and (5.B) gives the minimal projective resolution
0 → HomR (Y, N1 ) → HomR (Y, R1 ) → HomR (Y, R0 ) → HomR (Y, N1 ) → Acon → 0.
(3) We have pdA Acon = 3 by (2), and so chasing across the morita equivalence using 5.4,
# con = 3. The final statement follows.
pdΛ% Λ
!
Corollary 5.7. There are isomorphisms

ExttΛ (FAcon , T ) ∼= ExttΛ% (FAcon , T#) ∼= ExttA (Acon , S) ∼=and further Acon is a self-injective algebra.

$

C0

if t = 0, 3
else,

Proof. The first two isomorphisms are consequences of the fact that the Ext groups aresupported only at m. The third isomorphism is an immediate consequence of the minimal
projective resolution in 5.6, since HomA (Qi , S) = 0.
Now if the outer two terms in a short exact sequence are annihilated by an idempotent,
so is the middle term. Hence mod Acon is extension-closed in mod A, and it follows that∼ D Ext2 (Acon , S)
Ext1 (S, Acon ) = Ext1 (S, Acon ) =
(5.G)
Acon

A

A

where the last isomorphism holds since A is 3-sCY [IW10, 2.22(2)], pdA Acon < ∞ andS has finite length. Thus (5.G) shows that Ext1Acon (S, Acon ) = 0. Since Acon is finite
dimensional with unique simple S, it follows that Acon is self-injective.
!
5.3. On the mutation functor Φ. We retain the setup and notation from §5.1, namely# = EndR (Z). For the case of left mutation νY , there is a derived
A = EndR (Y ) and Λ
equivalence between EndR (Y ) and EndR (νY ) [IW10, §6] given by a tilting EndR (Y )module V constructed as follows. We consider the sequence (5.E)
b∗

0 → N1 −→ R1 → K1∗ → 0
obtained by dualizing (5.C). It is trivial that b∗ is a minimal left (add R)-approximation of
N1 . Applying HomR (Y, −), gives an induced map (b∗ ·) : HomR (Y, N1 ) → HomR (Y, R1 ),and our tilting A-module is by definition given by
V := HomR (Y, R) ⊕ Cok(b∗ ·).
∼ HomR (Y, K ∗ ), so in fact
By (5.F) we already know that Cok(b∗ ·) =1

V ∼= HomR (Y, R) ⊕ HomR (Y, K1∗ ) = HomR (Y, νY ).

NONCOMMUTATIVE DEFORMATIONS AND FLOPS

29

This gives rise to an equivalence [IW10, 6.8]
Φ := RHomA (V, −) : Db (mod A) → Db (mod EndR (νY )),which we call the mutation functor. By 5.5 we can mutate EndR (νY ) back to obtain A,
and in an identical way
W := HomR (νY, Y )is a tilting module giving rise to an equivalence which by abuse of notation we also denoteΦ := RHom(W, −) : Db (mod EndR (νY )) → Db (mod A).
Similarly
V ′ := HomR (Z, R⊕a0 ) ⊕ HomR (Z, (K1∗ )⊕a1 ) = HomR (Z, νZ)#
is a tiling Λ-module,
giving rise to an equivalence# → Db (mod EndR (νZ))Φ′ := RHomΛ% (V ′ , −) : Db (mod Λ)

and we mutate back via the tilting module W ′ := HomR (νZ, Z).
The following is easily seen, and is an elementary application of morita theory.
Lemma 5.8. The following diagram commutes:F

Db (mod A)#
Db (mod Λ)Φ′Φ

Db (mod EndR (νY ))moritaDb (mod EndR (νZ))Φ′Φ
F

Db (mod A)#
Db (mod Λ)For our purposes later, we need to be able to track the object Acon through thederived equivalence Φ. This is taken care of in the following lemma.
Lemma 5.9. Write B := EndR (νY ) and Bcon := B/[R], then(1) Φ(Acon ) ∼= Bcon [−1].
(2) Φ ◦ Φ(Acon ) ∼= Acon [−2].
(3) Φ′ ◦ Φ′ (FAcon ) ∼= FAcon [−2].
Proof. Note first that in fact Acon ∼= Bcon as algebras [IW10, 6.20], so dimC Bcon < ∞.
(1) We already know that V = HomR (Y, νY ). Now since K1∗ ∼= K0 by 5.5, splicing (5.A)
and (5.E) gives us an exact sequence of R-modules
c

b∗ ·a

d∗

0 → K1∗ → R0 −−→ R1 → K1∗ → 0
to which applying HomR (Y, −) gives a complex of A-modulesc·(b∗ ·a)·d∗ ·0 → HomR (Y, K1∗ ) −→ HomR (Y, R0 ) −−−−→ HomR (Y, R1 ) −−→ HomR (Y, K1∗ ) → 0 (5.H)
whereas applying HomR (νY, −) gives a complex of B-modulesc·(b∗ ·a)·d∗ ·0 → HomR (νY, K1∗ ) −→ HomR (νY, R0 ) −−−−→ HomR (νY, R1 ) −−→ HomR (νY, K1∗ ) → 0(5.I)
In both (5.H) and (5.I), we consider the last Hom term to be in degree zero. Since every
term in (5.H) is a summand of V , by reflexive equivalence it follows that under the derived
equivalence Φ, (5.H) gets sent to (5.I). Hence the statement in the lemma follows if we
can show that (5.H) is quasi-isomorphic to Acon [1] and that (5.I) is quasi-isomorphic to
Bcon [0].

30WILL DONOVAN AND MICHAEL WEMYSS

Consider first (5.H). Combining the exact sequences (5.F) and (5.B)
b∗ ·

d∗ ·

0 → HomR (Y, N1 ) −−→ HomR (Y, R1 ) −−→ HomR (Y, K1∗ ) → 0c·a·0 → HomR (Y, K1∗ ) −→ HomR (Y, R0 ) −→ HomR (Y, N1 ) → Acon → 0
shows that (5.H) is quasi-isomorphic to Acon [1].
Next, consider (5.I). Applying 5.6(1) with the module νY (instead of Y ), it follows
that
c·
a·
0 → HomR (νY, K1∗ ) −→ HomR (νY, R0 ) −→ HomR (νY, N1 ) → 0
is exact. Further, since trivially d∗ is a minimal (add R)-approximation, applying thefunctor HomR (νY, −) to (5.E) gives an exact sequence
b∗ ·

d∗ ·

0 → HomR (νY, N1 ) −−→ HomR (νY, R1 ) −−→ HomR (νY, K1∗ ) → Bcon → 0.
Splicing the last two exact sequences shows that (5.I) is quasi-isomorphic to Bcon [0].
(2) Since as above dimC Bcon < ∞, this follows since we can apply (1) twice.
(3) This follows from (2) using the commutativity of the diagram in 5.8.

!

We write IA for the ideal [R] of A = EndR (R⊕N1 ). The following gives a descriptionof the mutation–mutation functors in terms of the bimodules IA and I#con .
Theorem 5.10. There are functorial isomorphisms(1) Φ ◦ Φ ∼= RHomA (IA , −).
(2) Φ′ ◦ Φ′ ∼= RHomΛ% (I#con , −).
# with inverse − ⊗L I#con , andThus RHom % (I#con , −) is an autoequivalence of Db (mod Λ),Λ#
so I#con is a tilting Λ-module.

%Λ

Proof. (1) To avoid confusion we will write B = EndR (νY ), and also denoteΦDb (mod A) −→ Db (mod B) by Φ1 = RHomA (V, −),andΦ

Db (mod B) −→ Db (mod A) by Φ2 = RHomB (W, −).
L∼
The composition Φ2 ◦ Φ1 ∼= RHomA (W ⊗L
B V, −), so we show that W ⊗B V = IA .
As above, W := HomR (νY, Y ). Hence W is quasi-isomorphic to its projective resolution(c·0 ). . . → 0 → HomR (νY, K1∗ ) −−→ HomR (νY, R0 ) ⊕ HomR (νY, R) → 0 → . . .
(5.J)
But Φ1 is an equivalence that sends V to B. In fact, Φ1 sends
HomR (Y, R) *→ HomR (νY, R)
HomR (Y, K1∗ ) *→ HomR (νY, K1∗ ).
Hence applying the inverse of Φ1 , namely − ⊗L
B V , to (5.J) gives the complex(c·0 ). . . → 0 → HomR (Y, K1∗ ) −−→ HomR (Y, R0 ) ⊕ HomR (Y, R) → 0 → . . .
Then using K ∗ ∼= K0 and the fact that (5.B)

(5.K)

1

c·

0 → HomR (Y, K1∗ ) −→ HomR (Y, R0 ) → HomR (Y, N1 ) → Acon → 0
is exact, the complex (5.K) is quasi-isomorphic to. . . → 0 → A = HomR (Y, R) ⊕ HomR (Y, N1 ) → Acon → . . .
which is clearly quasi-isomorphic to IA .
(2) is similar. The final statements follow since the mutation functors Φ′ are always
derived equivalences [IW10, 6.8].
!
Corollary 5.11. Φ ◦ Φ(S) ∼= S[−2].

31

NONCOMMUTATIVE DEFORMATIONS AND FLOPS

Proof. Let J := Rad(Acon ) denote the Jacobson radical, then Acon /J ∼= S and soS∼= (Acon /J) ⊗A Acon ∼= (Acon /J) ⊗L Acon .
Acon

con

By 5.9(2) and 5.10 it follows that S[−2] ∼= (Acon /J) ⊗L
Acon RHomA (IA , Acon ), so the result
follows from the fact that
(Acon /J) ⊗L RHomA (IA , Acon ) ∼= RHomA (IA , (Acon /J) ⊗L Acon )
Acon

Acon

∼
= RHomA (IA , S),
where the first canonical isomorphism is standard (see e.g. [IR08, 2.12(2)]).

!

The equivalence RHomΛ% (I#con , −) in 5.10 above is the complete local version of ournoncommutative twist functor. The following is a corollary of the results in this section,
and will be used later.
Corollary 5.12. With the setup as above,(1) RHomΛ% (I#con , FAcon ) ∼= FAcon [−2].
∼###(2) RHomΛ% (Icon , T ) = T [−2].

5.10

5.9

Proof. (1) We have RHomΛ% (I#con , FAcon ) ∼= Φ′ ◦ Φ′ (FAcon ) ∼= FAcon [−2].
5.10

5.8

5.11

(2) RHomΛ% (I#con , T#) ∼= Φ′ ◦ Φ′ (T#) = Φ′ ◦ Φ′ (FS) ∼= F ◦ Φ ◦ Φ(S) ∼= FS[−2] = T#[−2]. !
6. Zariski Local Twists

We keep our running setup of a flopping contraction as in §4, with an irreducible
rational curve contracting to a point m. Since R is Gorenstein, R is a canonical Rmodule, but in this non-local setting canonical modules are not unique. Throughout we
set ωR := g ! C[−3], where g : Spec R → Spec C is the structure morphism. This may or
may not be isomorphic to R.
Throughout, as in §4, Λ := EndR (R ⊕ N ) and we already know that Λ ∈ CM R.
As before Icon := [R] and Λcon := Λ/Icon . Since the category CM R is no longer Krull–
Schmidt, we must be careful.
6.1. Local tilting. In this subsection, we show that the Λ-module Icon is still a tilting
module, with endomorphism ring Λ. First, we need the following lemma.
Lemma 6.1. Put I := Icon , then
(1) Viewing I as a right Λ-module, we have Λ ∼= EndΛ (I) under the map which sends
λ ∈ Λ to the map (λ·) : I → I given by i *→ λi.
(2) Under the isomorphism in (1), the bimodule EndΛ (I) IΛ coincides with the natural
bimodule structure Λ IΛ .
Proof. (1) First recall that Λ ∼= EndΛ (ΛΛ ) via the ring homomorphism which sends λ ∈ Λto (λ·) : Λ → Λ sending x *→ λx, left multiplication by λ. Now consider the short exact
sequence
0 → I → Λ → Λcon → 0(6.A)

of Λ-bimodules. Applying HomΛ (−, Λ) gives an exact sequence
0 → HomΛ (Λcon , Λ) → HomΛ (Λ, Λ) → HomΛ (I, Λ) → Ext1Λ (Λcon , Λ).

(6.B)

Since Λ ∈ CM R, it follows that Λ is singular d-CY by [IW10, 2.22(2)], hence
dimR Λcon = d − inf{i ≥ 0 | ExtiΛ (Λcon , Λ) ̸= 0}
by [IR08, 3.4(5)(ii)]. But R is normal, which implies that dimR Λcon ≤ d − 2 (see e.g.
[IW10, 6.19]), thus combining we see that HomΛ (Λcon , Λ) ∼= Ext1Λ (Λcon , Λ) = 0.
On the other hand, applying HomΛ (I, −) to (6.A) gives
0 → HomΛ (I, I) → HomΛ (I, Λ) → HomΛ (I, Λcon ) = 0.

(6.C)

32

WILL DONOVAN AND MICHAEL WEMYSS

Combining (6.B) and (6.C) shows that EndΛ (Λ) ∼= EndΛ (I). Chasing through the sequences, this isomorphism is given by (λ·) ∈ EndΛ (Λ) *→ (λ·) ∈ EndΛ (I), so the isomorphism is in fact a ring isomorphism.
(2) I is naturally a left EndΛ (I)-module via f · i := f (i). By the isomorphism in (1), thisis just left multiplication by λ.
!
We can now extend 5.10.
Theorem 6.2. Icon is a tilting Λ-module with EndΛ (Icon ) ∼= Λ. Furthermore(1) pdΛ Icon = 2, so pdΛ Λcon = 3.
(2) pdΛop Λcon = 3.
Proof. The fact that EndΛ (I) ∼= Λ is 6.1. Let n ∈ Max R, then the short exact sequence
0 → Icon → Λ → Λcon → 0 shows that (Icon )n ∼/ Supp Λcon = {m}. Further,= Λn if n ∈
5.10
#m ∼
ExtiΛ (Icon , Icon )m ⊗Rm R= ExtiΛ% (I#con , I#con ) = 0

for all i > 0. Thus, in either case, the completion of ExtiΛ (Icon , Icon ) at each maximal
ideal is zero, hence ExtiΛ (Icon , Icon ) = 0 for all i > 0. Similarly# con | n ∈ Supp Λcon = {m}} − 1pdΛ Icon = sup{pdΛ% Λwhich is 2 by 5.6(3). For generation, suppose that Y ∈ D(Mod Λ) with RHomΛ (Icon , Y ) =0. Then2.15# ∼0 = RHomΛ (Icon , Y )n ⊗ R= RHom % (I#con , Y# )
Λand so since I#con is tilting by 5.10, Y# = 0 for all n ∈ Max R. Hence Y = 0, proving
generation. Part (2) is shown similarly.
!

6.2. Tracking the contraction algebra. In this subsection we track the objects FAcon
and T across the derived equivalence induced by Icon (extending 5.12), and also under theaction of a certain Serre functor.
Proposition 6.3. Under the setup above,(1) RHomΛ (Icon , FAcon ) ∼= FAcon [−2] as Λ-modules.
(2) RHomΛ (Icon , T ) ∼= T [−2] as Λ-modules.
Proof. (1) We have
# ∼!
RHomΛ (Icon , FAcon )n ⊗ R= RHomΛ% (I#con , FA
con )
$
5.12
0
if n ∈/ Supp FAcon = {m}∼=FAcon [−2] if n ∈ Supp FAcon = {m}.

It follows that C := RHom(Icon , FAcon ) is a stalk complex in homological degree two, andfurther that the degree two piece is supported only on m. Thus
2.15

2 (C)[−2] ∼!
C∼= H 2 (C)[−2] ∼= H= FAcon [−2]as required.
(2) is similar, using RHomΛ% (I#con , T#) ∼= T# by 5.12.

!

In our possibly singular setting, we need the following RHom version of Serre functors.

Definition 6.4. We say that a functor S : Kb (proj Λ) → Kb (proj Λ) is a Serre functor
relative to ωR if there are functorial isomorphisms
RHomR (RHomΛ (a, b), ωR ) ∼= RHomΛ (b, S(a))
in D(Mod R) for all a ∈ Kb (proj Λ) and all b ∈ Db (mod Λ).

(6.D)

NONCOMMUTATIVE DEFORMATIONS AND FLOPS

33

Since under our assumptions in this section R is Gorenstein, the module ωR is adualizing module of R, i.e.
RHomR (−, ωR ) : D(mod R) → D(mod R)
gives a duality. Since HomR (−, ωR ) : mod Λ → mod Λop , this induces a duality
RHomR (−, ωR ) : D ± (mod Λ) → D ∓ (mod Λ),
and we define the functor SΛ to be the composition
RHomΛ (−,Λ)

RHomR (−,ωR )

D − (mod Λ) −−−−−−−−−→ D + (mod Λop ) −−−−−−−−−−→ D − (mod Λ).
As in the proof of [IR08, 3.5], functorially SΛ (−) ∼= RHomR (Λ, ωR ) ⊗LΛ −. The following
is known by [G06, 7.2.14] and [IR08, §3]. We include the proof for completeness.
Proposition 6.5. SΛ is a Serre functor relative to ωR .
Proof. First, since Λ = EndR (M ) ∈ CM R, it follows that
inj.dimΛ Λ = dim R = inj.dimΛop Λ
by combining [IW10, 2.22] and [IR08, 3.1(6)(2)]. Given this, the fact that SΛ maps
Kb (proj Λ) to Kb (proj Λ) is then [G06, 7.2.14(i)], and the fact that (6.D) holds is [G06,!
7.2.8] (see also [IR08, §3]).
The object FAcon does not change under the action of this Serre functor.
Proposition 6.6. SΛ (FAcon ) ∼= FAcon .
Proof. We haveSΛ (FAcon )n = RHomR (RHomΛ (FAcon , Λ), ωR )n
2.15∼= RHomRn (RHomΛn (FAcon n , Λn ), (ωR )n )which as above is isomorphic to RHomRn (Λn , (ωR )n ) ⊗L
Λn FAcon n . Now since R is Gorenstein (ωR )n ∼= Rn , and further RHomRn (Λn , Rn ) ∼= HomRn (Λn , Rn ) since Λ ∈ CM R. But
since Λn ∼= EndRn (Mn ) is a symmetric Rn -algebra [IR08, 2.4(3)], HomRn (Λn , Rn ) ∼= Λn
as Λn -bimodules. Thus$0
if n ∈/ Supp FAcon = {m}∼SΛ (FAcon )n ∼FA
= HomRn (Λn , Rn ) ⊗L
=
con
n
Λn
FAcon if n ∈ Supp FAcon = {m}.

It follows that SΛ (FAcon ) is a stalk complex in homological degree zero, and that degree
zero piece has finite length, supported only on m. Thus
2.15

SΛ (FAcon ) ∼= H 0 (SΛ (FAcon ))m ∼= FAcon .
= H 0 (SΛ (FAcon )) ∼!

6.3. Algebraic noncommutative twist functors. We can now give our definition ofalgebraic noncommutative twist functors. The ring homomorphism Λ → Λcon gives rise to
the standard extension–restriction of scalars adjunction. To ease notation, we temporarily
write M := Λ (Λcon )Λcon and N := Λcon (Λcon )Λ . The adjunctions then read
LA
Fcon
:=−⊗L
ΛM

Db (mod Λcon )Fcon :=RHomΛcon (M,−)=−⊗L
Λcon NDb (mod Λ)

(6.E)

RA
Fcon
:=RHomΛ (N,−)

where we make use of the following:RA
LA
Lemma 6.7. Fcon , Fcon
and Fcon
all preserve bounded complexes of finitely generated
modules.

34

WILL DONOVAN AND MICHAEL WEMYSSProof. Fcon preserves boundedness since restriction of scalars is exact on the abelian level.
RALA
We remark that if Y ∈ Mod Λ then Fcon
(Y ) and Fcon
(Y ) are both bounded since Λcon
has finite projective, hence finite flat, dimension as both a left and right Λ-module by 6.2.
RA
LA
The fact that Fcon
and Fconboth preserve boundedness then follows by induction on thelength of the bounded complex. The preservation of finite generation is obvious.
!
# con as algebras by 2.17, and there is a morita equivalence
Now Λcon ∼=Λ
mod Acon−⊗Acon FAcon

HomΛ% con (FAcon ,−)induced from 5.4.

# con
mod Λ

(6.F)

Lemma 6.8. Composing (6.E) with (6.F) gives adjunctions
F LA

Db (mod Acon )F :=−⊗L
Acon FAcon

Db (mod Λ)(6.G)

F RA :=RHomΛ (FAcon ,−)Proof. This follows since FAcon ⊗Λcon Λcon ∼= FAcon as Acon –Λ bimodules.

!

∼∼
Now since M ⊗LΛcon N = M ⊗Λcon N = Λ (Λcon )Λ as Λ-bimodules, we have functorial
isomorphisms
RA ∼∼F ◦ F RA ∼= Fcon ◦ Fcon
= RHomΛ (M ⊗LΛcon N, −) = RHomΛ (Λ (Λcon )Λ , −)

and
LA ∼
L
L
∼
F ◦ F LA ∼= Fcon ◦ Fcon
= − ⊗LΛ M ⊗Λcon N = − ⊗Λ Λ (Λcon )Λ .

Definition 6.9. Using the bimodule structure Λ IΛ we define(1) The twist functor Tcon : Db (mod Λ) → Db (mod Λ) by Tcon = RHomΛ (Icon , −).
∗∗(2) The dual twist functor Tcon
: Db (mod Λ) → Db (mod Λ) by Tcon
= − ⊗LΛ Icon .
The twist is an equivalence, and its inverse is the dual twist, by 6.2. The terminology‘twist’ is justified by the following lemma.
Lemma 6.10. We have functorial triangles as follows(1) F ◦ F RA → IdΛ → Tcon →∗(2) Tcon→ IdΛ → F ◦ F LA →Proof. (1) The short exact sequence (6.A) of Λ-bimodules gives rise to a triangle
Icon → Λ → Λcon →(6.H)

in the derived category of Λ-bimodules. For X ∈ Db (mod Λ), applying RHomΛ (−, X) to(6.H) gives a triangle
RHomΛ (Λcon , X) → RHomΛ (Λ, X) → RHomΛ (Icon , X) →in Db (mod Λ), which is simply
F ◦ F RA (X) → IdΛ (X) → Tcon (X) → .
(2) This follows by applying X ⊗L
Λ − to (6.H).

!

35

NONCOMMUTATIVE DEFORMATIONS AND FLOPS

6.4. Geometric local noncommutative twists. Again we keep the setup of §4. In thissubsection we define the geometric noncommutative twist purely in terms of the geometryof U , and use it as the local model for our global twist functor later.
To fix notation, as in (2.E) we write V := OU ⊕ N and thus we have a derived
equivalence
H LA :=RHomU (V,−)

Db (mod Λ)H:=−⊗LΛV

Db (coh U )(6.I)

H RA :=RHomU (V,−)

given by a tilting bundle V and Λ ∼= EndY (V). We know by 3.7 that EU (in degree zero)corresponds to FAcon across the derived equivalence.
Lemma 6.11. With notation and setup as in §4, composing (6.G) with (6.I) gives
GLA
U :=RHomAcon (RHomU (−,EU ),Acon )Db (mod Acon )GU :=−⊗L
Acon EUDb (coh U )

(6.J)

GRA
U :=RHomU (EU ,−)∼Proof. The composition H ◦ F is clear since FAcon ⊗LΛ V = EU by 3.7. Also by 3.7
RARA∼
RHomU (V, EU ) = FAcon , so the composition F◦H
is functorially isomorphic to
RHomΛ (FAcon , RHomU (V, −)) ∼= RHomΛ (RHomU (V, EU ), RHomU (V, −))∼= RHomU (EU , −).
Lastly,RHomAcon (F LA ◦ H LA (−), Acon ) ∼= RHomU (−, H ◦ F (Acon ))∼= RHomU (−, Acon ⊗L EU )Acon∼
= RHomU (−, EU ).
But since Acon is self-injective RHomAcon (−, Acon ) = HomAcon (−, Acon ) is a duality, so
dualizing both sides gives
F LA ◦ H LA (−) ∼= RHomAcon (RHomU (−, EU ), Acon ),as required.

!

Remark 6.12. We remark that (6.J) is intrinsic to the geometry of U , and does notdepend on the choice of Λ.
Definition 6.13.

(1) We define TEU as the composition
RHomU (V,−)

T

−⊗L V

Db (coh U ) −−−−−−−−−→ Db (mod Λ) −−con−→ Db (mod Λ) −−−Λ−→ Db (coh U )and call it the geometric noncommutative twist functor.
∗(2) We obtain an inverse twist TE∗U by composing similarly with Tcon.
Being a composition of equivalences, the geometric noncommutative twist is also an
equivalence. We remark that the definition of the geometric noncommutative twist functor
relies on the tilting equivalence, but below in 6.16 we show that it can be formulated asa Fourier–Mukai transform, by making use of the following setup.

36

WILL DONOVAN AND MICHAEL WEMYSS

Consider the commutative diagram
p1

U ×U
p2

U

U

πU

(6.K)

πU

πU ×U

pt

∗
and set V ∨ " V = p∗1 V ∨ ⊗L
OU ×U p2 V. Then there is an induced derived equivalence

Db (coh U × U )RHomU ×U (V ∨ !V,−)∨−⊗LΛe (V !V)Db (mod Λ ⊗C Λop )(6.L)

as in [BH], where we denote the enveloping algebra by Λe := Λ ⊗C Λop .
Notation 6.14. If Y is a C-scheme and Γ is a C-algebra, we define
OYΓ := OY ⊗C Γ,
where C and Γ are the constant sheaves associated to C and Γ.
To be slightly more pedantic, and armed with the notation above, we think of V ∨ "ΛeV as a left OU×U
-module, and the lower functor in (6.L) is the derived version of thecomposition

mod Λ−⊗OΛe (V ∨ !V)
U ×U
Λe−−−−−−→ mod OU×U −−−−−−−−−−−→ mod OU×U .

∗e πU ×U (−)

Now, we may view the tilting equivalences in 6.13(1) as Fourier–Mukai functors between Db (coh U ) and Db (mod Λ), given by V ∨ and V respectively, considered as objects inΛ(op)Db (mod OU). The lower functor in (6.L) should then be seen as taking a Λ-bimodule,
and composing it with these Fourier–Mukai functors as in 6.13(1). Hence, taking W ′ tobe the object in Db (coh U × U ) corresponding to the Λ-bimodule RHomΛ (Icon , Λ), we willfind in 6.16(1) below that our twist TEU is itself a Fourier–Mukai functor with kernel W ′ .
This motivates the following definition and lemma.
Definition 6.15.

(1) We define a geometric twist kernel W ′ as follows∗eW ′ := πU×U
(RHomΛ (Icon , Λ)) ⊗L
OΛ

U ×U

∗
(p∗1 V ∨ ⊗L
OU ×U p2 V).

(2) We obtain an inverse twist kernel W as follows∗e
W := πU×U
Icon ⊗LOΛ

U ×U∗(p∗1 V ∨ ⊗L
OU ×U p2 V).

The following lemma describes the twist and inverse twist.
Lemma 6.16. There are functorial isomorphisms.
′(1) TEU ∼= FM(W ′ ) := Rp2 ∗ p∗1 (−) ⊗L
OU ×U W ,(2) TE∗U ∼= FM(W).

(6.M)

37

NONCOMMUTATIVE DEFORMATIONS AND FLOPS

Proof. Put I := Icon . We prove (2) first. We have∗
L
TE∗U = πU(RHomOU (V, −) ⊗LΛ I) ⊗O Λ V(6.13(2))

∨
L
L
∗
∼
(RΓ(− ⊗L= πU
Λ V
OU V ) ⊗Λ I) ⊗OU(definition of RHom)

U

∨
L
∗
L
∼
(base change for (6.K))= Rp2 ∗ p∗1 (− ⊗LΛ πU I ⊗O Λ V
OU V ) ⊗OU
U
5
6
∨
L
∗
∼
p∗2 (πU
I ⊗L(projection formula)= Rp2 ∗ p∗1 (− ⊗LΛΛ V)
OU V ) ⊗OU
OU
×U
5
5
66
∗ ∨
L
∗
∼
p∗2 πUI ⊗L
p∗2 V= Rp2 ∗ p∗1 (−) ⊗LΛΛ
OU ×U p1 V ⊗OU
OU
×U
×U
66
5
5
∗
L
∗ ∨
L
∗
∼
e
p
V)
π
I
⊗
(p
V
⊗
= Rp2 ∗ p∗1 (−) ⊗L
Λ
OU ×U
U×U
1
OU ×U 2
OU ×U
- ∗
.
L
∼
(by (6.M))= Rp2 ∗ p1 (−) ⊗OU ×U W .

Since Tcon = RHomΛ (I, −) = (−) ⊗LΛ RHomΛ (I, Λ), a similar calculation using 6.13(1)gives (1).
!
For our purposes later we must track the object EU under a certain Serre functor.
Again, since U is possibly singular, this involves the RHom version. Throughout, if Y is a
C-scheme of dimension d, although canonical modules are not unique, we write DY := g ! C
for the dualizing complex of Y , where g : Y → Spec C is the structure morphism. If further
Y is normal CM of dimension d, we will always write ωY for DY [−d], and refer to ωY as
the geometric canonical.
Definition 6.17. Suppose that f : Y → Spec T is a morphism where T is a normal CM
C-algebra with canonical module ωT . Typically• T = C (in the case Y = X is projective), or• T = R with Spec R a 3-fold and f is also birational (in the case Y = U of thelocal model).
We say that a functor S : per(Y ) → per(Y ) is a Serre functor relative to ωT if there arefunctorial isomorphisms
RHomT (RHomY (F , G), ωT ) ∼= RHomY (G, S(F ))in D(Mod T ) for all F ∈ per(Y ), G ∈ Db (coh Y ).
Lemma 6.18. Suppose that f : Y → Spec T is a projective morphism, where Y andSpec T are both Gorenstein varieties. ThenSY := − ⊗Y ωY [dim Y − dim T ] : per(Y ) → per(Y )is a Serre functor relative to ωT .
Proof. First, since Y is Gorenstein, ωY is a line bundle, so tensoring does preserve perfectcomplexes. The remainder is just Grothendieck duality: since f ! ωT = ωY [dim Y − dim T ]we have
RHomY (G, SY (F )) ∼= RHomY (RHomY (F , G), f ! ωT ) ∼= RHomT (RHomY (F , G), ωT )in D(Mod T ) for all F ∈ per(Y ), G ∈ Db (coh Y ).

!

We now revert back to the assumptions and setup of §4. The following are geometricversions of 6.6 and 6.3.
Lemma 6.19. SU (EU ) ∼= EU , i.e. EU ⊗ ωU ∼= EU .
Proof. If we temporarily denote the derived equivalence by Ψ := RHomU (V, −) withinverse Φ := − ⊗LΛ V, then on Λ, Ψ ◦ SU ◦ Φ is a Serre functor relative to ωR , so since Serrefunctors are unique with respect to a fixed canonical, SΛ ∼= Ψ ◦ SU ◦ Φ. Hence
3.7

6.6

3.7

SU (EU ) ∼= SU (ΦFAcon ) ∼= ΦΨSU (ΦFAcon ) ∼= ΦSΛ (FAcon ) ∼= ΦFAcon ∼= EU .

!

38

WILL DONOVAN AND MICHAEL WEMYSS

Proposition 6.20. With notation as above,(1) TEU (EU ) ∼= EU [−2] and TEU (E) ∼= E[−2].
∗(2) TEU (EU ) ∼= EU [2] and TE∗U (E) ∼= E[2].
Proof. (1) By definition TEU is the composition
RHomU (V,−)

T

−⊗L V

Db (coh U ) −−−−−−−−−→ Db (mod Λ) −−con−→ Db (mod Λ) −−−Λ−→ Db (coh U ).
We know that the first functor takes EU to FAcon by 3.7(1), the second functor takes
FAcon to FAcon [−2] by 6.3(1), and the third functor takes FAcon [−2] to EU [−2], again by
3.7(1). Tracking E is similar.
(2) This follows immediately from (1), since as we have already observed, TE∗U is theinverse of TEU .
!
7. Noncommutative Twist Functors: Global CaseIn this section we globalize the previous sections to obtain noncommutative twists ofprojective varieties. We keep the assumptions and setup of §4.
7.1. Global Ext vanishing. Using 3.6, we can deduce Ext vanishing on X by reducing
the problem to Ext vanishing on U , which we have already solved in 5.7. Also, we remarkthat the following shows that although i∗ E need not be perfect, its noncommutative
thickening E automatically is.
Theorem 7.1. We keep the assumptions as above, in particular X is a projective normal
3-fold X with at worst Gorenstein terminal singularities. Then E is a perfect complex andfurther$
C if t = 0, 3
ExttX (E, i∗ E) =0 else.
Proof. Now the first assertion in the statement is local, so it suﬃces to check that Ex has
finite projective dimension for all closed points x ∈ X. Restricting E to U , by 3.6 it isclear that i∗ E = EU , which across the derived equivalence (2.E) corresponds to FAcon by
3.7. Since FAcon is supported only on the maximal ideal m ∈ Max R by 2.13, it follows
that pdΛ FAcon = pdΛ% FAcon = pdA Acon which by 5.6 is finite. Hence back across theequivalence i∗ E is a perfect OU -module. Thus Eu has finite projective dimension for allu ∈ U . Since Ex = 0 for all x ∈/ U , this implies that E is perfect.
For the second assertion, we have
(2.B)

ExttX (E, i∗ E) ∼= ExttU (EU , E) ∼= ExttΛ (FAcon , T )and so the result follows from 5.7.

!

The following result will be needed later. Again 3.6 reduces the proof to the local
model, allowing us to use 6.19.
Proposition 7.2. E ⊗X ωX ∼= E.
Proof. Since X is Gorenstein, ωX is a line bundle and so
3.6

6.19

3.6

L ∗
L
∼
∼
∼
∼
E ⊗ X ωX ∼= Ri∗ EU ⊗L
X ωX = Ri∗ (EU ⊗U i ωX ) = Ri∗ (EU ⊗U ωU ) = Ri∗ (EU ) = E.

!

NONCOMMUTATIVE DEFORMATIONS AND FLOPS

39

7.2. Global inverse twist. We now extend the definition of the local inverse twist TE∗U
on the open neighbourhood U (from 6.13) to a global inverse twist TE∗ on X. It turns out
to be technically much easier to extend TE∗U rather than to extend TEU , so we do this first.
We return to the problem of defining the global noncommutative twist in §7.5.
Throughout, we keep the setting and assumptions of §4. We begin with a slightrefinement of the results of §6.4. Consider the triangle of Λ-bimodules (6.H)
Icon →Λ → Λcon →,∨then applying the functor − ⊗L
Λe (V " V) we obtain an exact triangle
φU

WU → O∆,U −−→ QU →(7.A)

of Fourier–Mukai kernels on U × U .
Definition 7.3. The inverse twist TE∗ is defined to be the Fourier–Mukai functor FM(WX )given by the kernel WX := Cone(φX )[−1], where φX is the compositionη∆R(i×i)∗ φU

O∆,X −−→ R(i × i)∗ O∆,U −−−−−−−→ R(i × i)∗ QU =: QX .

(7.B)

Here η∆ is defined to beR∆X∗ η
R∆X∗ OX −−−−−→ R∆X∗ Ri∗ i∗ OX ∼= R(i × i)∗ R∆U∗ OU ,

(7.C)

∗

with η : OX → Ri∗ i OX the unit morphism.
Since R(i×i)∗ does not in general preserve coherence, we must work hard to establishthat WX , QX ∈ Db (coh X × X). To do this, we first establish the triangle representation
for TE∗ in (7.F) below, and thence show that the Fourier–Mukai functors for WX and QX
preserve Db (coh X).
Consider the adjunctions in (6.J), which were used to describe the local inverse twist
TE∗U . These adjunctions are induced from adjunctions on the unbounded level, and composing them with (2.B) yields the diagram of adjoints
LA ∗GLA
X :=GU ◦i

D(Mod Acon )D(Qcoh X)

GX :=Ri∗ (−⊗L
Acon EU )

(7.D)

RA !
GRA
X :=GU ◦i

By the projection formula we have GX ∼= − ⊗LAcon Ri∗ EU , so by 3.6 and adjunction, theabove yields
GLA
X

D(Mod Acon )L
GX ∼=−⊗Acon E

D(Qcoh X)

(7.E)

∼
GRA
X =RHomX (E,−)

We remark that the above diagram is intrinsic to the geometry of X. The following lemma
establishes, amongst other things, that the inverse twist preserves the bounded derived
category.
Lemma 7.4. Under the setup above,(1) FM(QX ) ∼= GX ◦ GLA
X , and so for all x ∈ D(Qcoh X), there is a triangleTE∗ (x) → x → GX ◦ GLA
X (x) → .
(2) Both FM(QX ) and the inverse twist TE∗ preserve Db (coh X).

(7.F)

40

WILL DONOVAN AND MICHAEL WEMYSSProof. (1) From the definition 7.3, there is a functorial triangle
TE∗ → IdX → FM(QX ) → .
Now since QX = R(i × i)∗ QU by definition (7.B), necessarily
FM(QX ) ∼= Ri∗ ◦ FM(QU ) ◦ i∗by [H06, 5.12]. By the argument of 6.16, we have that FM(QU ) ∼= GU ◦ GLA
U , hence
∗ ∼
LA
∗
∼
FM(QX ) = Ri∗ ◦ FM(QU ) ◦ i = Ri∗ ◦ G ◦ G ◦ i = G ◦ GLA
(7.G)
U

U

X

X

and so the statement follows.
(2) Since Acon is a finite dimensional algebra, its derived category Db (mod Acon ) is generated by its simple module S. Note that GX sends S to i∗ E ∈ Db (coh X), thus it follows
LA∗that GX sends Db (mod Acon ) to Db (coh X). Since GLA
X is the composition GU ◦ i , and
LA
b
b
LA
b
by (6.J) GU takes D (coh X) to D (mod Acon ), it follows that GX sends D (coh X) tobb
Db (mod Acon ). Combining, we see that GX ◦ GLA
X takes D (coh X) to D (coh X), so thebclaim that the inverse twist preserves D (coh X) follows immediately from the triangle in(1), using the two-out-of-three property.
!
7.3. Existence of adjoints. The existence of left and right adjoints of our inverse twist
follows directly from the following known theorem [HMS1, HMS2]. Recall that for projections p1 , p2 : Y × Y → Y , an object G ∈ D(Qcoh Y × Y ) is said to have finite projective dimension with respect to pi , if RHomOY ×Y (G, p!i F ) ∈ Db (Qcoh Y × Y ) for all
F ∈ Db (Qcoh Y ).
Theorem 7.5. Suppose that Y is a projective variety with only Gorenstein singularities,∗band G ∈ Db (Qcoh Y × Y ). If FM(G) := Rp2∗ (G ⊗L
OY ×Y p1 (−)) preserves D (coh Y ), then(1) G ∈ Db (coh Y × Y ).
(2) G has finite projective dimension with respect to p1 .
∗b
If furthermore Rp1∗ (G ⊗L
OY ×Y p2 (−)) also preserves D (coh Y ), then(3) G has finite projective dimension with respect to both p1 and p2 , and FM(G) admits
left and right adjoints which also preserve Db (coh Y ). The right adjoint of FM(G)is
FM(G ∨ ⊗ p∗1 ωY [dim Y ]).
Proof. (1) Take a very ample bundle O(1) on Y , then p∗1 O(1) is very ample with respect to∗the morphism p2 . Since by assumption Rp2∗ (G ⊗L
OY ×Y p1 O(r)) has coherent cohomology
for all r, it follows that G ∈ Db (coh Y × Y ) by [HMS2, remark below 2.5].
(2) By (1) G ∈ Db (coh Y × Y ), so since by assumption its Fourier–Mukai functor FM(G)takes Db (coh Y ) to Db (coh Y ), by [HMS2, 2.7] G must have finite homological dimension
with respect to p1 (for the definition of this, see [HMS1, 1.3]). Hence by [HMS1, 1.6] G
has finite projective dimension with respect to p1 .
(3) With the additional assumption, applying (2) with the roles of p1 and p2 exchanged
we see that G also has finite projective dimension with respect to p2 . Thus by [HMS1,
1.17] FM(G) now has both left and right adjoints, which also preserve Db (coh Y ), and theright adjoint is of the form above.
!
The following is a consequence of 7.4 and 7.5.
Corollary 7.6. We keep the assumptions as in §4, and consider the triangle
WX → O∆,X → QX →from 7.3. Then∗L∗b(1) Rp2∗ (QX ⊗L
OX×X p1 (−)) and Rp1∗ (QX ⊗OX×X p2 (−)) both preserve D (coh X).
(2) WX , O∆,X and QX all belong to Db (coh X × X) and have finite projective dimension over both factors. Hence the corresponding Fourier–Mukai functors haveright and left adjoints, which preserve Db (coh X).

41

NONCOMMUTATIVE DEFORMATIONS AND FLOPS∗
Proof. (1) We have FM(QX ) = Rp2∗ (QX ⊗L
OX×X p1 (−)), which we established preserves
Db (coh X) in 7.4(2). For the other, we note that by [H06, 5.12]
∗
L
∗
∗
∼
Rp1∗ (QX ⊗L
OX×X p2 (−)) = Ri∗ ◦ Rp1∗ (QU ⊗OU ×U p2 (−)) ◦ i(7.H)

where by abuse of notation pi also denote the projections U ×U → U . But again using the∗argument in 6.16 it follows immediately that Rp1∗ (QU ⊗L
OU ×U p2 (−)) is the compositionV ∨ ⊗L −Λcon ⊗L −RHomU (V ∨ ,−)Db (coh U ) −−−−−−−−−−→ Db (mod Λop ) −−−−−Λ−→ Db (mod Λop ) −−−−Λ−→ Db (coh U )so every object in the image is a bounded complex of coherent sheaves filtered by E. Since
Ri∗ E = i∗ E by (2.B) and is certainly bounded coherent, it follows that (7.H) preserves
Db (coh X).
(2) The fact that WX , QX ∈ Db (coh X × X) is a consequence of 7.4(2) and 7.5(1). Thusby (1) and 7.5(3), QX has finite projective dimension over both factors. Since O∆,X also
has this property, by two-out-of-three so does WX . The remaining statements now follow,!
again using 7.5(3).
Remark 7.7. If one could establish that the inverse twist TE∗ has both left and rightadjoints that preserve Db (coh X) under the weaker assumption that X is only quasiprojective, then the proof that TE∗ is an autoequivalence in the next subsection goes
through in that level of generality.
7.4. Proof of equivalence for global flops. In this subsection, keeping the assumptions
and setting of §4, we give the proof that the inverse twist TE∗ is a derived equivalence inthe situation of a flopping contraction. As above, our strategy is to reduce as much as
possible to the local case and apply the results of §6. With this in mind, the following isuseful.
Proposition 7.8. The following diagram is naturally commutative:
D(Qcoh U )Ri∗

D(Qcoh X)

TE∗

TE∗

U

D(Qcoh U )Ri∗

D(Qcoh X)

Proof. Translating the question into Fourier–Mukai kernels, by [H06, 5.12] we require(i × 1)∗ WX ∼= R(1 × i)∗ WU .

(7.I)

We evaluate the left-hand side of (7.I). By definition (7.3), WX = Cone(φX )[−1], whereφX factors through the morphismη∆O∆,X −−→ R(i × i)∗ O∆,U .
By definition (7.C), the morphism η∆ is given (up to isomorphism) as a pushforward
R∆X∗ η of a unit morphism η associated to the adjunction i∗ ⊣ Ri∗ . A triangular identity
for this adjunctions yields that i∗ η is an isomorphism, and so using flat base change aroundthe following Cartesian square
U

1×i

i×1

i

X

U ×X

∆X

we find that (i × 1)∗ η∆ is also an isomorphism.

X × X,

42

WILL DONOVAN AND MICHAEL WEMYSSIt then follows, using the definition of φX (7.B), that(i × 1)∗ WX ∼= (i × 1)∗ Cone(R(i × i)∗ φU )[−1]∼= (i × 1)∗ R(i × i)∗ Cone(φU )[−1]∼= (i × 1)∗ R(i × 1)∗ R(1 × i)∗ Cone(φU )[−1].
Now i×1 is an open embedding and thence (i×1)∗ R(i×1)∗ = Id, and WU ∼= Cone(φU )[−1]!
by (7.A), so we have shown (7.I), and the statement is proved.
We now build up our next main result in 7.12 via a series of lemmas. Recall thatΩ ⊆ Db (coh X) is called a spanning class if(1) RHomX (a, c) = 0 for all c ∈ Ω implies that a = 0.
(2) RHomX (c, a) = 0 for all c ∈ Ω implies that a = 0.
To obtain a spanning class, usually one uses a Serre functor, which for us is SX from
6.18. However in our setting, since E need not be perfect, obtaining a spanning class isa little delicate since the usual candidate E ∪ E ⊥ will not do. The solution is to use E,
which we already know is perfect.
Lemma 7.9. With the assumptions as in §4, defineE ⊥ := {a ∈ Db (coh X) | RHom(E, a) = 0},⊥

E := {a ∈ Db (coh X) | RHom(a, E) = 0}.

Then E ⊥ = ⊥ E, and Ω := E ∪ E ⊥ is a spanning class in Db (coh X).
Proof. For the first statement, note that
7.2

6.18

RHomX (a, E) = 0 ⇐⇒ RHomX (a, E ⊗ ωX ) = 0 ⇐⇒ RHomX (a, SE) = 0.
On the other hand
6.17

RHomX (E, a) = 0 ⇐⇒ RHomT (RHomX (E, a), ωT ) = 0 ⇐⇒ RHomX (a, SE) = 0
where T = C. Combining gives E ⊥ = ⊥ E.
Now we check that Ω := E ∪ E ⊥ is a spanning class. If RHomX (a, c) = 0 for all c ∈ Ω,
then a ∈ ⊥ E and so by the above a ∈ E ⊥ . Thus taking c := a gives RHomX (a, a) = 0 andhence a = 0. Also, if RHomX (c, a) = 0 for all c ∈ Ω, in particular RHomX (E, a) = 0 andso a ∈ E ⊥ . As above, this gives a = 0.
!
The following is expected from the standard theory of Seidel–Thomas twists. We notehere that in our setting the proof follows immediately from the local model, bypassing
subtleties with the octahedral axiom in the ‘usual’ proof [T07, p124].
Lemma 7.10. TE∗ (E) ∼= E[2], TE∗ (i∗ E) ∼= i∗ E[2] and functorially Id |E ⊥ ∼= TE∗ |E ⊥ .
Proof. The first and second assertions follow from our local mutation calculation:
(2.B)

7.8

6.20

7.8

6.20

(2.B)

TE∗ (E) ∼= Ri∗ TE∗U (EU ) ∼= Ri∗ EU [2] ∼= TE∗ (Ri∗ EU ) ∼= E[2].
(2.B)

(2.B)

TE∗ (i∗ E) ∼= TE∗ (Ri∗ E) ∼= Ri∗ TE∗U (E) ∼= Ri∗ E[2] ∼= i∗ E[2].
For the last assertion, since any morphism between kernels induces a natural transformation between the resulting Fourier–Mukai functors, the morphism WX → O∆,X in∗
7.3 induces a natural transformation β : TE∗ → Id. Since FM(QX ) = Ri∗ ◦ GU ◦ GLA
U ◦i
by (7.G), and we have
6.11

(2.B)

∗
∗
∼
∼
GLA
U (i a) = RHomAcon (RHomU (i a, EU ), Acon ) = RHomAcon (RHomX (a, E), Acon )which is zero for all a ∈ ⊥ E, we see that βa : TE∗ (a) → a is an isomorphism for all a ∈ ⊥ E.
!
Thus functorially Id |⊥ E ∼= TE∗ |⊥ E . Since E ⊥ = ⊥ E by 7.9, the result follows.

NONCOMMUTATIVE DEFORMATIONS AND FLOPS

43

We are almost ready to prove our main result. The key point below is that we checkfully faithfulness on the spanning class from 7.9, where the hard case (case 1 in the proofof 7.12 below) is already taken care of by the known derived equivalence on the local
model (§6). Once we know our functor is fully faithful, the problem then is that the usual
Serre functor trick to establish equivalence does not work in our possibly singular setting.
However we are able to bypass this with the following.
Lemma 7.11. Let C be a triangulated category, and F : C → C an exact fully faithful
functor with right adjoint F RA . Suppose that there exists an object c ∈ C such that
F (c) ∼= c[i] for some i, and further F (x) ∼= x for all x ∈ c⊥ . Then F is an equivalence.
Proof. Since F is fully faithful, by [H06, 1.24] we have∼Id −→ F RA ◦ F.

(7.J)

Now we know that F (a) ∼= a for all a ∈ c⊥ , hence it follows from (7.J) that F RA (a) ∼=a⊥for all a ∈ c .
Now by [H06, 1.50], F is an equivalence provided that F RA (x) = 0 implies thatx = 0. Hence suppose that F RA (x) = 0, then trivially HomC (c, F RA (x)[j]) = 0 for allj ∈ Z. By adjunction HomC (F (c), x[j]) = 0 for all j ∈ Z, so since F (c) ∼= c[i], we have
HomC (c, x[j]) = 0 for all j ∈ Z, i.e. x ∈ c⊥ . But then F RA (x) = x, so x = 0, and we are
done.
!
Theorem 7.12. With the assumptions as in §4, the inverse twistTE∗ : Db (coh X) → Db (coh X)
is an equivalence.
Proof. We know that TE∗ : Db (coh X) → Db (coh X) by 7.4, and has left and right adjoints
that preserve Db (coh X) by 7.6. Combining 7.10 with 7.11, it suﬃces to show that TE∗ isfully faithful.
We know from 7.9 that Ω = E ∪ E ⊥ is a spanning class, so by [H06, 1.49] we just needto check thatTE∗ : HomD(X) (G1 , G2 [i]) → HomD(X) (TE∗ (G1 ), TE∗ (G2 )[i])(7.K)

is a bijection for all G1 , G2 ∈ Ω and all i ∈ Z. This is taken care of in the following fourcases.
Case 1: G1 = G2 = E. Since Ri∗ EU = E, by 7.8 we have a commutative diagram
HomD(U) (EU , EU [i])

Ri∗
∼

TE∗

TE∗

U

HomD(U) (TE∗U (EU ), TE∗U (EU )[i])HomD(X) (E, E[i])Ri∗∼HomD(X) (TE∗ (E), TE∗ (E)[i])We already know that the left-hand map is a bijection since TE∗U is an equivalence, hence
the right-hand map is also a bijection.
Case 2: G1 = E, G2 ∈ E ⊥ . The left-hand side of (7.K) is H i RHomX (E, G2 ), which is zero.
By 7.10, the right-hand side of (7.K) is Hom(E[2], G2 [i]) = H i−2 RHomX (E, G2 ), which
also equals zero.
Case 3: G1 ∈ E ⊥ , G2 = E. Since E ⊥ = ⊥ E by 7.9, we obtain RHomX (G1 , E) = 0. As in
Case 2, this implies that both sides of (7.K) are zero.
Case 4: G1 , G2 ∈ E ⊥ . Since functorially Id |E ⊥ ∼= TE∗ |E ⊥ by 7.10, (7.K) is bijective for all⊥
G1 , G2 ∈ E .
!

44

WILL DONOVAN AND MICHAEL WEMYSS

7.5. The global noncommutative twist. In this subsection we define the noncommutative twist functor, and show it sits in the expected functorial triangle. We consider the
triangle of Fourier–Mukai kernels
φX

WX → O∆,X −−→ QX →(7.L)

from 7.3, which by 7.4 give the Fourier–Mukai functors TE∗ , Id and GX ◦ GLA
X respectively,
which all preserve Db (coh X). Set D := (−)∨ ⊗ p∗1 ωX [dim X]. By 7.5(3) and 7.6(2),applying D to (7.L) induces the right adjoints, which also preserve Db (coh X).
Definition 7.13. We define the noncommutative twist TE : Db (coh X) → Db (coh X) to
be FM(D(WX )).
Proposition 7.14. With the assumptions as in §4,(1) TE is the inverse functor to TE∗ . In particular, TE is an autoequivalence.
(2) There is a functorial triangle
GX ◦ GRA
X → Id → TE →L∼
on Db (coh X), where GX ◦ GRA
X = RHomX (E, −) ⊗Acon E.

Proof. (1) Since TE∗ is an equivalence, its right adjoint TE is its inverse.
(2) By 7.5, the triangle
D(QX ) → D(O∆,X ) → D(WX ) →gives the right adjoints. The Fourier–Mukai functor of D(QX ) gives the right adjoint toRAL
GX ◦ GLA
X , namely GX ◦ GX , which by (7.E) is isomorphic to RHomX (E, −) ⊗Acon E.
The Fourier–Mukai functor of D(O∆,X ) gives the right adjoint to the identity, which is
the identity. By definition, the Fourier–Mukai functor of D(WX ) gives TE . The result
follows.
!
7.6. Relation with flop–flop functors. Finally, in this subsection we show that thenoncommutative twist TE is inverse to an autoequivalence naturally associated with theflop, namely the flop–flop functor FF. We thence obtain an intrinsic description of FF interms of the noncommutative deformation theory of the curve, without assuming that theflop exists.
Recall that the category 0 Per X of perverse sheaves is defined as⎧(⎫(
H i (a) = 0 if i ̸= 0, −1⎨⎬(
0
Per X := a ∈ Db (coh X) (( f∗ H −1 (a) = 0, R1 f∗ H 0 (a) = 0
,
(7.M)
⎩
( Hom(c, H −1 (a)) = 0 for all c ∈ C ⎭with C := {c ∈ coh X | Rf∗ c = 0}.
The following two straightforward lemmas, 7.15 and 7.16, record the properties of TE
and FF that we will need. In 7.17, we deduce that the functor Ψ := TE ◦ FF preserves
the category 0 Per X above, and preserves the numerical equivalence class of skyscraper
sheaves. The proof that TE is inverse to FF then concludes in 7.18, following an argument
given by Toda.
Lemma 7.15. The twist functor TE has the following properties:(1) Rf∗ ◦ TE ∼= Rf∗ .
(2) TE (OX ) ∼= OX .
(3) TE (E) ∼= E[−2].
Proof. Property (1) may be deduced by applying Rf∗ to the triangle representation of
TE in 7.14(2), using the fact that Rf∗ ◦ GX ∼= 0, which we prove now. The commutative

45

NONCOMMUTATIVE DEFORMATIONS AND FLOPS

diagramiUfSpec R

X
f

icon

Xcon

gives
Rf∗ ◦ GX = Rf∗ Ri∗ (− ⊗LAcon EU )∼ Ricon∗ Rf∗ (− ⊗L EU )
=
Acon

∼
= Ricon∗ (− ⊗LAcon Rf∗ EU ).
As EU is filtered by E, it follows that Rf∗ EU ∼= 0, and the property follows.
To obtain (2), we note that f ! OXcon = OX since f is crepant, so
RHomX (E, OX ) = RHomX (E, f ! OX ) ∼= RHomX (Rf∗ E, OX ) = 0,
con

con

con

again using Rf∗ E ∼= 0. Hence TE (OX ) ∼= OX by 7.14(2).
The last property (3) follows from 7.10.

!

Lemma 7.16. The flop–flop functor FF has the following properties:(1) Rf∗ ◦ FF ∼= Rf∗ .
(2) FF(OX ) ∼= OX .
(3) FF(E) ∼= E[2].
Proof. We denote the flop by X + . Property (1) follows immediately since the individualflop functors commute with the pushdown by [B02, (4.4)]. Further, by [B02, (4.8)] noting
the correction to the sign in [T14, (27)], the flop functor induces an equivalence
0

Per X → −1 Per X + .

By morita theory, under this equivalence projectives must go to projectives, and simples
must go to simples. By Van den Bergh’s description of these objects [V04, 3.2.7, 3.5.7,
3.5.8], and the fact that the flop functor commutes with pushdown, it follows that the flopfunctor must take OX to OX + , and E to E[1]. Applying the flop functor again, properties!
(2) and (3) follow.
The following observations are used in the proof of 7.18, where we will show that thefunctor Ψ defined below is in fact isomorphic to the identity.
Proposition 7.17. The functor Ψ := TE ◦FF is an autoequivalence of Db (coh X), preserving the category 0 Per X of perverse sheaves given in (7.M). It also preserves the numericalequivalence class of skyscraper sheaves, that is for points p ∈ X we haveχ(L, Op ) = χ(L, Ψ(Op ))for all locally free sheaves L on X, where χ is the Euler characteristic. For definitions,see for instance [BBH, 7.61].
Proof. Note that Ψ is an equivalence by construction, using 7.14(1), and also that combining 7.15 and 7.16 gives(1) Rf∗ ◦ Ψ ∼= Rf∗ ,
(2) Ψ(E) ∼= E.
We first argue that Ψ preserves the category 0 Per X of perverse sheaves. As explained in[B02, Lemma 3.2], this category is obtained as the intersection, for perversity p = 0, of
the subcategoriespA≤0 := {a ∈ Db (coh X) | Rf∗ a ∈ Db (coh Xcon )≤0 , a ∈ ⊥ (C >p )},pA≥0 := {a ∈ Db (coh X) | Rf∗ a ∈ Db (coh Xcon )≥0 , a ∈ (C <p )⊥ },46

WILL DONOVAN AND MICHAEL WEMYSS

where C >p := {c ∈ Db (coh X) | Rf∗ c = 0, H i (c) = 0 for i ≤ p} and C <p is definedsimilarly. Now the property (1) implies that Ψ−1 preserves {c ∈ Db (coh X) | Rf∗ c = 0},
and using property (2) it is easy to see that Ψ−1 preserves the induced standard t-structureon {c ∈ Db (coh X) | Rf∗ c = 0}, and hence preserves C >p and C <p . From this, it follows
that Ψ−1 preserves p A≤0 and p A≥0 , and hence 0 Per X, as claimed.
Next we argue that Ψ preserves the numerical equivalence class of skyscraper sheaves.
Taking a sheaf Op for a point p ∈ X, we consider two cases. If p ̸∈ C, then f is an
isomorphism at p. Property (1) above then gives that Ψ(Op ) ∼= Op , and so there isnothing to prove in this case. If p ∈ C, then Op is no longer simple in 0 Per X but it isfiltered by the simples {ωC [1], E}. Thus we have a series of exact triangles
a1

b0

a2

...

an−1

Op
(7.N)

b1

b2

bn

where all bi ∈ {ωC [1], E}. But Ψ is an equivalence preserving 0 Per X, so it must permute
the simples. Since we already know that Ψ fixes E and Ox for x ∈ X\C, it follows thatΨ also fixes ωC [1] and hence fixes all the simples.
Because of this, we also have a series of exact triangles
Ψa1

b0

Ψa2

...

ΨOp

Ψan−1

(7.O)
b1

b2

bn

Using additivity of χ(L, −) on triangles, the left-hand triangles in (7.N) and (7.O) imply
that χ(L, a1 ) = χ(L, Ψa1 ). Inducting to the right on (7.N) and (7.O) gives χ(L, Op ) =χ(L, ΨOp ), as required.
!
The following is the main result of this subsection.
Proposition 7.18. TE is an inverse of FF, the Bridgeland–Chen flop–flop functor of §1.4.
Proof. We use the argument indicated in [T07, end of proof of 3.1], but write the proof infull to show that there is nothing particular about commutative deformations or smoothness used in loc. cit. that aﬀects the proof. As above, consider the autoequivalenceΨ := TE ◦ FF.
Using 7.15 and 7.16 we know that Ψ(OX ) ∼= OX , and by 7.17 Ψ preserves the category
Per X of perverse sheaves, and preserves the numerical equivalence class of skyscraper
sheaves.
We now follow the argument in [T06, end of proof of 6.1] to deduce that in factΨ∼
= Id. We first show that for points p ∈ X, Ψ(Op ) is a skyscraper sheaf. Most of thework here is to show that Ψ(Op ) is a sheaf, as the result will then follow from the above
remark on numerical equivalence classes. We take the short exact sequence of sheaves
0

0 → Ip → OX → Op → 0,
and observe that it is also a short exact sequence in 0 Per X (for this, we need only checkthat R1 f∗ = 0 for each sheaf). Since Ψ preserves 0 Per X, and Ψ(OX ) ∼= OX ,
0 → Ψ(Ip ) → OX → Ψ(Op ) → 0
is also an exact sequence in 0 Per X. The perversity gives that H 1 Ψ(Ip ) = 0, and so theassociated long exact sequence yields a surjection
OX → H 0 Ψ(Op ).

(7.P)

NONCOMMUTATIVE DEFORMATIONS AND FLOPS

47

We claim that the target of this surjection must be non-trivial. Otherwise, the perversity
implies that Ψ(Op ) = H −1 Ψ(Op )[1], which then implies that for L suitably anti-ample
7.17

1 = χ(L, Op ) = χ(L, Ψ(Op )) = −χ(L, H −1 Ψ(Op )) ≤ 0,
which is a contradiction. Hence the map (7.P) above is non-trivial. Furthermore, it isadjoint to a map OXcon → f∗ H 0 Ψ(Op ), which must consequently also be non-trivial, andso we find thatf∗ H 0 Ψ(Op ) ̸= 0.
(7.Q)>1
Now the fibres of f have dimension at most 1, so R f∗ = 0 and therefore theGrothendieck spectral sequence
E p,q = Rp f∗ (H q Ψ(−)) → Rp+q (f∗ ◦ Ψ)(−) ∼= Rp+q f∗ (−)
2

collapses. Hence the exact sequence of low-degree terms applied to Op gives a short exact
sequence
0 → R1 f∗ H −1 Ψ(Op ) → f∗ (Op ) → f∗ H 0 Ψ(Op ) → 0.
The middle term is just Of (p) . This surjects onto the last term, which we know by (7.Q)
to be non-zero. We deduce that R1 f∗ H −1 Ψ(Op ) = 0. Recall that Ψ(Op ) ∈ 0 Per X, andhence f∗ H −1 Ψ(Op ) = 0 also, so we conclude thatRf∗ H −1 Ψ(Op ) = 0,
so that H −1 Ψ(Op ) ∈ C, the category used in the definition of 0 Per X. Using once more
the fact that Ψ(Op ) ∈ 0 Per X, we see from the definition that
HomX (H −1 Ψ(Op ), H −1 Ψ(Op )) = 0,
and thence that H −1 Ψ(Op ) = 0, and that Ψ(Op ) is a sheaf. This sheaf is numericallyequal to a skyscraper sheaf by 7.17, and thence by [BBH, 7.62(2)] is itself a skyscraper
sheaf.
Finally, we therefore have that Ψ(Op ) ∼= Oφ(p) for all p and some function φ. We
deduce that φ is an automorphism because Ψ is an equivalence. ThereforeΨ∼= L ⊗ φ∗ (−)
for some line bundle L, which must be trivial as Ψ(OX ) ∼= OX . We must have f ◦ φ = fbecause Ψ commutes with the contraction, and so φ = Id. This completes the proof. !

48

WILL DONOVAN AND MICHAEL WEMYSSAppendix A. List of Notations
η, ε
(−)ab
(−)e
pd
Mod A (mod A)
add M
CM R
CArt
D (Db )
D−, D+FM(W)F RA , F LA
F, F′FF
Alg1 (CAlg1 )Art1 (CArt1 )Γ
a, b ∈ A
Pairs(A, Γ)
DefaA (cDefaA )
C⊂X
C red
E
p ∈ Xcon
f
Ucon
R
U
i
V
N
T
0
Per U
Λ
Λcon
Icon
[R]
#
U
# R
R,
#
Λ

OU% ⊕ N1
N1
F
A
Acon
E, EU
F , FU
Adef
wid(C), cwid(C)µ, νΦunit and counit morphisms
abelianizationenveloping algebra
projective dimension(finitely generated) right A-modules
summands of finite sums of module M
category of maximal Cohen–Macaulay R-modules
category of local commutative artinian C-algebras
derived category (bounded)
derived category bounded above, and below
Fourier–Mukai functor with kernel W
right and left adjoints, for a functor F
flop functors
flop–flop functor
category of 1-pointed (commutative) C-algebras
category of 1-pointed artinian (commutative) C-algebras
1-pointed artinian C-algebra
objects of abelian category A
Γ-objects in A
noncommutative (commutative) deformation functor
curve C in a 3-fold X
reduced subscheme associated to C (∼= P1 )
red
sheaf OP1 (−1) supported on C
singular point p in contracted 3-fold Xconcontraction morphismaﬃne open neighbourhood of singular point pC-algebra, with Ucon ∼= Spec R:= f −1 (Ucon ), open neighbourhood of curve C
open embedding of U in X:= OU ⊕ N , tilting bundle on U:= f∗ N:= HomU (V, E)
perverse sheaves on U
endomorphism algebra EndU (V)contraction algebra associated to Λ
ideal associated to Λconideal of morphisms of Λ which factor through add R
formal fibre of f over singular point p
completion of R at pcompletion of Λ#
tilting bundle on U:= f∗ N1
morita equivalence#basic algebra, morita equivalent to Λ
contraction algebra
noncommutative universal sheaf on X, and on U
commutative universal sheaf on X, and on U:= EndX (E), noncommutative deformation algebra
width, and commutative width, of C
right and left mutation
mutation functor

§1.4
§1.4
2.1
2.1
2.1
§2
2.3
2.4
§2.2
§2.2
§2.2
§2.2
§2.2
§2.2
§2.2
§2.2
§2.2
§2.3
§2.3
§2.3
§2.3
§2.3
2.9
2.9
2.9
§2.4
§2.4
§2.4
§2.4
§2.4
§2.4
2.12
2.12
§3.2
§3.2
3.8
3.11
5.1
§5.3

NONCOMMUTATIVE DEFORMATIONS AND FLOPS

S∗
Tcon , Tcon∗
T EU , T EU
OYΓ
WU , Q UTE∗WX , Q X
TE

Serre functor
algebraic twist, and inverse
noncommutative twist on U , and inverse:= OY ⊗C Γ
FM kernel for TE∗U , and related kernel
inverse twist on X
FM kernel for TE∗ , and related kernel
noncommutative twist on X

49

6.4
6.9
6.13
6.14
§7.2
7.3
7.3
7.13

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[K94]
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[LV]
[P83]
[R83]
[S08]
[ST01]
[S13]
[T06]
[T07]
[T14]
[V04]
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Will Donovan, The Maxwell Institute, School of Mathematics, James Clerk Maxwell
Building, The King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ, UK.
E-mail address: Will.Donovan@ed.ac.uk
Michael Wemyss, The Maxwell Institute, School of Mathematics, James Clerk Maxwell
Building, The King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ, UK.
E-mail address: wemyss.m@googlemail.com