NONCOMMUTATIVE ENHANCEMENTS OF CONTRACTIONSarXiv:1612.01687v1 [math.AG] 6 Dec 2016

WILL DONOVAN AND MICHAEL WEMYSS
Abstract. Given a contraction of a variety X to a base Y , we enhance the locus in Y
over which the contraction is not an isomorphism with a certain sheaf of noncommutative rings D, under mild assumptions which hold in the case of (1) crepant partial
resolutions admitting a tilting bundle with trivial summand, and (2) all contractions
with fibre dimension at most one. In all cases, this produces a global invariant. In
the crepant setting, we then apply this to study derived autoequivalences of X. We
work generally, dropping many of the usual restrictions, and so both extend and unify
existing approaches. In full generality we construct a new endofunctor of the derived category of X by twisting over D, and then, under appropriate restrictions on
singularities, give conditions for when it is an autoequivalence. We show that these
conditions hold automatically when the non-isomorphism locus in Y has codimension 3
or more, which covers determinantal flops, and we also control the conditions when
the non-isomorphism locus has codimension 2, which covers 3-fold divisor-to-curve
contractions.

Contents
1. Introduction
2. Global Thickenings
3. A Universal Sheaf E
4. Spherical Properties via Cohen–Macaulay Modules
5. Global Twist Functors
6. One-Dimensional Fibre Applications
References

1
5
12
15
19
22
25

1. Introduction
It has become increasingly clear that various aspects of birational geometry should beenhanced into a mildly noncommutative setting. One example of this, namely associating
noncommutative algebras to certain contractions, has recently yielded many new results,
including: algorithmic ways to relate minimal models based on cluster theory [W], newinvariants for flips and flops [DW1] linked to Gopakumar–Vafa invariants [T2], the braidingof flop functors [DW3], noncommutative versions of curve counting [T3], a full conjectural
analytic classification of 3-fold flops [DW1, HT, BW], and, in addition, the first new
examples of 3-fold flops since 1983 [BW].
However, a major technical and philosophical drawback of many of these constructions
is that they are local in nature, or apply only to contractions of curves. In this paper, we
work in a much more general setting. We take a contraction f : X → Y satisfying mild
characteristic-free assumptions, and enhance the non-isomorphism locus Z in Y with acertain sheaf of noncommutative rings. Amongst other things, we use this sheaf to give a
new class of derived autoequivalences.
2010 Mathematics Subject Classification. Primary 14F05; Secondary 14D15, 14E30, 16E45, 16S38, 18E30.
The first author was supported by World Premier International Research Center Initiative (WPI), MEXT,
Japan, and JSPS KAKENHI Grant Number JP16K17561. The second author was supported by EPSRC
grant EP/K021400/1.
1

2

WILL DONOVAN AND MICHAEL WEMYSSIn this paper we accomplish the following.
• In an axiomatic framework sketched in §1.1, we construct a noncommutative
ringed space (Z, D) where Z is the non-isomorphism locus in Y . In 3-folds thisapplies to flopping, flipping, and divisor-to-curve contractions, but also works inarbitrary dimension, and for contractions with higher-dimensional fibres.
• In a crepant setting given in §1.2, using the sheaf of noncommutative rings D
above, we construct an endofunctor TwistX of D(Qcoh X) associated to the contraction f , and establish criteria for it to be an autoequivalence.
The main benefit of this approach is its generality: we show in 2.5(1) that the axiomatic framework applies when f is a crepant resolution of a Gorenstein d-fold Y , withmild restrictions which hold in many known settings, including those of Haiman [H1],Bezrukavnikov–Kaledin [BK], Procesi bundles [L], Toda–Uehara [TU], Springer resolutions of determinantal varieties [BLV], and all known 3-fold crepant resolutions including
derived McKay correspondence [BKR]. We note in 2.7 that it covers all 3-fold projective
crepant resolutions, if the Craw–Ishii conjecture holds. Further we show in 2.5(2) that
our framework applies to all contractions with fibres of dimension at most one.
As the construction of D is the most subtle part of the paper, we first briefly outline
the local-to-global problems that arise in one specific example. Consider the following
crepant divisor-to-curve contraction, given explicitly in 2.25. Y is singular along a onedimensional locus Z, and above every point in Z is an irreducible curve.

X
f
Y

Z

khhx,yii
x2 ,y 20 zk[[x]]

At every closed point z ∈ Z, [DW2] constructs a noncommutative deformation algebra
Acon,z , which induces a universal sheaf Ez on the formal fibre above z. In the above
example, at every closed point of Z away from the origin, Acon,z is isomorphic to k[[x]],and at the origin it has the form shown above. The question is whether there exists a
global sheaf of algebras D on Z which specialises, complete locally, to the algebras Acon,z .
Similarly one can ask whether the universal sheaves Ez , as z varies over Z, can be glued
into a single coherent sheaf E on X.
The key insight in this paper is that this can be done, but not on the nose; weconstruct a global D that recovers the algebras Acon,z up morita equivalence, and in (1.B)
a global E that recovers the Ez up to additive equivalence. In the above example thismeans that D completed at a point z away from the origin is the ring of 2 × 2 matrices
over k[[x]], instead of k[[x]] itself. This is rather harmless, since to extract local invariants,
we pass to the basic algebra [DW1, §3], and for construction of derived autoequivalences,passing through morita equivalences is a mild and necessary procedure.
1.1. Construction of invariant (Z, D) and global E. We sketch this briefly, before
describing our main results and applications. The construction does not use deformation
theory, or any restrictions on singularities or fibre dimension. Full details are given in §2.
By a contraction, we mean a projective birational map f : X → Y between ddimensional normal varieties over an algebraically closed field k, satisfying Rf∗ OX = OY ,where we assume that Y is quasi-projective. Write Z for the locus of points of Y above
which f is not an isomorphism.

NONCOMMUTATIVE ENHANCEMENTS OF CONTRACTIONS

3

We then assume that there is a vector bundle P on X containing OX as a summand,such that(1) The natural map f∗ EndX (P) → EndY (f∗ P) is an isomorphism.
(2) The functor
Rf∗ RHomX (P, −) : Db (coh X) → Db (mod B)(1.A)

is an equivalence, where B := f∗ EndX (P) is considered as a sheaf of OY -algebras.
Given this setup, we write Q := f∗ P, so that B ∼= EndY (Q). We then define asubsheaf I of B consisting of local sections that at each stalk factor through a finitely
generated projective module, in 2.8. We show in 2.9 that I is naturally a subsheaf of
two-sided ideals of B.
Thus we may also view I as a subsheaf of B op , and define D := B op /I, which wecall the sheafy contraction algebra. For most purposes, taking the opposite ring structure
can be ignored. In 2.11 we give two alternative descriptions of I, which are importantsince they allow us to control local sections of D, to such an extent that we can prove thefollowing.
Theorem 1.1 (=2.16, Global Contraction Theorem). SuppY D = Z.
It follows that D is naturally a sheaf of algebras on the locus Z, and thus we canview the ringed space (Z, D) as a noncommutative enhancement of Z.
Remark 1.2. When the locus Z can be realized as a moduli spaces M of stable sheaves,Toda [T4] constructs, under restrictions on dim Z, another noncommutative enhancementof Z which is different to ours, since it is unusual for the stalks of D to be local rings; seealso 2.20 and 2.26.
The noncommutative sheaf of rings D constructed is naturally a sheaf on the base
Y of the contraction f : X → Y . We next lift this to give an object on X, by observing
that by construction D is a factor of B op , so it also carries the natural structure of aB-bimodule. In particular, we can view D as an object of Db (mod B op ), and hence, acrossa dual version of (1.A), it gives an object∗
E := f −1 D ⊗L
f −1 Bop P(1.B)

of Db (coh X). In general E need not be a sheaf. In our most general setup, we prove thefollowing.
Proposition 1.3 (=3.2). Rf∗ E = 0. In particular, if E is a sheaf then SuppX E lies inthe exceptional locus.
The case of one-dimensional fibres is particularly pleasant, and a typical example issketched below.

X

Supp EY

Supp DThe following are our main results in this setting. In particular, this globalises the local
noncommutative deformation theory as studied in [DW1, DW2, BB, K1].
Theorem 1.4. Suppose that f : X → Y is a contraction where the fibres have dimension
at most one. Then the following hold.

4

WILL DONOVAN AND MICHAEL WEMYSS(1) (=2.24) The completion of D at a closed point z ∈ Z is morita equivalent tothe algebra that prorepresents noncommutative deformations of the reduced fibre
over z.
(2) (=3.5) E is a sheaf.
(3) (=3.7) SuppX E equals the exceptional locus of f .
(4) (=3.8) The restriction of E to the formal fibre above a closed point z ∈ Z recovers
the universal sheaf Ez from noncommutative deformation theory, up to additive
equivalence.
The fact that in the one-dimensional fibre setting E is a sheaf is our main motivation
for considering B op instead of B.
1.2. Applications to autoequivalences TwistX . We next turn our attention to crepant
contractions and autoequivalences. One benefit of the construction of E on X is that, using
the natural exact sequence
0 → I → B op → D → 0,
we can construct in §5, under our most general setup in §1.1, a functor TwistX , which sits
in a functorial triangle
f −1 Rf∗ RHomX (E, a) ⊗L
f −1 D E → a → TwistX (a) → .
For a precise description of the terms in these expressions, we refer the reader to §5,but remark here that when Z is a point, TwistX reduces to a noncommutative twist
over a contraction algebra, as studied in [DW1, DW3]. The existence of a global sheafy
contraction algebra D allows us to avoid delicate local-to-global gluing arguments.
In general, TwistX is not an equivalence, but we have the following criterion in termsof a Cohen–Macaulay property of D on Y , under a restriction to hypersurface singularities.
There are two main reasons for this restriction on singularities (although it is not usedeverywhere), outlined in 2.20 below: note however that it holds automatically in the twomain applications in this paper, namely to Springer resolutions of determinantal varietiesof n × n matrices in 1.6, and to 3-fold divisor-to-curve contractions in 1.8.
Theorem 1.5 (=5.7). Under the general assumptions of 2.3, assume that f is crepant,bY,z are hypersurfaces for all closed points z ∈ Z. Then the following are equivalent.
and O(1) D is a Cohen–Macaulay sheaf on Y , and E is a perfect complex on X.
(2) D is relatively spherical (in the sense of 5.6) for all closed points z ∈ Z.
If these conditions hold, and they are automatic provided that dim Z ≤ dim Y − 3, thenthe functor TwistX is an autoequivalence of Db (coh X).
We remark that both parts of 1.5(1) can fail in general, and indeed if Z is not
equidimensional, then D is not relatively spherical for some z ∈ Z.
The last statement in 1.5 ensures that our framework recovers previous autoequivalences associated to flopping contractions [T1, DW1, BB], since the assumptions of 1.5
are satisfied in the one-dimensional fibre flops setting described in [V1, Theorem C], but
the main advantage of 1.5 is that it includes other interesting settings, which we briefly
outline here.
Our first new application is to the varieties of singular n × n matrices, namely the determinantal varieties k[xij ]/(det x). For each such variety, the Springer resolution admits
a suitable tilting bundle by work of Buchweitz, Leuschke, and Van den Bergh [BLV].
Corollary 1.6 (=5.8). Consider the Springer resolution X → Y of the variety of singularn × n matrices. Then TwistX is an autoequivalence of X.
We then establish results for one-dimensional fibre contractions which are not an
isomorphism in codimension two, where the conditions in 1.5 are not automatic. Amongst
others, this includes partial resolutions of Kleinian singularities, and 3-fold crepant divisorto-curve examples. Leveraging our control of E in this setting, which by 1.4 is a sheaf

NONCOMMUTATIVE ENHANCEMENTS OF CONTRACTIONS

5

with support coinciding with the exceptional locus, we reformulate the first part of 1.5(1),namely D being a Cohen–Macaulay sheaf on Y , in terms of a Cohen–Macaulay property
for E on X.
Theorem 1.7 (=6.2). With the general setup in 1.5, suppose further that the fibres of f
have dimension at most one. Then the following are equivalent.
(1) D is a Cohen–Macaulay sheaf on Y .
(2) For all y ∈ Z, and all x ∈ f −1 (y), we have Ex ∈ CMdim Zy +1 OX,x .
In particular, if these conditions hold, then above every y ∈ Z, the exceptional locus isequidimensional of dimension dim Zy + 1.
Combining with 1.5, the dimension criterion above gives an easy-to-check obstruction
to D being relatively spherical, and we demonstrate this in various examples, such as 6.4.
Furthermore, the conditions above enable us to prove that TwistX is a equivalence in the
motivating divisor-to-curve contraction example.
Theorem 1.8 (=6.5). With the setup in 1.7, suppose in addition that X is smooth anddim X = 3, such that every reduced fibre above a closed point in Z contains precisely one
irreducible curve. Then TwistX is an autoequivalence of X.
This generalises results on twists in families, for instance as in [H3, DS].
Remark 1.9. We expect that, when TwistX is an autoequivalence, it can be expressed
as a twist of a spherical functor as in [A2, A1, AL, K2]. However, we do not address thisquestion in this paper, as it would not simplify our proofs.
1.3. Acknowledgements. We are grateful for conversations with Arend Bayer, Agnieszka Bodzenta, and Yukinobu Toda.
1.4. Conventions. Throughout we work over an algebraically closed field k. Unqualified uses of the word ‘module’ refer to right modules, and mod A denotes the categoryof finitely generated right A-modules. We use the functional convention for composing
homomorphisms, so f ◦ g means g then f . In particular, naturally this makes M ∈ mod A
into an EndA (M )op -module.
For a ∈ A an abelian category, we let add a denote all possible summands of finite
direct sums of a. Given two objects a, b ∈ A where a is a summand of b, we then write [a]for the two-sided ideal of EndA (b) consisting of all morphisms that factor through add a.
2. Global Thickenings
In this section we will noncommutatively enhance the non-isomorphism locus of acontraction f : X → Y which satisfies some mild conditions. We first recall what thismeans, and fix the setting.
Definition 2.1. By a contraction, we will mean a projective birational morphism f : X →Y between normal varieties over k, satisfying Rf∗ OX = OY . It will be assumed that Y
is quasi-projective.
Notation 2.2. Given a contraction f : X → Y , write Z for the locus of (not necessarilyclosed) points of Y above which f is not an isomorphism.
Given a contraction, we will furthermore assume the following condition.
Assumption 2.3. For a contraction f : X → Y , where d = dim X ≥ 2, assume that
there is vector bundle P = OX ⊕ P0 on X, such that the following conditions hold.
(1) The natural map f∗ EndX (P) → EndY (f∗ P) is an isomorphism.

6

WILL DONOVAN AND MICHAEL WEMYSS(2) The bundle P is tilting relative to Y , namely
Rf∗ EndX (P) = f∗ EndX (P) =: B,
and furthermore there is an equivalence∼

Rf∗ RHomX (P, −) : Db (coh X) −→ Db (coh(Y, B)).

(2.A)

Here B is considered as a sheaf of OY -algebras in the sense of [KS, §18.1], and the boundedderived category of modules over B is denoted by Db (coh(Y, B)).
With mind to our applications, we will show in 2.5 below that the assumption israther mild, and holds in general settings. This requires the following easy consequenceof 2.3(2).
Lemma 2.4. Suppose that 2.3(2) holds, V is an affine open subset of Y , and U :=f −1 (V ). Then P|U is a tilting bundle on U .
Proof. Setting Λ := EndU (P|U ), the following diagram commutes.
Db (coh X)

Rf∗ RHomX (P,−)∼

Db (coh(Y, B))|U|V

Db (coh U )RHomU (P|U ,−)Db (mod Λ)In particular, RHomU (P|U , P|U ) ∼= Λ, which gives Ext vanishing. For generation,= B|V ∼
suppose that a ∈ D(Qcoh U ) with RHomU (P|U , a) = 0. We must show that a = 0.
By adjunction RHomX (P, Ri∗ a) = 0 where i : U ֒→ X is the open inclusion. Since P
compactly generates D(Qcoh X), it follows that Ri∗ a = 0, and thus a = 0 since Ri∗ isfully faithful.
Proposition 2.5. Assumption 2.3 is guaranteed to hold in the following two settings.
(1) Y is a Gorenstein d-fold, f is crepant, and X admits a relative tilting bundle
containing OX as a summand.
(2) X is a d-fold, and the fibres of f have dimension at most one.
Proof. (1) Take P to be the relative tilting bundle on X, with P = OX ⊕ P0 . Condition 2.3(2) is immediate. Condition 2.3(1) is local, therefore it suffices to consider affine
Y = Spec R, in which case the condition translates into showing that
EndX (P) → EndR (f∗ P),(2.B)

is an isomorphism. Note that EndX (P) is a maximal Cohen–Macaulay R-module since
f is crepant [IW3, 4.8]. We next claim that f∗ P is also maximal Cohen–Macaulay. The
tilting property for P implies that H i (P0 ) = 0 = H i (P0∗ ) for all i > 0, and so∼ RHomR (Rf∗ P0 , OR )RHomR (f∗ P0 , R) =∼= Rf∗ RHomX (P0 , f ! OR )∼= Rf∗ RHomX (P0 , OX )= Rf∗ (P0∗ ) = f∗ (P0∗ ),where we use Grothendieck duality and crepancy. Thus RHomR (f∗ P0 , R) is concentrated
in degree zero, hence f∗ P0 is maximal Cohen–Macaulay, and thence f∗ P is also. In
particular, (2.B) is a morphism between reflexive R-modules, so since it is an isomorphismin codimension two, it must be an isomorphism.
(2) Let P denote the bundle constructed by Van den Bergh in [V1, 3.3.2], which is relatively
generated by global sections. Condition 2.3(2) is [V1, 3.3.1]. For condition 2.3(1), note
first that f∗ gives rise to the natural morphism
B = f∗ EndX (P) → EndY (f∗ P).

NONCOMMUTATIVE ENHANCEMENTS OF CONTRACTIONS

7

It suffices to check that this is an isomorphism on open affines V of Y , on which it is justthe following, where we write U := f −1 (V ).
f∗ : EndU (P|U ) → EndV (f∗ P|V )(2.C)

By 2.4 P|U is tilting, so Ext1U (P|U , P|U ) = 0. Since further Y is normal, f is projective
birational, and P|U is generated by global sections, it follows from [DW2, 4.3] that (2.C)
is an isomorphism.
Remark 2.6. The relative tilting assumption from 2.5(1) reduces, in the case of affine
Y = Spec R, to the condition that ExtiX (P, P) = 0 for i > 0, and P generates.
Remark 2.7. We remark that if d = 3 in 2.5(1), the assumption that X admits a tilting
bundle is automatic if X is constructed as the moduli of an NCCR of an affine Y [V2,§6]. It is expected that the tilting bundle condition in 2.5(1) for d = 3 always holds, as aconsequence of the Craw–Ishii conjecture and work of Iyama and the second author [IW2,4.18(2)].
2.1. Construction of I. In this subsection we work under the general assumptions of
2.3, and define an ideal subsheaf I of B := f∗ EndX (P) ∼= EndY (f∗ P). To ease notation,
we put Q := f∗ P, so that B ∼
End
(Q).
For
F
a
sheaf,s ∈ F (V ), and v ∈ V , we write=Yhs, V iv for s viewed in the stalk Fv .
Definition 2.8. With notation as above, for each open subset V of Y , define
hs,V iv

I(V ) := {s ∈ EndV (Q|V ) | Qv −−−−→ Qv factors through add OY,v for all v ∈ V },which is a two-sided ideal of B(V ). That is, I consists of local sections of B that at eachstalk factor through a finitely generated projective OY,v -module.
Proposition 2.9. I is a subsheaf of B consisting of two-sided ideals.
Proof. We just need to prove that I is a subsheaf. For opens U ⊆ V of Y , denote therestriction morphisms of B by ρV U . Note that if s ∈ I(V ) then sincehs, V iu = hρV U (s), U iu ,and this factors through a projective for all u ∈ U , it follows that ρV U takes I(V ) toI(U ). The presheaf axioms of B then imply that I is a subpresheaf of B, so it suffices tocheck the two sheaf axioms. S
Suppose then that U = Ui and s ∈ I(U ) is such that s|Ui = 0 for all i. Viewing
this in B, it follows that s = 0.S
Lastly, suppose that U = Ui and there is a collection si ∈ I(Ui ) ⊆ B(Ui ) such thatsi |Ui ∩Uj = sj |Ui ∩Uj

for all i, j. Then since B is a sheaf certainly there exists s ∈ B(U ) such that s|Ui = si
for all i. We claim that s ∈ I(U ), that is hs, U iu factors through add OY,u for all u ∈ U .
Thus let u ∈ U , then certainly u ∈ Ui for some i. But thenhs, U iu = hs|Ui , Ui iu ,which factors through a projective since s|Ui = si ∈ I(Ui ).

We now show that on affine open subsets, I takes a particularly nice form. To
simplify notation, for an affine open subset V = Spec R of Y , write Q for the R-module
corresponding to Q|Spec R , so that B(Spec R) ∼= EndR (Q). Choose a surjection h : F ։ Qwith F a free R-module, and write
α:=(h◦)

HomR (Q, F ) −−−−−→ HomR (Q, Q).
It is standard that this localises toαp =(hp ◦)HomRp(Qp , Fp ) −−−−−−→ HomRp(Qp , Qp ).

8

WILL DONOVAN AND MICHAEL WEMYSS

By the defining property of projective modules, any morphism from an object inP ∈ add R to Q has to factor as
P

F

h

Q,

and any morphism from an object of P ∈ add Rp to Qp has to factor as
P

Fp

hp

Qp .

Hence
{g : Q → Q | g factors through add R} = Im(α),

(2.D)

and
I(Spec R) = {g : Q → Q | gp factors through add Rp for all p ∈ Spec R}= {g : Q → Q | gp ∈ Im(αp ) for all p ∈ Spec R}.

(2.E)

The following lemma says that a morphism factors stalk-locally if and only if it factors
affine-locally.
Lemma 2.10. If Spec R is an affine open subset of Y , thenI(Spec R) = {g : Q → Q | g factors through add R}.
Proof. By (2.D) and (2.E), it suffices to prove thatg ∈ Im α ⇐⇒ gp ∈ Im αp
for all p ∈ Spec R. The direction (⇒) is clear. For the (⇐) direction, suppose thatg ∈ Im(αp ) for all p ∈ Spec R, and consider g + Im α ∈ EndR (Q)/ Im α. Then stalk-locallythis element is zero, hence it is zero (see e.g. [E, Lemma 2.8(a)]), and thus g ∈ Im α. It follows that we can define I in various equivalent ways, without referring to stalks.
Corollary 2.11. Suppose that V is an open subset of Y , and s ∈ B(V ). Then thefollowing conditions are equivalent.
(1) s ∈ I(V ).
S(2) There exists an open affine cover V = Vi such that s|Vi factors through add OVi
for all i.
S(3) For every open affine cover V = Vi , s|Vi factors through add OVi for all i.

Proof. Since I is a sheaf by 2.9, if s ∈ I(V ) then s|Vi ∈ I(Vi ). Hence (1)⇒(3) follows
from 2.10. (3)⇒(2) is obvious, and (2)⇒(1) is clear.

2.2. Sheafy Contraction Algebras. In this subsection we continue to work under thegeneral assumptions of 2.3, but consider the dual bundle V := P ∗ , and A := f∗ EndX (V).
Although not strictly necessary (up to taking opposite algebras, we show in the proof of
2.15 below that it gives the same objects), taking the dual is convenient since in the setting
2.5(2) of fibre dimension at most one, it will allow us to relate the above construction to
our previous work, and deformation theory, in an easier way.
By 2.9, I is a sheaf of two-sided ideals of B := f∗ EndX (P). Since A ∼= B op , we canalso view I as a subsheaf of two-sided ideals of A. Thus we can consider the presheaf
quotient of A by I, which is naturally a presheaf of algebras, and hence its sheafification
D := A/I is a sheaf of algebras on Y .
Definition 2.12. We call D := A/I the sheaf of contraction algebras on Y .
Note that there is a natural OY -algebra structure on D given by a morphism ofsheaves of rings OY → A, as in [KS, §18.1], however this morphism is not injective, as aconsequence of 2.16 below. Locally, the sections of D have a particularly easy form, which
we now describe. For an affine neighbourhood V in Y , consider the following Zariski local
setup obtained by base change.

NONCOMMUTATIVE ENHANCEMENTS OF CONTRACTIONS

9

Setup 2.13. (Zariski local) Suppose that f : U → V = Spec R is a projective birational
morphism between normal varieties over k, with Rf∗ OU = OV .
We set ΛV := EndU (V|U ). The following is an extension of [DW2, §3.1], which only
considered contractions with fibre dimension at most one.
Definition 2.14. Write Icon for the two-sided ideal of ΛV consisting of those morphisms
ϕ : V|U → V|U which factor through an object F ∈ add OU , as follows.
V|U

ϕ

V|U

F
Then the contraction algebra for ΛV is defined to be (ΛV )con := ΛV /Icon .
The following proposition is important, and will be heavily used later.
Proposition 2.15. If V = Spec R is an affine open subset of Y , then D(V ) ∼= (ΛV )con .
Proof. For all affine open subsets V = Spec R of Y
0 → I(V ) → A(V ) → D(V ) → 0
is exact. It follows that, for U := f −1 (V ),op
D(V ) = A(V )/I(V ) = B(V )/I(V )op∼= EndR (f∗ P|U )/[R]op= EndU (P|U )/[OU ]which is EndU (V|U )/[OU ] = (ΛV )con .

(by (2.C), 2.10)(by 2.3(1))
A benefit of our global construction of D, which we will make use of later, is thatthe contraction theorem in [DW2] can be globalised. Although [DW2, 4.4–4.7] was statedin the one-dimensional fibre setting, the proof works word-for-word in the more general
setup here.
Corollary 2.16 (Global Contraction Theorem). SuppY D = Z.
Proof. By [DW2, 4.7], which does not require the one-dimensional fibre assumption, we seethat SuppV (ΛV )con = Z ∩V for all affine opens V in Y . Since SuppV (ΛV )con = SuppV D|V
by 2.15, the result follows.
Remark 2.17. By construction D is a sheaf of algebras on Y . By 2.16 it is naturally asheaf of algebras on Z, and thus we can view the ringed space (Z, D) as a noncommutative
enhancement of the locus Z. Note further that coh(Y, D) = coh(Z, D).
Example 2.18. Consider R := C[x, y, z, t]/(x2 t + y 3 − z 2 ) and Y = Spec R. This issingular along x = y = z = 0, and has a crepant resolution sketched below, obtained byblowing up the ideal (x, y, z).

X

Y

10

WILL DONOVAN AND MICHAEL WEMYSSIn this example X is derived equivalent to Λ := EndR (R ⊕ M ), where M is the cokernelof the following 4 × 4 matrix.
x z −y 0z xt 0y y 2 0 xt
z 

2
0
y
−z
−x
R4 −−−−−−−−−−−−−−−−−−→ R4







(2.F)

We now calculate the stalks of the sheaf D at the closed points of Z. It is clear that,away from the origin, complete locally R is given by the equation x2 + y 3 − z 2 . This is
the A2 surface singularity crossed with the affine line, and thus the generic fibre is justthe minimal resolution of A2 crossed with the affine line. Since M has rank two, it follows
that for all points of Z away from the origin, the completion of the stalk of D is given by
the completion of the following quiver with relations.
a◦s=t◦ab◦t=s◦b
a◦b=0
b◦a=0

a
s

t

b

On the other hand, the origin is a cD4 point, and calculating the stalk of D there isa little trickier. Using the matrix in (2.F), it is easy to first present Λ as a quiver withrelations, as in [AM, §4], then calculate the contraction algebra, as in [DW1, 1.3]. Doing
this, one finds that the completion of D at the origin is the completion of
Cha, bi
ab + ba, a2
at the ideal (a, b).
By 2.17 there is a noncommutative ringed space (Z, D), and it is natural to compare
this to the usual commutative ringed space (Z, OZ ), where OZ is the structure sheaf thatendows Z with its reduced subscheme structure. A key property of (Z, OZ ) is that forevery point z ∈ Z, the stalk OZ,z is local, that is, it admits only one simple module.
This is not true in general for the stalk Dz . However, in the crepant setting of 2.5(1), thefollowing holds.
Proposition 2.19. Suppose the crepant setting of 2.5(1), let z ∈ Z be a closed point,bz is local, then O
bY,z is aand furthermore assume that d = 3 and X is smooth. If D
hypersurface.
bY,z , and write D
bz = EndR (R ⊕ M ). Since by assumption D
bz isProof. Put R := Olocal, necessarily M must be indecomposable. Then by [IW2, 6.25(3)] mutation at theindecomposable summand M must be an involution, and so Ω2 M ∼= M . But this implies
that the complexity of M must be less than or equal to one, which easily (see for examplethe proof of [W, 6.14]) implies that R is a hypersurface.
Remark 2.20. The converse to 2.19 is false, since if R is complete locally a hypersurface,bz need not be local. Nevertheless, being the only situation with smooth X where a local
D
bz is possible, the situation where Y is complete locally a hypersurface has special status.
D
It is also the only situation where our D can possibly coincide with Toda’s noncommutative
thickening [T4]. This partially motivates the hypersurface assumptions in §4 below.
2.3. D and Deformations. In the setting 2.5(2) when f has fibres of dimension at mostone, in this subsection we briefly relate D to noncommutative deformation theory. For this,recall that an n-pointed k-algebra Γ is an associative k-algebra equipped with k-algebramorphismsipkn → Γ → kn

NONCOMMUTATIVE ENHANCEMENTS OF CONTRACTIONS

11

which compose to the identity. Note that the morphisms p and i allow us to lift the
canonical idempotents {e1 , . . . , en } of kn to Γ. We refer to Γ as artinian if it is finitedimensional as a vector space over k, and the ideal Ker p is nilpotent. Such algebras
naturally form a category Artn , and we write pArtn for the category of pro-artinian algebras(see for instance [DW2, §2] for the precise construction).
Recall that a DG category is a graded category A whose morphism spaces have thestructure of DG vector spaces. We write DGn for the category which has as objects theDG categories with precisely n objects. Given a DG category A ∈ DGn and an algebraΓ ∈ Artn , we now recall an appropriate notion of deformations over Γ, according to thestandard Maurer–Cartan formulation. Writing n = Ker p ⊂ Γ, first consider the DG
category A ⊗ n ∈ DGn with objects {1, . . . , n}, morphisms
HomA ⊗ n (i, j) := HomA (i, j) ⊗ k (ej nei ),and differential induced from A. (Note that the convention used here for notating compositions in Γ is opposite to that of [DW2].) Given a degree-1 morphism ξ ∈ (A ⊗ n)1
consider the Maurer–Cartan equation
MC(ξ) := dξ + 21 [ξ, ξ] ∈ (A ⊗ n)2 = 0
where d is the differential, and the bracket is induced from the commutator brackets on
A and n.
Definition 2.21. Given A ∈ DGn , the associated deformation functor is given by
Def A : Artn → Sets
Γ 7→ ξ ∈ (A ⊗ n)1 | MC(ξ) = 0 / ∼where the standard gauge equivalence relation ∼ is given in e.g. [DW2, 2.6].
Given objects E1 , . . . , En ∈ coh(X), and injective resolutions Ei → I•i , thenM A := EndDG
I•inaturally lies in DGn , and the associated deformation functor Def A describes the simultaneous noncommutative deformations of the collection {Ei }.
Definition 2.22. A functor F : Artn → Sets is said to be prorepresentable by Γ ∈ pArtn
if the restriction of HompArtn(Γ, −) to Artn is naturally isomorphic to F .
Given a contraction f with at most one dimensional fibres as Sin 2.5(2), for everynz ∈ Z, C = f −1 (z) is a curve, and recall that we write C red = i=1 Ci where eachCi ∼= P1 . We put Ei = OCi (−1) and form A as above.
The following theorem was shown in [DW2, 1.1].
Theorem 2.23. In the setting 2.5(2), for each closed point z ∈ Z the functor of noncommutative deformations Def A is represented by an algebra Acon,z .
The precise form of Acon,z is not important for now, but we do explain this in §4.2.
Recall from §2.2 that for an affine open V of Y , and U := f −1 (V ), we write ΛV :=EndU (P ∗ |U ). It was shown in [DW2, 1.1, 1.2] that the completion of D|V = (ΛV )con at a
closed point z ∈ Z ∩ V is morita equivalent to Acon,z . The following is then an immediate
corollary of 2.15.
Corollary 2.24. In the setting 2.5(2), let z ∈ Z be a closed point. Then the completionof the stalk Dz is morita equivalent to Acon,z .
The morita equivalence above is illustrated in the following example.

12

WILL DONOVAN AND MICHAEL WEMYSS

Example 2.25. Consider R := C[x, y, z, t]/(x3 − xyt − y 3 + z 2 ) and Y = Spec R, which
is singular along the t-axis. It has a crepant resolution sketched as follows.

X

Y
In this example X is derived equivalent to Λ = EndR (R ⊕ M ), where M is the cokernelof the following 4 × 4 matrix.




x
z
−y
0

0
y2
−z
x 


−z x2 − yt
0
y 

2
2
−y
0
x − yt z4
R −−−−−−−−−−−−−−−−−−−−−−−−−→ R4







Away from the origin, the singular locus is just the A1 surface singularity crossedwith the affine line, so since M has rank two, complete locally away from the origin it
must split into two isomorphic copies of the same rank-one CM module L, so thatb∼Λ
= EndR (R ⊕ L⊕2 ).

Hence, away from the origin, the completion of the stalk of D is the 2 × 2 matrices overC[t], since C[t] is the contraction algebra of EndR (R ⊕ L).
At the origin, complete locally M is indecomposable, so that the stalk of D at theorigin must be a local algebra. Using the same method as in 2.18, it is not difficult toshow that the completion of the stalk is isomorphic to the completion of Cha, bi/(a2 , b2 )at the ideal (a, b).
Remark 2.26. This remark explains why our thickening is different to the one constructed by Toda [T4]. In the above example, the generic exceptional fibre consists of asingle smooth projective curve, and so pick its structure sheaf and consider its Hilbert
polynomial, which gives rise to a moduli space of simple sheaves. All generic fibres havethis Hilbert polynomial (by flatness), and the moduli space has a point for each one ofthese, plus a point corresponding to the origin, to give a pure dimension one scheme. The
completion of Toda’s sheaf D′ at each point abelianizes to give the completion of the stalkof the structure sheaf of the moduli space at that point. In particular, since the moduli
space has pure dimension one, this stalk cannot be finite dimensional. But D abelianizes
at the central point to give C[[a, b]]/(a2 , b2 ), which is finite dimensional, and hence thesheaves D and D′ are different.
3. A Universal Sheaf E
In this section we use D from 2.12 to construct a complex of sheaves E on X, and we
present some of its basic properties. In the case of fibre dimension at most one, 2.5(2),we will show that E is a sheaf, with support equal to the exceptional locus.
3.1. General Construction of E. In this subsection we work under the general assumptions of 2.3. By definition, 2.12, D is a sheaf on Y . However, in the setting 2.5(2), consider
V := P ∗ , which induces an equivalence
Db (coh X)

Rf∗ RHomX (V,−)∼

Db (coh(Y, A)).

(3.A)

NONCOMMUTATIVE ENHANCEMENTS OF CONTRACTIONS

13

The sheaf of algebras D := A/I constructed in 2.12 is in particular an A-module, andthus an object of Db (coh(Y, A)).
Notation 3.1. We write E for the object in Db (coh X) which corresponds to D acrossthe equivalence (3.A).
Proposition 3.2. Under the general assumptions of 2.3, for an affine open V = Spec R ⊆Y , set U = f −1 (V ) and write f ′ : U → V for the restricted map. Then the following hold.
(1) Rf∗′ (E|U ) = 0.
(2) Rf∗ E = 0.
In particular, if E is a sheaf then SuppX E is contained in the exceptional locus.
Proof. (1) There is a commutative diagram as follows.
Db (coh X)

Rf∗ RHomX (V,−)∼|U

Db (coh U )Db (coh(Y, A))
(3.B)

|V
RHomU (V|U ,−)∼Db (mod ΛV )Since E corresponds to D on the top equivalence, it follows that E|U corresponds to D(V )across the bottom equivalence. Hence by 2.15 E|U corresponds to (ΛV )con .
Let e denote the idempotent in ΛV corresponding to R. Then the following diagram
commutes.
Db (coh U )Rf∗′

Db (coh Spec R)

RHomU (V|U ,−)∼Db (mod ΛV )
(−)e

(3.C)

Db (mod R)

Since (ΛV )con ∈ mod ΛV satisfies (ΛV )con e = 0, it follows that E|U satisfies Rf∗′ (E|U ) = 0.
(2) Since Ri f∗ E|V = Ri f∗′ (E|U ) by flat base change (see e.g. [H2, III.8.2]), it follows from
(1) that Rf∗ E = 0.
It may be the case that SuppX E always equals the exceptional locus, but this seems
tricky to prove without more control over where the simple (ΛV )con -modules go under thederived equivalence. Controlling the support of E is important, since later it will give aneasy-to-check obstruction to D being relatively spherical.
3.2. E and Fibre Dimension One. This subsection considers the setting of 2.5(2) whenf has fibres of dimension at most one, and proves that E is a sheaf whose support SuppX E
equals the exceptional locus.
In the setting of 2.5(2), recall from [B, §3] and [V1, §3.1] that perverse coherent
sheaves on X may be defined as follows.
Definition 3.3. The category −1 Per(X, Y ), respectively 0 Per(X, Y ), consists of thoseobjects a ∈ Db (coh X) such that
(1) H i (a) = 0 if i 6= 0, −1,(2) f∗ H −1 (a) = 0, R1 f∗ H 0 (a) = 0,
(3) HomX (H 0 (a), c) = 0, respectively HomX (c, H −1 (a)) = 0, for all c ∈ C 0
where C := ker Rf∗ and C 0 denotes the full subcategory of C whose objects have cohomology
only in degree 0.
It follows from [V1, 3.3.1, proof of 3.3.2] that the P in 2.5(2) is a local progeneratorof −1 Per(X/Y ), and V = P ∗ likewise for 0 Per(X/Y ).
To show that E is a sheaf requires the following lemma, which is well known.
Lemma 3.4. Suppose that a ∈ Db (coh X) such that Rf∗ a = 0.

14

WILL DONOVAN AND MICHAEL WEMYSS(1) a ∈ −1 Per(X, Y ) if and only if a is concentrated in degree −1.
(2) a ∈ 0 Per(X, Y ) if and only if a is concentrated in degree 0.
Proof. (1) (⇒) By definition H i (a) = 0 unless i = −1 or 0, so we need only show H 0 (a) =0. As f has at most one-dimensional fibres, Rf∗ a = 0 implies that Rf∗ H 0 (a) = 0 using[B, 3.1]. But HomX (H 0 (a), c) = 0 for all sheaves c such that Rf∗ c = 0, in particular forc = H 0 (a), and the claim follows.
(⇐) This follows immediately from the definition.
(2) The proof is similar.

Corollary 3.5. In the setting of 2.5(2), the following statements hold.
(1) E|U ∼= (ΛV )con ⊗Λ V|U .
(2) E is a sheaf in degree zero.
Proof. Both results follow from 3.2 and 3.4(2) applied to 0 Per(U, V ).



For a closed point z ∈ Z, consider an affine neighbourhood V = Spec R of z, andconsider U := f −1 (V ) and f |U : U → V = Spec R. The fibre above each closed point in Z,
with reduced scheme structure, decomposes into curves Ci say, each isomorphic to P1 .
Proposition 3.6. In the setting of 2.5(2), if x is a closed point in the exceptional locus
of f |U , then x ∈ SuppU E|U .
Proof. Certainly x sits on some Ci , so x ∈ Supp OCi (−1). Write T for the ΛV -module
corresponding to OCi (−1) across the bottom equivalence in (3.B), then by [V1, 3.5.8] T isa simple ΛV -module. But since Rf∗ OCi (−1) = 0, it follows from (3.C) that T is also a(ΛV )con -module, and hence a simple (ΛV )con -module. As such, there exists a surjection(ΛV )con ։ T , and so back across the bottom equivalence in (3.B) there is a short exact
sequence
0 → K → E|U → OCi (−1) → 0
in 0 Per(U, V ). Since the last two terms are sheaves, it follows from the long exact sequencein ordinary cohomology that so too is K. Hence since passing to stalks is exact, the stalkof E at x must be non-zero, since it surjects onto the stalk of OCi (−1) at x, which is
non-zero. It follows that x ∈ Supp E|U .

Corollary 3.7. In the setting of 2.5(2), SuppX E is equal to the exceptional locus.
Proof. By 3.6, by varying z ∈ Z we see that the set of closed points in the support of
E contains the set of closed points in the exceptional locus. Since Rf∗ E = 0 by 3.2(2),Supp E does not contain any further closed points. Since X is a variety, and so Zariski
locally every prime ideal is the intersection of maximal ideals, we are done.
We next show that the global E recovers the universal sheaves from noncommutative
deformation theory in the setting of 2.5(2). For a closed point z ∈ Z, write Rz for thecompletion of the structure sheaf of Y at z, and consider the following flat base change
diagram.
Xz

jz

X

fz

Spec Rz

f
iz

Y

The deformation theory in §2.3 gives rise to a universal sheaf Ez ∈ coh Xz for every z ∈ Z.
Proposition 3.8. In the setting of 2.5(2), for every z ∈ Z, add jz∗ E = add Ez .
Proof. This is implicit in [DW2, 3.7], but we sketch the argument here. To ease notation,
write Λ = ΛV .

NONCOMMUTATIVE ENHANCEMENTS OF CONTRACTIONS

15

b is morita equivalent to an algebra A, via a functor F,
As in [DW2, 3.7] the algebra Λband furthermore there are decompositions as Λ-modules
FAcon = P1 ⊕ . . . ⊕ Pn

b con = P ⊕a1 ⊕ . . . ⊕ P ⊕an
and Λ
n
1

(3.D)

b con -modules. Now
for some ai ≥ 1, where the Pi are the projective Λb
RHomXz (jz∗ V, −) : Db (coh Xz ) → Db (mod Λ)b con .
is an equivalence, and it is easy to see using flat base change that jz∗ E corresponds to Λ
As in [W, 4.14] and [DW1, §3.2], Ez corresponds across the equivalence to FAcon . Sinceb con = add FAcon , it follows that add j ∗ E = add Ez .
(3.D) implies that add Λz
4. Spherical Properties via Cohen–Macaulay Modules

This section considers the Zariski local crepant contractions of 2.5(1) and characterises, under some assumptions on singularities, when the noncommutative enhancement
D is relatively spherical. These results are globalised in §5. All this involves the Cohen–
Macaulay property, which we now review.
4.1. Cohen–Macaulay modules. Recall that if (R, m) is a commutative local noetherian ring, with M ∈ mod R, then the depth of M is defined to be
depthR M := min{i | ExtiR (R/m, M ) 6= 0},
and for all p ∈ Ass(M ) there is a chain of inequalities
depthR M ≤ dim R/p ≤ dimR M.
Then M is called a Cohen–Macaulay module if either M = 0, or M 6= 0 and
depthR M = dimR M.
We write CMS R for the category of such modules. It is clear that if M is Cohen–
Macaulay, necessarily M is equidimensional (i.e. dim R/p = dimR M for every minimal
prime p ∈ Supp M ), and has no embedded primes (i.e. every associated prime is minimal).
If R is local Gorenstein, and M 6= 0, then by local duality (see e.g. [BH, 3.5.11])
M ∈ CMS R ⇐⇒ RHomR (M, R) is concentrated in degree dim R − dimR M.
When R is not necessarily local, but is still Gorenstein and equi-codimensional, we saythat M ∈ mod R is CM if Mm ∈ CMS Rm for all maximal ideals m of R, and again write
CMS R for the category of CM R-modules. Since dimRm Mm can vary between maximal
ideals, RHomR (M, R) may have cohomology in more than one degree.
We say that M ∈ mod R is maximal Cohen–Macaulay if depthRm Mm = dim Rm
for all m ∈ Max R, and we write MCM R for the category of maximal Cohen–Macaulay
R-modules.
4.2. Setting. The remainder of this section considers the following refinement of theaffine crepant contraction setting of 2.5(1).
Setup 4.1. Suppose that f : X → Y is a contraction as in 2.1, where in addition Y =Spec R is affine and Gorenstein, d = dim X ≥ 2, f is crepant, and X admits a tilting
bundle P = OX ⊕ P0 .
Necessarily R is equi-codimensional by [E, 13.4], and a Gorenstein normal domain byassumption.
Notation 4.2. We set M := f∗ P0 , which is an R-module, and consider Λ := EndR (R ⊕M ), which is isomorphic to EndX (P) by 2.5(1). Further, since f is crepant necessarilyΛ ∈ MCM R by [IW3, 4.8], and thus M ∼= HomR (R, M ) ∈ MCM R, being a summandof Λ. We write Λcon := Λ/Icon , where Icon := [R].

16

WILL DONOVAN AND MICHAEL WEMYSS

We will often reduce arguments about Λcon to the completions of closed points, inwhich case the following notation will be useful. For a closed point z = m ∈ Spec R, write
R for the completion of R at z, and consider the Krull–Schmidt decompositionc = R⊕a0 ⊕ M ⊕a1 ⊕ . . . ⊕ M ⊕an .
Mn1

Then we write K = M1 ⊕ . . . ⊕ Mn , and A := EndR (R ⊕ K). This depends on z, but wesuppress this from the notation. We define
Acon,z := A/[R],but throughout this section, to ease notation we will usually refer to this as simply Acon .
Definition 4.3. We say that Λcon is t-relatively spherical if

k if j = 0, t
ExtjΛ (Λcon , T ) ∼=0 else,for all simple Λcon -modules T . There is no requirement that Λcon is perfect.
There is an obvious variant of 4.3 for Acon . The following two subsections characterise
when Λcon and Acon are relatively spherical, under the assumption that complete locally
R has only hypersurface singularities. This additional assumption is motivated in partby 2.20, in part by the fact that in characteristic zero 3-dimensional Gorenstein terminal
singularities have this property [R83, 0.6(I)], and in part since one of our main applications
later in §6.2 will be to crepant divisor-to-curve contractions of 3-folds, in which case thehypersurface singularity condition holds automatically.
4.3. Spherical via CM R-modules I. The following three results are elementary.
Lemma 4.4. In the setup 4.1, with p ∈ Spec R, the following are equivalent.
(1) p ∈ Z.
(2) p ∈ Supp Λcon .
(3) Mp is a non-free Rp -module.
Proof. (1)⇔(2) is the local version of 2.16.
(2)⇔(3) (Λcon )p ∼= EndRp (Mp ), which is zero if and only if Mp is free.

The following is also elementary, and is a simple consequence of the depth lemma.
Lemma 4.5. In the setup 4.1, if dimR Λcon ≤ d − 3, then Ext1R (M, M ) = 0.
Proof. By the assumption and 4.4, for p ∈ Spec R with ht p = 2, Mp is a free Rp -module,
and hence certainly Ext1Rp (Mp , Mp ) = 0. This implies that for all q ∈ Spec R with
ht q = 3, Ext1Rq (Mq , Mq ) is a finite length Rq -module.
But since EndRq (Mq ) ∈ MCM Rq , by the depth lemma Ext1Rq (Mq , Mq) = 0. In turn,this implies that the Ext group for height four primes has finite length. Again, by thedepth lemma, this must be zero. By induction the result follows.

Corollary 4.6. In the setup 4.1, if dimR Sing R ≤ d − 3, then Ext1R (M, M ) = 0.
Proof. Since M ∈ CM R, it is clear from 4.4 that dimR Λcon ≤ dimR Sing R. The resultthen follows immediately from 4.5.
The remainder of this subsection considers the case when Λcon does not have maximal
dimension, that is when dimR Λcon ≤ d − 3. The case dimR Λcon = d − 2 is trickier, andwill be dealt with in the next subsection.
Lemma 4.7. In the setup 4.1, suppose further that Y is complete locally a hypersurface.
If dimR Λcon ≤ d − 3, then at every closed point z ∈ Z,
(1) pdA Acon,z = 3.
(2) Acon,z ∈ CMd−3 R, in particular dim Acon,z = d − 3.

NONCOMMUTATIVE ENHANCEMENTS OF CONTRACTIONS

17

Furthermore, the following statements hold.
(3) The stalk of OY at every closed point of Z has dimension d − 3.
(4) pdΛ Λcon = 3.
(5) Λcon is 3-relatively spherical.
Proof. (1)(2) By matrix factorisation there is an exact sequence
0 → K → F → F → K → 0,
where F is a free R-module. Further by assumption and 4.5, we know that Ext1R (K, K) =0, hence applying HomR (R ⊕ K, −) to the above sequence gives a projective resolution
0 → PK → P0⊕b → P0⊕b → PK → Acon → 0.
Thus pdA Acon ≤ 3, from which Auslander–Buchsbaum implies that depth Acon ≥ d − 3.
It thus follows that Acon ∈ CMd−3 R, and so pdA Acon = 3.
(3) By (2) dimR Acon = d − 3, and by 4.4 dimR Acon = dimR Z.
(4) This follows from (1), since projective dimension can be checked complete locally.
(5) This follows by checking complete locally, and using the projective resolution of Acon
given above.

4.4. Spherical via CM R-modules II. Seeking a more general version of 4.7 whenΛcon has maximal dimension is subtle, for two reasons. First, pdΛ Λcon = ∞ can occur,and second the dimension of Z at closed points may vary, in which case asking for Λcon ∈CMS R is more natural than specifying a particular CMd−t R.
Before extracting a global statement, we first work complete locally, and extend 4.7
as follows.
Theorem 4.8. In the setup 4.1, suppose further that R is complete locally a hypersurface.
Then Acon is t-relatively spherical if and only if(1) pdA Acon < ∞.
(2) Acon ∈ CMd−t R.
In this case necessarily t = d − dim Acon , which is either 2 or 3, and furthermore theassumptions (1) and (2) hold when dimR Acon ≤ d − 3.
Proof. (⇐) Case t = 2: By Auslander–Buchsbaum, pdA Acon = 2. It then follows from
[W, A.3] that ΩK ∼= K, and so applying HomR (R ⊕ K, −) to the exact sequence
0 → ΩK → F → K → 0
gives the minimal projective resolution
0 → PK → P0⊕a → PK → Acon → 0.
Hence Acon is 2-relatively spherical, and since by assumption Acon ∈ CMd−2 R, t =d − dim Acon .
Case t ≥ 3: Since Acon ∈ CMt R, necessarily dim Acon ≤ d − 3, so by 4.7 Acon is 3relatively spherical, and t = d − dim Acon .
(⇒) Case t = 2: Consider the beginning of the minimal projective resolution of Acon .
Since Acon is 2-relatively spherical, this has the formψ

→ PK → P0⊕a −→ PK → Acon → 0
Certainly Ker ψ = HomR (R ⊕ K, ΩK), hence the projective cover of Ker ψ is obtained byapplying HomR (R ⊕ K, −) to a minimal add(R ⊕ K)-approximationf : U → ΩK.
But since the projective cover of Ker ψ is PK , it follows that HomR (R ⊕ K, U ) ∼= PK :=
HomR (R ⊕ K, K), so by reflexive equivalence U ∼= K. Further, since f is in particular an
add R-approximation, it is necessarily surjective. But R is a hypersurface, so the rank of18

WILL DONOVAN AND MICHAEL WEMYSSK equals the rank of ΩK, hence Ker f = 0 and thus f is an isomorphism. In turn, this
implies that Ker ψ ∼= PK , and hence pdA Acon = 2.
By Auslander–Buchsbaum, depthR Acon = d− 2. But as in 4.6, and since R is normald − 2 = depthR Acon ≤ dim Acon ≤ dim Sing R ≤ d − 2.
Hence equality holds throughout, and Acon ∈ CMd−2 R.
Case t ≥ 3: Again, consider the beginning of the minimal projective resolution of Acon ,which now has the formφψ

→ P0⊕b −→ P0⊕c −→ PK → Acon → 0.
The morphism P0⊕b → Ker ψ = HomR (R ⊕ K, ΩK) is induced from a morphism of theform f : F → ΩK (where F is free), which is a minimal add(R⊕K)-approximation. Again,
necessarily this f is surjective, and its kernel is ΩΩK ∼= K, since R is a hypersurface. But
then, applying HomR (R ⊕ K, −) to the exact sequencef

0→K→F −
→ ΩK → 0
and using the fact that f is an add(R ⊕ K)-approximation,
0 → HomR (R ⊕ K, K) → HomR (R ⊕ K, F ) ։ HomR (R ⊕ K, ΩK) → Ext1R (K, K) → 0
is exact. It follows that Ext1R (K, K) = 0, and Ker φ = PK . Consequently, pdA Acon = 3
and Acon is 3-relatively spherical, thus t = 3.
By Auslander–Buchsbaum, depthR Acon = d − 3. We claim that Acon ∈ CMd−3 R,
so we just need to prove that dim Acon 6= d − 2. If there exists p ∈ Supp Acon with height
two, then by 4.4 Kp is a non-free Rp -module. But by the above
HomCM Rp(Kp , Kp [1]) ∼= Ext1Rp(Kp , Kp ) = 0,
and further since Rp is a Gorenstein surface with only an isolated singular point, CM Rp is
1-CY. This implies that HomRp(Kp , Kp ) = 0, and so Kp is free, which is a contradiction.
Hence such a p with height two cannot exist, and so Acon ∈ CMd−3 R.
The last statement is 4.7.

Remark 4.9. Neither condition (1) or (2) in 4.8 is guaranteed if dim Acon = d − 2; see
4.12 below and also [W, 4.18].
Remaining complete local, gives the following tilting consequence.
Proposition 4.10. In the setup 4.1, suppose further that R is complete locally a hypersurface. If the equivalent conditions of 4.8 hold, then IA is a tilting A-module.
Proof. By 4.8 there are only two cases, t = 2 and t = 3. When t = 2, the fact that IA
is tilting is just [W, A.3, A.5]. When t = 3, the argument is identical to [DW1, 5.10(1)],since although in that proof d = 3, the fact that by 4.5 Ext1R (K, K) = 0 means that
[DW1, (5.F)] is still exact, so exactly the same proof works.
It is the global version of 4.10 that interests us the most. The fact that Acon can be
relatively spherical at all closed points, but that the value of the spherical parameter canvary, is problematic; similarly the projective dimension of IA can vary over the maximal
ideals. Hence we do not seek an autoequivalence condition globally in terms of one parameter. Instead, in condition (3) below we ask for Λcon to belong to CMS R, which gives
the required flexibility. The following is one of our main results.
Theorem 4.11. In the setup 4.1, suppose further that R is complete locally a hypersurface. Then the following are equivalent.
(1) Acon,z is a relatively spherical A-module for all z ∈ Max R.
(2) pdA Acon,z < ∞ and Acon,z ∈ CMS R for all z ∈ Max R.
(3) pdΛ Λcon < ∞ and Λcon ∈ CMS R.

NONCOMMUTATIVE ENHANCEMENTS OF CONTRACTIONS

19

If any of these conditions hold, and they are automatic if dim Sing R ≤ d − 3, then Icon isba tilting Λ-module, and − ⊗LΛ Icon preserves D (mod Λ).
Proof. (1)⇔(2) is 4.8, since d − t = d − (d − dim Acon ) = dim Acon .
(2)⇔(3) just follows since finite projective dimension can be checked complete locally at
maximal ideals, and Λcon ∈ CMS R is again defined locally.
The statement about the assumptions being automatic is 4.7. The fact Icon is a tilting
module then follows since being a tilting module can be checked complete locally (see e.g.
the proof of [DW1, 6.2]), and the complete local statement is 4.10.
Example 4.12. Consider M := (u, x) ⊕ (u, x2 ) for R = C[u, v, x, y]/(uv − x2 y). Then
Λ := EndR (R ⊕ M ) is an NCCR of R. Complete locally at the origin, since ΩM ≇ M , as
/ CMS R. In fact, this can be seen directly, since the minimalin 4.8 it follows that Acon ∈projective resolution of Acon has the form
0 → P2⊕2 → P0⊕3 ⊕ P1 → P0⊕4 → P1 ⊕ P2 → Acon → 0,
which is evidently not relatively spherical. Note that since Acon ∈/ CMS R, it follows thatΛcon ∈/ CMS R. However, the other condition in 4.11(3), namely pdΛ Λcon < ∞, does holdsince Λ is an NCCR.
A more conceptual geometric explanation of the above example is given in 6.4 below.
5. Global Twist Functors
In this section, under our most general d-fold contraction assumptions of 2.3, where
additionally f is crepant, we produce an endofunctor TwistX of Db (coh X). After restricting singularities, we then give a criterion for TwistX to be an autoequivalence, before giving
our first application. Further applications will appear in §6.
5.1. Twist construction. By the definition of D in 2.12, there is an exact sequence ofA-bimodules
0 → I → A → D → 0.
(5.A)
The bimodule I induces a functor
RHomA (I, −) : D(Mod A) → D(Mod EndA I),
and the following lemma, which is simply the global version of [DW1, 6.1], ensures that
it furthermore yields an endofunctor of D(Mod A).
Lemma 5.1. Under the general assumptions of 2.3, suppose further that f is crepant.
∼ EndA I.
(1) There is an isomorphism of sheaves of algebras A =(2) Under this isomorphism the (EndA I, A)-bimodule structure on I coincides with
the natural A-bimodule structure.
Proof. (1) There is a canonical morphism
A → EndA I(5.B)

given on any open subset V of Y by
A(V ) → HomA(V ) (I(V ), I(V ))λ 7→ αλwith αλ : i 7→ λi. By our assumptions 2.3, dim X ≥ 2, so since f is crepant the proof of[DW1, 6.1(1)] shows that this is an isomorphism for affine V , where 2.10 ensures that
I(V ) is the ideal considered in [DW1]. It follows that (5.B) is an isomorphism.
(2) This is a formal consequence of (1), as in [DW1, 6.1(2)].
20WILL DONOVAN AND MICHAEL WEMYSS

Composing the above endofunctor with the equivalences
G=f −1 (−) ⊗LVf −1 A

D(Mod A)

D(Qcoh X)(5.C)

GRA =Rf∗ RHomX (V,−)

leads to the following definition.
Definition 5.2. Under the general assumptions of 2.3, suppose further that f is crepant.
The twist and dual twist endofunctors are defined to be
TwistX , Twist∗X : D(Qcoh X) → D(Qcoh X)
TwistX = G ◦ RHomA (I, −) ◦ GRA ,RA.
Twist∗X = G ◦ (− ⊗LA I) ◦ G

We will show that TwistX preserves Db (coh X) in 5.5(2) below, under the condition
that D is a perfect A-module, or equivalently E from 3.1 is an object in Perf(X). To dothis, we first describe some additional structure on E.
By definition E = f −1 D ⊗L
f −1 A V is a complex of sheaves on X. Since this expression
can be computed using both factors, by resolving the second factor we see that E ∈Db (mod f −1 Dop ).
Next, consider the following adjoint functors
G=f −1 (−) ⊗LVf −1 A− ⊗L
D D

D(Mod D)

D(Mod A)
RHomA (D,−)

D(Qcoh X).
GRA =Rf∗ RHomX (V,−)(5.D)

By construction and assumption, D and V are sheaves. We will write F for the compositionof the top functors, and F RA for the composition of the bottom functors. Note that F
and F RA can be expressed easily as
L
F = f −1 (− ⊗LD D) ⊗f −1 A V∼ f −1 (−) ⊗L−1 f −1 D ⊗L−1 V
=
f D
f A
−1
L
∼
f
(−)
⊗
V,
=
−1
f

(since f −1 distributes over tensor)

A

F RA = RHomA (D, Rf∗ RHomX (V, −))∼= Rf∗ RHomf −1 A (f −1 D, RHomX (V, −))(by adjunction, c.f. [KS, (18.3.2)])∼= Rf∗ RHomX (f −1 D ⊗L
f −1 A V, −)(by adjunction)= Rf∗ RHomX (E, −).
Remark 5.3. Regardless of whether E is a sheaf or a complex, below and throughout
when we write Rf∗ RHomX (E, −) we will mean the functor F RA , which takes values inD(Mod D). In particular, f −1 Rf∗ RHomX (E, −) takes values in D(Mod f −1 D).
Proposition 5.4. Under the general assumptions of 2.3, suppose further that f is crepant.
Then TwistX fits into a functorial triangle
f −1 Rf∗ RHomX (E, −) ⊗L
f −1 D E → Id → TwistX → .
Proof. For any object a ∈ D(Mod A), simply applying RHomA (−, a) to the sequence(5.A) gives a functorial triangle in D(Mod A)
RHomA (D, a) → a → RHomA (I, a) → .

(5.E)

We may reinterpret the left-hand term as RHomA (D, a) ⊗L
D DA , in other words, it is given
by a composition of the left-hand adjoint pair in the diagram (5.D).

NONCOMMUTATIVE ENHANCEMENTS OF CONTRACTIONS

21

Hence precomposing (5.E) with GRA from (5.C), and postcomposing with G, gives afunctorial triangle
F ◦ F RA → a → TwistX (a) → .
Using the expressions for F and F RA above, the result follows.

Proposition 5.5. Under the general assumptions of 2.3, suppose further that f is crepant.
If D ∈ Perf(A), or equivalently E ∈ Perf(X), then
(1) RHomI (D, −) preserves Db (mod A).
(2) TwistX preserves Db (coh X).
Proof. (1) Since D ∈ Perf(A), the functor RHomA (D, −) preserves bounded complexes.
It clearly preserves coherence, and so the result follows using the two-out-of-three property
for the triangles (5.E).
(2) Again, since D ∈ Perf(A), the functor RHomA (D, −) in (5.D) preserves bounded
coherent complexes, and all the other functors in (5.D) also preserve bounded coherence.
Hence F and F RA preserve bounded coherence, thus so does TwistX by 5.4, again using
the two-out-of-three property.

5.2. Conditions for Equivalence. This subsection uses the Zariski local tilting result
4.11 to give a condition for when TwistX is an equivalence globally. Recall that D is a
Cohen–Macaulay sheaf if it is Cohen–Macaulay at each closed point, as defined in §4.1.
The following notion, a translation of 4.3, will be used.
Definition 5.6. We say that D is t-relatively spherical for a closed point z ∈ Z if
k if j = 0, tj∼b
ExtAb (Dz , T ) =0 else,zbz -modules T . Here D
bz is the completion of the stalk of D at z.
for all simple D
S
To set notation, choose an affine open cover Y = Vi , and for any V = Vi consider
RHomA|V (I|V , −) : D(Mod A|V ) → D(Mod A|V ).

As V is affine, say V = Spec R, we may use setup 4.1, where now A|V corresponds to Λ,
and I|V to the Λ-bimodule Icon by 2.10. It follows that the above functor is simply
RHomΛ (Icon , −) : D(Mod Λ) → D(Mod Λ).
The following is the main theorem of this section.
bY,zTheorem 5.7. Under the general assumptions of 2.3, assume that f is crepant, and O
are hypersurfaces for all closed points z ∈ Z. Then the following are equivalent.
(1) D is a Cohen–Macaulay sheaf on Y , and E is a perfect complex on X.
(2) D is relatively spherical for all closed points z ∈ Z.

If these conditions hold, and they are automatic provided that dim Z ≤ d − 3, then thefunctor TwistX is an autoequivalence of Db (coh X).
Proof. Since being Gorenstein is an open condition [M, 24.6], Y is Gorenstein in a neighbourhood N of Z.
Conditions (1) and (2) can both be checked locally. Since D is supported on Z by 2.16,
and recalling from 3.1 that E is defined as the image of D under an equivalence, it suffices
to show that they are equivalent after restricting to N . But there, the required result is
4.11, which also shows that the conditions are automatic provided that dim Z ≤ d − 3.
For the final statement, since G and GRA are equivalences by definition, 5.2, it suffices
to prove that
RHomA (I, −) : Db (mod A) → Db (mod A)

22

WILL DONOVAN AND MICHAEL WEMYSS

is an equivalence. We write η and η̃ respectively for the counits of the following adjunctions− ⊗L
A I ⊣ RHomA (I, −)− ⊗LΛ Icon ⊣ RHomΛ (Icon , −).
Consider, for a ∈ D(Mod A), the distinguished triangle
ηa

RHomA (I, a) ⊗L
A I −→ a −→ Cone(ηa ) −→ .
This restricts to each V = Vi along the inclusion j : V ֒→ Y , to givej ∗ ηa−−→ j ∗ a −→ j ∗ Cone(ηa ) −→ .
RHomΛ (Icon , j ∗ a) ⊗LΛ Icon −By inspection, the counit is defined locally, soSj ∗ ηa = η̃j ∗ a .
We may take our affine open cover Y = Vi to be the union of a cover of N ⊃ Z,
and a cover of Y \Z. If V ⊆ Y \Z then Icon = Λ. If, on the other hand, V ⊆ N then we
have that V is Gorenstein, and then RHomΛ (Icon , −) is an equivalence by 4.11, condition(3). In either case, it follows that j ∗ ηa is an isomorphism. Hence j ∗ Cone(ηa ) = 0 for all
V , so Cone(ηa ) = 0 for all a, and thence η is an isomorphism.
The argument for the unit ǫ is similar, since it too by inspection is defined locally,and so it follows from standard adjoint functor results that− ⊗LAI

D(Mod A)

D(Mod A)
RHomA (I,−)

is an equivalence.
By 5.5(1) we already know that RHomA (I, −) preserves Db (mod A), so it suffices toprove that − ⊗L
A I also has this property.
Consider a ∈ Db (mod A), and a ⊗L
A I. Restricting to V ⊆ N , we see thatL∼ ∗j ∗ (a ⊗LA I) = j a ⊗Λ Icon ,bwhich belongs to Db (mod Λ) since j ∗ a does and − ⊗LΛ Icon preserves D (mod Λ) by 4.11.
L
On the other hand, for V ⊆ Y \Z, we have Icon = Λ, and so − ⊗Λ Icon certainly preserves
Db (mod Λ). Since the cover is finite, and the restriction of a ⊗L
A I to each piece is bounded
coherent, it follows that a ⊗L
A I is bounded coherent.

5.3. Application to Springer resolutions. The following is a corollary of 5.7.
Corollary 5.8. Consider the Springer resolution X → Y of the variety of singular n × nmatrices. Then TwistX is an autoequivalence of X.
Proof. Y is a hypersurface cut out by the determinant. It is well known that Y is smoothin codimension two [BLV, §5.1], and it is one of the main results of [BLV] that X admits
a tilting bundle with trivial summand [BLV, C].

6. One-Dimensional Fibre Applications
6.1. Relative spherical via CM sheaves. The first half of this subsection is general. If
X is a Gorenstein variety with tilting bundle T , necessarily Λ := EndX (T ) is an Iwanaga–
Gorenstein ring, namely it is noetherian with finite injective dimension on both the left
and right [KIWY, 4.4]. It follows that Λ with its natural bimodule structure gives rise toa duality functor
RHomΛ (−, Λ) : Db (mod Λ) → Db (mod Λop ).

NONCOMMUTATIVE ENHANCEMENTS OF CONTRACTIONS

23

On the other hand, since X is Gorenstein, it too has a good duality functor given by
RHomX (−, OX ), and so we can consider the following diagram.
Db (coh X)

RHomX (T ,−)∼Db (mod Λ)

RHomX (−,OX )RHomΛ (−,Λ)

Db (coh X)

RHomX (T ∗ ,−)∼Db (mod Λop )Proposition 6.1. Suppose that X is a Gorenstein variety with tilting bundle T . Thenfor all a ∈ Db (coh X), there is an isomorphism
RHomX (T ∗ , RHomX (a, OX )) ∼= RHomΛ (RHomX (T , a), Λ).
Proof. Since RHomX (T , −) is a dg equivalence, there is an isomorphism
RHomX (a, b) ∼= RHomΛ (RHomX (T , a), RHomX (T , b))
for all a, b ∈ Db (coh X). Hence setting b = T gives an isomorphism
RHomX (a, T ) ∼= RHomΛ (RHomX (T , a), Λ)(6.A)

for all a ∈ Db (coh X).
Next, since X is Gorenstein (−)∨ := RHomX (−, OX ) is a duality on Db (coh X), andfurther since T ∗ = T ∨ , there is an isomorphism
RHomX (a, T ) ∼(6.B)= RHomX (T ∗ , a∨ )for all a ∈ Db (coh X). Combining (6.A) and (6.B) yields the result.



The following is the main result of this subsection, and is specific to the setting of
one-dimensional fibres. It relates a homological condition about D on the base Y to ahomological condition about E at closed points of X.
Theorem 6.2. In the setting of 2.5(2), assume that Y is Gorenstein in a neighbourhoodof Z, and f is crepant. Then the following conditions are equivalent.
(1) D ∈ CMS Y .
(2) For all y ∈ Z, we have Dy ∈ CMdim Zy OY,y .
(3) For all y ∈ Z, and all x ∈ f −1 (y), we have Ex ∈ CMdim Zy +1 OX,x .
In particular, if these conditions hold, then above every y ∈ Z, the exceptional locus isequidimensional of dimension dim Zy + 1.
Proof. (1)⇔(2) This is the definition of CMS Y , together with the fact that dim Zy =dim Dy by the contraction theorem 2.16.
(2)⇔(3) For y ∈ Z, choose a Gorenstein affine neighbourhood V = Spec R of y. To
simplify notation, write Λ = ΛV , k = dim Zy and d = dim Ry .
Since f is crepant, Λy is a maximal CM Ry -module [IW3, 4.14], so necessarily it issingular Calabi–Yau [IW2, 2.22(2)]. Hence(6.C)
Exti (M, Ry ) ∼= Exti (M, Λy )ΛyRyby [IR, 3.4], for all M ∈ mod Λy and all i ≥ 0. Thus,
2.15

Dy ∈ CMk OY,y ⇐⇒ (Λcon )y ∈ CMk Ry⇐⇒ RHomRy ((Λcon )y , Ry ) is concentrated in degree d − k⇐⇒ RHomΛy ((Λcon )y , Λy ) is concentrated in degree d − k.
Now there is a chain of isomorphisms
RHomΛ ((Λcon )y , Λy ) ∼= RHomΛ (RHomX (V|X , E|X ), Λy )
y

y

y

y

y

∼
= RHomXy (V|∗Xy , RHomXy (E|Xy , OXy )),24

WILL DONOVAN AND MICHAEL WEMYSS

where the first follows since Λcon corresponds across the equivalence to E, and the second
follows by 6.1 since Xy is Gorenstein (since Spec Ry is), and V|Xy is a progenerator of0
Per(Xy , Ry ), so it is tilting on Xy , with endomorphism ring Λy (e.g. [IW3, 4.3(2)]).
Combining the above, we see that Dy ∈ CMk OY,y if and only if
RHomXy (V|∗Xy , RHomXy (E|Xy , OXy )[d − k]) is concentrated in degree 0.
But now V|Xy progenerates 0 Per(Xy , Ry ), and its dual progenerates −1 Per(Xy , Ry ), andhence the above condition holds if and only if
RHomXy (E|Xy , OXy )[d − k] ∈ −1 Per(Xy , Ry ).
But by crepancy and Grothendieck duality∼ Rf∗ RHomX (E|X , f ! OR )Rf∗ RHomX (E|X , OX ) =
y

y

y

y

y

y

3.2
∼
= RHomRy (Rf∗ E|Xy , ORy ) = 0,

so it follows from 3.4(1) that Dy ∈ CMk OY,y if and only if
RHomXy (E|Xy , OXy )[d − (k + 1)] is concentrated in degree 0,
which holds if and only if Ex ∈ CMk+1 OXy ,x for all x ∈ f −1 (y). Since OXy ,x ∼= OX,x , theresult follows.
Since SuppX E equals the exceptional locus by 3.7, it follows for all x ∈ f −1 (y),dim Ex equals the dimension of the exceptional locus at the point x. Hence the condition
Ex ∈ CMdim Zy +1 OX,x forces the exceptional locus to have dimension dim Zy + 1 at all
points x above y, and so the last claim follows.

Corollary 6.3. In the one-dimensional fibre setting of 2.5(2), assume that f is crepant,bY,z are hypersurfaces for all closed points z ∈ Z. Then the following are equivalent.
and O(1) D is relatively spherical for all closed points z ∈ Z.
(2) D ∈ Perf(A) and D ∈ CMS Y .
(3) E ∈ Perf(X) and D ∈ CMS Y .
(4) E ∈ Perf(X), and for all y ∈ Z and x ∈ f −1 (y), we have Ex ∈ CMdim Zy +1 OX,x .
If these conditions hold, and they are automatic provided that dim Z ≤ d − 3, then thefunctor TwistX is an autoequivalence of Db (coh X).
Proof. The assumptions force Y to be Gorenstein in a neighbourhood of Z [M, 24.6].
Recalling from 3.1 that E is defined as the image of D under an equivalence, the equivalenceof (1)–(4) follows from 6.2. The rest follows from 5.7.
Example 6.4. Consider R = C[u, v, x, y]/(uv − x2 y), and Y = Spec R. Then Y has threecrepant resolutions, sketched below.

Y

Z

Each gives a thickening of Z, which we write D1 , D2 , and D3 , respectively. Above theorigin, the exceptional locus of the outer two resolutions is not equidimensional of dimension two, since in both cases there is a curve poking out of the surface. Thus, by 6.2 and
6.3, D1 and D3 are not relatively spherical at the origin. Note that Λ in 4.12 is derived
equivalent to the left-hand resolution, and so the failure of the exceptional locus to beequidimensional explains geometrically why D1 = Λcon ∈/ CMS R in 4.12.

NONCOMMUTATIVE ENHANCEMENTS OF CONTRACTIONS

25

The exceptional locus of the middle resolution is equidimensional, but this does notguarantee that D2 ∈ CMS Y . However, to see that this indeed holds, note that the middle
resolution is derived equivalent to Λ2 = EndR (R ⊕ N ), where N = (u, x) ⊕ (u, xy). SinceΩN ∼= N , applying HomR (N, −) to the exact sequence
0 → N → R4 → N → 0
gives an exact sequence
0 → P1 → P0⊕4 → P1 → D2 → 0.
Completing the above at every closed point of Z, we see that D2 is 2-relatively sphericalat each closed point, so as in 4.11, pdΛ2 D2 = 2 and D2 ∈ CMS R. Thus TwistX is an
autoequivalence on the derived category of the middle resolution.
6.2. The Single Curve Fibre Case. In the case when there is a single curve in eachfibre, and X is smooth, we show here that the assumptions in 6.3 hold. This can arise in
the setting of moduli of simple sheaves.
The following result covers both divisor-to-curve contractions and flops.
Theorem 6.5. In the one-dimensional fibre setting of 2.5(2), suppose that d = 3, X issmooth, Y is Gorenstein, and f is crepant such that every reduced fibre above a closed
point in Z contains precisely one irreducible curve. Then
(1) Dz ∈ CMS OY,z for all closed points z ∈ Z.
(2) D ∈ CMS Y .
(3) TwistX is an autoequivalence of X.
Proof. Since there is only one curve above each point z ∈ Z, each Acon,z is local and so
bY,z is a hypersurface.
by 2.19 every O(1) The assertion can be checked at the completion, thus we can assume that Y = Spec R,
with maximal ideal m lying in Z. We just need to check that Acon ∈ CMS R. But since
X is smooth, pdA Acon < ∞, and also by assumption dim R = 3, hence it follows by e.g.
[IW2, 6.19(4)] that inj.dimAcon Acon ≤ 1. This being the case, since further Acon is local,
by Ramras [R, 2.15] there is a chain of equalities
depthR Acon = dimR Acon = inj.dimAcon Aconand hence Acon ∈ CMS R.
(2) Since the support of D equals Z by 2.16, this is an immediate corollary of (1).
(3) Since X is smooth, automatically E ∈ Perf(X). Hence the result follows by combining (2) and 6.3(3).
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Will Donovan, Kavli IPMU (WPI), UTIAS, The University of Tokyo, Kashiwa, Chiba 2778583, Japan.
E-mail address: will.donovan@ipmu.jp
Michael Wemyss, School of Mathematics and Statistics, University of Glasgow, 15 University Gardens, Glasgow, G12 8QW, UK.
E-mail address: michael.wemyss@glasgow.ac.uk