Singular Derived Categories of Q-factorial terminalizations and Maximal Modification Algebras
aa r X i v : . [ m a t h . AG ] A p r SINGULAR DERIVED CATEGORIES OF Q -FACTORIALTERMINALIZATIONS AND MAXIMAL MODIFICATION ALGEBRAS OSAMU IYAMA AND MICHAEL WEMYSS
Abstract.
Let X be a Gorenstein normal 3-fold satisfying (ELF) with local ringswhich are at worst isolated hypersurface (e.g. terminal) singularities. By using thesingular derived category D sg ( X ) and its idempotent completion D sg ( X ), we givenecessary and sufficient categorical conditions for X to be Q -factorial and completelocally Q -factorial respectively. We then relate this information to maximal modi-fication algebras(=MMAs), introduced in [IW2], by showing that if an algebra Λ isderived equivalent to X as above, then X is Q -factorial if and only if Λ is an MMA.Thus all rings derived equivalent to Q -factorial terminalizations in dimension threeare MMAs. As an application, we extend some of the algebraic results in [BIKR] and[DH] using geometric arguments. Contents
1. Introduction 11.1. Overview 11.2. Results on Q -factoriality 21.3. Results on MMAs 31.4. Application to cA n singularities 31.5. Conjectures 42. Preliminaries 52.1. d -CY and d -sCY algebras 52.2. MM and CT modules 62.3. Homologically finite complexes 83. Singular derived categories and Q -factorial terminalizations 93.1. Singular derived categories of Gorenstein schemes with isolated singularities 93.2. Rigid-freeness and Q -factoriality 124. MMAs and NCCRs via tilting 134.1. The endomorphism ring of a tilting complex 134.2. Cohen–Macaulayness and crepancy 154.3. A counterexample in dimension four 185. cA n singularities via derived categories 195.1. Crepant modifications of cA n singularities 19References 231. Introduction
Overview.
The broader purpose of this work, in part a continuation of [IW2], is tounderstand (and run) certain aspects of the minimal model program (=MMP) in dimen-sion three using categorical and noncommutative techniques. As part of the conjecturalunderlying picture, it is believed in dimension three (see 1.8 below) that the theory ofnoncommutative minimal models (=MMAs below) should ‘control’ the geometry of com-mutative minimal models (= Q -factorial terminalizations) in the same way noncommuta-tive crepant resolutions (=NCCRs) control the geometry of crepant resolutions [V04b]. Ifthis is to have any chance of being true, we must first be able to extend the well-knowncharacterization of smoothness by the singular derived category to be zero to a charac-terization of Q -factoriality. This involves moving from smooth schemes to singular ones, and it is unclear geometrically what form such a derived category characterization shouldtake.However, from the noncommutative side, in the study of MMAs one homologicalcondition, which we call ‘rigid-freeness’, completely characterizes when ‘maximality’ hasbeen reached. The purpose of this paper is to show that this very same condition gives anecessary condition for a scheme with isolated Gorenstein singularities to be Q -factorial.If furthermore the singularities are hypersurfaces (e.g. Gorenstein terminal singularities)this condition is both necessary and sufficient. The ability of the noncommutative side togive new results purely in the setting of algebraic geometry adds weight to the conjecturesoutlined in § Results on Q -factoriality. We denote by CM R the category of CM (=maximalCohen-Macaulay) R -modules, and by CM R its stable category (see § sg ( X ) := D b (coh X ) / per( X ) (see § F : T → T ′ an equivalence up to direct summands if it is fully faithful and any object s ∈ T ′ is isomorphic to a direct summand of F ( t ) for some t ∈ T . The first key technicalresult is the following. Note that the latter part on the idempotent completion also followsfrom work of Orlov [O09], and is well-known to experts [BK]. Theorem 1.1. (=3.2) Suppose that X is a Gorenstein scheme of dimension d satisfying(ELF) (see 3.2 for full details and explanation), with isolated singularities { x , . . . , x n } .Then there is a triangle equivalence D sg ( X ) ֒ → n M i =1 CM O X,x i up to direct summands. Thus taking the idempotent completion gives a triangle equivalence D sg ( X ) ≃ n M i =1 CM b O X,x i . Recall that a normal scheme X is defined to be Q -factorial if for every Weil divisor D , there exists n ∈ N for which nD is Cartier. If we can always take n = 1, we say that X is locally factorial . Note that these conditions are Zariski local, i.e. X is Q -factorial(respectively, locally factorial) if and only if O X,x is Q -factorial (respectively, factorial)for all closed points x ∈ X . It is well–known (and problematic) that, unlike smoothness, Q -factoriality cannot in general be checked complete locally. Thus we define X to be complete locally Q -factorial (respectively, complete locally factorial ) if the completion b O X,x is Q -factorial (respectively, factorial) for all closed points x ∈ X .Now we recall a key notion from representation theory. Let T be a triangulatedcategory with a suspension functor [1]. An object a ∈ T is called rigid if Hom T ( a, a [1]) =0. We say that T is rigid-free if every rigid object in T is isomorphic to the zero object.Applying 1.1, we have the following result. Theorem 1.2. (=3.10, 3.11) Suppose that X is a normal 3-dimensional Gorensteinscheme over a field k , satisfying (ELF), with isolated singularities { x , . . . , x n } . (1) If D sg ( X ) is rigid-free, then X is locally factorial. (2) If D sg ( X ) is rigid-free, then X is complete locally factorial. (3) If O X,x i are hypersurfaces for all ≤ i ≤ n , then the following are equivalent: (a) X is locally factorial. (b) X is Q -factorial. (c) D sg ( X ) is rigid-free. (d) CM O X,x is rigid-free for all closed points x ∈ X . (4) If b O X,x i are hypersurfaces for all ≤ i ≤ n , then the following are equivalent: (a) X is complete locally factorial. (b) X is complete locally Q -factorial. (c) D sg ( X ) is rigid-free. (d) CM b O X,x is rigid-free for all closed points x ∈ X . -FACTORIAL TERMINALIZATIONS AND MMAS Thus the technical geometric distinction between a Q -factorial and complete locally Q -factorial variety is explained categorically by the existence of a triangulated categoryin which there are no rigid objects, but whose idempotent completion has many. Notealso that 1.2 demonstrates that by passing to the idempotent completion of D sg ( X ) welose information on the global geometry (see e.g. 2.13), and so thus should be avoided.1.3. Results on MMAs.
The homological characterization of rigid-freeness in 1.2 origi-nated in the study of modifying and maximal modifying modules (see 2.6 for definitions).Recall that if M is a maximal modifying R -module, we say that End R ( M ) is a maximalmodification algebra (=MMA) . Proposition 1.3. (=2.14, 2.3) Suppose that R is normal, Gorenstein, equi-codimensional,3-dimensional ring, and let M be a modifying R -module. Assume that End R ( M ) has onlyisolated singularities. Then End R ( M ) is an MMA if and only if the category D sg (End R ( M )) is rigid-free. Throughout this paper we will use the word variety to mean • a normal integral scheme, of finite type over a field k , satisfying (ELF) (see § X and Y are varieties, then a projective birational morphism f : Y → X iscalled crepant if f ∗ ω X = ω Y (see § Q -factorial terminalization of X is a crepant projective birational morphism f : Y → X such that Y has only Q -factorialterminal singularities. When Y is furthermore smooth, we call f a crepant resolution .Maximal modification algebras were introduced in [IW2] with the aim of generaliz-ing NCCRs to cover the more general situation when crepant resolutions do not exist.Throughout this paper, we say that Y is derived equivalent to Λ if D b (coh Y ) is equiv-alent to D b (mod Λ). The following result ensures that all algebras derived equivalent toa variety Y above are of the form End R ( M ), which relate the geometry to NCCRs andMMAs. Theorem 1.4. (=4.5) Let Y → Spec R be a projective birational morphism between d -dimensional varieties. Suppose that Y is derived equivalent to some ring Λ . If Λ ∈ ref R ,then Λ ∼ = End R ( M ) for some M ∈ ref R . Using this, we have the following result, which explains the geometric origin of thedefinition of modifying modules:
Theorem 1.5.
Let f : Y → Spec R be a projective birational morphism between d -dimensional Gorenstein varieties. Suppose that Y is derived equivalent to some ring Λ ,then (1) (=4.14) f is crepant ⇐⇒ Λ ∈ CM R . (2) (=4.15) f is a crepant resolution ⇐⇒ Λ is an NCCR of R .In either case, Λ ∼ = End R ( M ) for some M ∈ ref R . We stress that 1.5(1) holds even if Y is singular. The following consequence of 1.2,1.3 and 1.5 is one of our main results. Theorem 1.6. (=4.16) Let f : Y → Spec R be a projective birational morphism, where Y and R are both Gorenstein varieties of dimension three. Assume that Y has (at worst)isolated singularities { x , . . . , x n } where each O Y,x i is a hypersurface. If Y is derivedequivalent to some ring Λ , then the following are equivalent (1) f is crepant and Y is Q -factorial. (2) Λ is an MMA of R .In this situation, all MMAs of R have isolated singularities, and are all derived equivalent. As an important consequence, when k = C and Y has only terminal singularities andderived equivalent to some ring Λ, then Y is a Q -factorial terminalization of Spec R if andonly if Λ is an MMA. See 4.17 for more details.1.4. Application to cA n singularities. As an application of the above, we can ex-tend some results of [BIKR] and [DH]. We let K denote an algebraically closed field ofcharacteristic zero, and suppose that f , . . . , f n ∈ m := ( x, y ) ⊆ K [[ x, y ]] are irreduciblepolynomials, and let R := K [[ u, v, x, y ]] / ( uv − f . . . f n ) . OSAMU IYAMA AND MICHAEL WEMYSS
When K = C then R is a cA m singularity for m := ord( f . . . f n ) − R is not assumed to be an isolated singularity, so f , . . . , f n are not necessarilypairwise distinct. For each element ω in the symmetric group S n , define T ω := R ⊕ ( u, f ω (1) ) ⊕ ( u, f ω (1) f ω (2) ) ⊕ . . . ⊕ ( u, f ω (1) . . . f ω ( n − ) . Our next result generalizes [BIKR, 1.5] and [DH, 4.2], and follows more or less immediatelyfrom 1.6. This is stronger and more general since the results in [BIKR, DH] assume that R is an isolated singularity, and only study cluster tilting objects. Theorem 1.7. (=5.4) Let f , . . . , f n ∈ m := ( x, y ) ⊆ K [[ x, y ]] be irreducible polynomialsand R = K [[ u, v, x, y ]] / ( uv − f . . . f n ) . Then (1) Each T ω ( ω ∈ S n ) is an MM R -module which is a generator. The endomorphismrings End R ( T ω ) have isolated singularities. (2) T ω is a CT R -module for some ω ∈ S n ⇐⇒ T ω is a CT R -module for all ω ∈ S n ⇐⇒ f i / ∈ m for all ≤ i ≤ n . We remark that the proof of 1.7 also provides a conceptual geometric reason for thecondition f i / ∈ m . We refer the reader to § Conjectures.
One of the motivations for the introduction of MMAs is the followingconjecture, which naturally generalizes conjectures of Bondal–Orlov and Van den Bergh:
Conjecture 1.8.
Suppose that R is a normal Gorenstein 3-fold over C , with only canon-ical (equivalently rational, since R is Gorenstein [R87, (3.8)]) singularities. Then(1) R admits an MMA.(2) All Q -factorial terminalizations of R and all MMAs of R are derived equivalent.A special case of 1.8 is the long–standing conjecture of Van den Bergh [V04b, 4.6],namely all crepant resolutions of Spec R (both commutative and non-commutative) arederived equivalent. The results in this paper (specifically 1.6) add some weight to 1.8.We note that 1.8 is true in some situations: Theorem 1.9.
Suppose that R is a normal Gorenstein 3-fold over C whose Q -factorialterminalizations Y → Spec R have only one dimensional fibres (e.g. R is a terminalGorenstein singularity, or a cA n singularity in § The Q -factorial terminalizations are connected by a finite sequence of flops [K89],so they are derived equivalent by [C02]. By [V04a] there is a derived equivalence between Y and some algebra End R ( M ) with End R ( M ) ∈ CM R . Thus by 1.6 End R ( M ) is anMMA. On the other hand, we already know that MMAs are connected by tilting modules[IW2, 4.14], so they are all derived equivalent. (cid:3) As it stands, the proof of 1.9 heavily uses the MMP. We believe that it should bepossible to prove 1.9 (for more general situations) without using the MMP, instead buildingon the categorical techniques developed in this paper, and generalizing Van den Bergh’s[V04a] interpretation of Bridgeland–King–Reid [BKR]. Indeed, it is our belief and long-term goal to show that many of the results of the MMP come out of our categorical picture(along the lines of [B02], [BKR]), allowing us to both bypass some of the classificationsused in the MMP, and also run some aspects of the MMP in a more efficient manner.
Conventions.
Throughout commutative rings are always assumed to be noetherian, and R will always denote a commutative noetherian ring. All modules will be left modules, sofor a ring A we denote mod A to be the category of finitely generated left A -modules, andMod A will denote the category of all left A -modules. Throughout when composing maps f g will mean f then g . Note that with these conventions Hom R ( M, X ) is a End R ( M )-module and Hom R ( X, M ) is a End R ( M ) op -module. For M ∈ mod A we denote add M to be the full subcategory consisting of direct summands of finite direct sums of copiesof M , and we denote proj A := add A to be the category of finitely generated projective A -modules. We say that M ∈ mod R is a generator if R ∈ add M . For an abelian category A , we denote by D b ( A ) the bounded derived category of A .If X is a scheme, O X,x will denote the localization of the structure sheaf at the closedpoint x ∈ X . We will denote by b O X,x the completion of O X,x at the unique maximal -FACTORIAL TERMINALIZATIONS AND MMAS ideal. For us, locally will always mean Zariski locally, that is if we say that R has locallyonly isolated hypersurface singularities, we mean that each R m is an isolated hypersurfacesingularity (or is smooth). When we want to discuss the completion, we will always referto this as complete locally .Throughout, k will denote an arbitrary field. Rings and schemes will not be assumedto be finite type over k , unless specified. When we say ‘over k ’ we mean ‘of finite typeover k ’. Acknowledgements.
Thanks are due to Vanya Cheltsov, Anne-Sophie Kaloghiros,Raphael Rouquier and Ed Segal for many invaluable suggestions and discussions.2.
Preliminaries
For a commutative noetherian local ring ( R, m ) and M ∈ mod R , recall that the depth of M is defined to be depth R M := inf { i ≥ iR ( R/ m , M ) = 0 } . We say that M ∈ mod R is maximal Cohen-Macaulay (or simply, CM ) if depth R M = dim R . Now let R be a (not necessarily local) commutative noetherian ring. We say that M ∈ mod R is CM if M p is CM for all prime ideals p in R , and we say that R is a CM ring if R is a CM R -module. Denoting ( − ) ∗ := Hom R ( − , R ) : mod R → mod R , we say that X ∈ mod R is reflexive if the natural map X → X ∗∗ is an isomorphism.We denote ref R to be the category of reflexive R -modules, and we denote CM R tobe the category of CM R -modules. Throughout this section, we stress our conventionthat commutative rings are always assumed to be noetherian, and R always denotes acommutative noetherian ring.2.1. d -CY and d -sCY algebras. When dealing with singularities, throughout this pa-per we use the language of d -sCY algebras, introduced in [IR, § R be a commutative ring with dim R = d and let Λ be a module-finite R -algebra (i.e. an R -algebra which is a finitely generated R -module). For any X ∈ mod Λ,denote by E ( X ) the injective hull of X , and put E := L m ∈ Max R E ( R/ m ). This gives riseto Matlis duality D := Hom R ( − , E ). Note that if R is over an algebraically closed field k ,then D coincides with Hom k ( − , k ) on the category fl Λ by [O76, 1.1, 1.2]. We let D b fl (Λ)denote all bounded complexes with finite length cohomology. Definition 2.1.
For n ∈ Z , we call Λ n -Calabi-Yau (= n -CY) if there is a functorialisomorphism Hom
D(Mod Λ) ( X, Y [ n ]) ∼ = D Hom
D(Mod Λ) ( Y, X ) for all X ∈ D b fl (Λ) and Y ∈ D b (mod Λ) . Similarly we call Λ singular n -Calabi-Yau (= n -sCY) if the above functorial isomorphism holds for all X ∈ D b fl (Λ) and Y ∈ K b (proj Λ) . We will use the following characterizations.
Proposition 2.2. (1) [IR, 3.10]
A commutative ring R is d -sCY (respectively, d -CY) ifand only if R is Gorenstein (respectively, regular) and equi-codimensional, with dim R = d . (2) [IR, 3.3(1)] Let R be a commutative d -sCY ring and Λ a module-finite R -algebra whichis a faithful R -module. Then Λ is d -sCY if and only if Λ ∈ CM R and Hom R (Λ , R ) m ∼ = Λ m as Λ m -bimodules for all m ∈ Max R . Suppose that R is a commutative d -sCY ring, and let Λ be a module-finite R -algebrawhich is d -sCY. We say that X ∈ mod Λ is maximal Cohen-Macaulay (or simply, CM ) if X ∈ CM R . We denote by CM Λ the category of CM Λ-modules. Then X ∈ mod Λ is CMif and only if Ext iR ( X, R ) = 0 for all i ≥
1, which is equivalent to Ext i Λ ( X, Λ) = 0 for all i ≥ stable category , where wefactor out by those morphisms which factor through projective Λ-modules.On the other hand, when Λ is a noetherian ring we denoteD sg (Λ) := D b (mod Λ) / K b (proj Λ)to be the singular derived category of Λ. Since any d -sCY algebra satisfies inj . dim Λ Λ = d and inj . dim Λ op Λ = d by [IR, 3.1(6)(2)], we have the following equivalence by a standardtheorem of Buchweitz [B86, 4.4.1(2)]. OSAMU IYAMA AND MICHAEL WEMYSS
Theorem 2.3.
Suppose that R is a commutative d -sCY ring and Λ is a module-finite R -algebra which is d -sCY, then there is a triangle equivalence D sg (Λ) ≃ CM Λ . By 2.3 we identify D sg (Λ) and CM Λ in the rest of this paper. Now we recall twoimportant notions. Definition 2.4. [A84]
Let R be a commutative d -sCY ring and Λ is a module-finite R -algebra which is d -sCY. (1) We say that Λ is non-singular if gl . dim Λ = d . (2) We say that Λ has isolated singularities if gl . dim Λ p = dim R p for all non-maximalprimes p of R . Proposition 2.5.
Suppose that R is a commutative d -sCY ring and Λ is a module-finite R -algebra which is d -sCY. Then (1) Λ is non-singular if and only if CM Λ = 0 . (2) Λ has isolated singularities if and only if all Hom-sets in CM Λ are finite length R -modules.Proof. These are well-known (e.g. [Y90, 3.3], [A84]), but for convenience of the reader wegive the proof.(1) See for example [IW2, 2.17(1) ⇔ (3)].(2) ( ⇒ ) Since Hom CM Λ ( X, Y ) ∼ = Ext (Ω − X, Y ) and Ω − X ∈ CM Λ, we know thatSupp R Hom
CM Λ ( X, Y ) consists of maximal ideals (e.g. [IW2, 2.6]). Thus the assertionfollows.( ⇐ ) Let p be a prime ideal of R with ht p < d and let X ∈ mod Λ p . Certainly we can find Y ∈ mod Λ with Y p ∼ = X and so consider a projective resolution0 → K → P d − → . . . → P → P → Y → ∈ CM R , by localizing and using the depth lemma we see that K ∈ CM R , i.e. K ∈ CM Λ. Consider0 → Ω K → P d → K → , (2.A)then since Ext ( K, Ω K ) ∼ = Hom CM Λ (Ω K, Ω K ) has finite length, it is supported onlyon maximal ideals. Hence (2.A) splits under localization to p and so K p is free. Thusproj . dim Λ p ( X ) < d . By Auslander–Buchsbaum, gl . dim Λ p = dim R p . (cid:3) MM and CT modules.
Recall that M ∈ ref R gives an NCCR
Λ := End R ( M ) of R if Λ ∈ CM R and gl . dim Λ = d [V04b]. In this case Λ is d -sCY by 2.8(2) below, and bydefinition Λ is non-singular.The following more general notions are quite natural from a representation theoreticviewpoint. Definition 2.6. [IW2]
Let R be a d -sCY ring. Then(1) M ∈ ref R is called a rigid module if Ext R ( M, M ) = 0 .(2) M ∈ ref R is called a modifying module if End R ( M ) ∈ CM R .(3) We say M ∈ ref R is a maximal modifying (=MM) module if it is modifying andfurther if M ⊕ Y is modifying for Y ∈ ref R , then Y ∈ add M . Equivalently, add M = { X ∈ ref R | End R ( M ⊕ X ) ∈ CM R } . (4) We call M ∈ CM R a CT module if add M = { X ∈ CM R | Hom R ( M, X ) ∈ CM R } . The following results will be used extensively. As in [IW2], if X ∈ mod R , we denote fl X to be the largest finite length sub- R -module of X . Lemma 2.7. [IW2, 2.7, 5.12]
Suppose that R is a 3-sCY ring, and let M ∈ ref R . (1) If M is modifying, then fl Ext R ( M, M ) = 0 . The converse holds if M ∈ CM R . (2) Assume that R is an isolated singularity. If M is modifying, then it is rigid. Theconverse holds if M ∈ CM R . -FACTORIAL TERMINALIZATIONS AND MMAS Lemma 2.8.
Suppose that R is a d-sCY normal domain, and M ∈ ref R . Let Λ =End R ( M ) . Then (1) [IR, 2.4(3)] Λ ∼ = Hom R (Λ , R ) as Λ -bimodules. (2) [IW2, 2.22(2)] M is modifying if and only if Λ is d -sCY . We remark that the property CT can be checked complete locally [IW2, 5.5], that is M is a CT R -module if and only if c M m is a CT b R m -module for all m ∈ Max R . In ourcontext in §
3, this corresponds to the fact that a scheme X is non-singular if and only ifcomplete locally it is non-singular.In contrast, the property of being MM cannot be checked complete locally. If c M m isan MM b R m -module for all m ∈ Max R , then M is an MM R -module. However the converseis not true (see e.g. 2.13 below). This corresponds to the difference between Q -factorialand complete locally Q -factorial singularities in § Proposition 2.9.
Suppose R is 3-sCY normal domain, with isolated singularities. Con-sider the statements (1) R is an MM module. (2) Every modifying R -module is projective. (3) CM R is rigid-free. (4) R is locally factorial. (5) R is Q -factorial.Then we have (1) ⇔ (2) ⇔ (3) ⇒ (4) ⇒ (5) .Proof. Since R is an isolated singularity, modifying modules are rigid by 2.7(2).(1) ⇒ (2) Let M ∈ ref R be any modifying module. Since R is MM, there exists an exactsequence 0 → F → F f → M with F i ∈ add R such that f is a right (add R )-approximation, by [IW2, 4.12]. Then clearly f is surjective, and so we have proj . dim R ( M ) ≤
1. By [AG, 4.10], if proj . dim R ( M ) = 1then Ext R ( M, M ) = 0, a contradiction. Hence M is projective.(2) ⇒ (3) is clear, since Hom CM R ( M, M [1]) ∼ = Ext R ( M, M ).(3) ⇒ (1) Suppose that End R ( R ⊕ X ) ∈ CM R for some X ∈ ref R . Then X = Hom R ( R, X )is a CM R -module which is a rigid object in CM R . Hence X is zero in CM R , i.e. X ∈ add R .(2) ⇒ (4) Since R is a normal domain, all members of the class group have R as endomor-phism rings, and thus they are modifying R -modules. Hence every member of the classgroup is projective, so the result follows.(4) ⇒ (5) is clear. (cid:3) Below we will use the following result from commutative algebra.
Theorem 2.10. [D1, 3.1(1)]
Let S be a regular local ring of dimension four containinga field k , and let R = S/ ( f ) be a hypersurface. If R is a Q -factorial isolated singularity,then the free modules are the only modifying modules.Proof. This is the equi-characteristic version of [D1, 3.1(1)], but since loc. cit. is in some-what different language than used here, so we sketch Dao’s proof. Let N be a modifying R -module. Since R is isolated, N is rigid by 2.7(2). On the other hand, by [D2], N isa Tor-rigid R -module. These two facts imply, via a result of Jothilingham [J75, MainTheorem] (see also [D1, 2.4]), that N is a free R -module. (cid:3) Now we have the following result, which strengthens 2.9.
Theorem 2.11.
Let R be a 3-sCY normal domain over k , which locally has only isolatedhypersurface singularities. Then the following are equivalent: (1) R is an MM module. (2) Every modifying R -module is projective. (3) CM R is rigid-free. OSAMU IYAMA AND MICHAEL WEMYSS (4) R is locally factorial. (5) R is Q -factorial.Proof. By 2.9 we only have to show (5) ⇒ (1). Suppose that M ∈ ref R with End R ( R ⊕ M ) ∈ CM R . Then for any m ∈ Max R we have End R m ( R m ⊕ M m ) ∈ CM R m with R m satisfying the assumptions of 2.10, hence M m is a free R m -module. Thus M m ∈ add R m for all m ∈ Max R , so M ∈ add R [IW2, 2.26]. (cid:3) Remark 2.12.
The corollary is false if we remove the isolated singularities assumption,since then (4) does not necessarily imply (1). An example is given by R = C [ x, y, z, t ] / ( x + y + z ). Also, note that in the isolated singularity case it is unclear if the hypersurfaceassumption is strictly necessary; indeed Dao conjectures that 2.10 still holds in the case ofcomplete intersections [D3, § Example 2.13.
Consider the element f = x + x + y − uv in the ring C [ u, v, x, y ], andset R := C [ u, v, x, y ] / ( f ). Then R is an MM R -module, but b R m is not an MM b R m -module,where m = ( u, v, x, y ). Proof.
Since R has an isolated singularity only at m := ( u, v, x, y ) and √ x + 1 does notexist in R m , we have that R m is a factorial hypersurface singularity for all m ∈ Max R .Hence if M is a modifying R -module, then M m is a free R m -module for any m ∈ Max R by 2.10. Thus M is projective.For the last statement, since √ x + 1 exists (and is a unit) in the completion, b R m isisomorphic to C [[ u, v, x, y ]] / ( uv − xy ), for which b R m ⊕ ( u, x ) is a modifying module. (cid:3) Now we give a general categorical criterion for a given modifying module to be max-imal, generalizing 2.9.
Proposition 2.14.
Suppose that R is normal 3-sCY, let M be a modifying R -module andset Λ := End R ( M ) . If M is an MM R -module, then CM Λ is rigid-free. Moreover theconverse holds if Λ has isolated singularities.Proof. By [IW2, 4.8(1)], M is an MM R -module if and only if there is no non-zero object X ∈ CM Λ such that fl Hom
CM Λ ( X, X [1]) = 0. This clearly implies CM Λ is rigid-free.By 2.5(2), the converse is true if Λ has isolated singularities. (cid:3)
Remark 2.15.
We conjectured in [IW2] that in dimension three MMAs always haveisolated singularities (see 4.16 for some evidence), so the key property from 2.14 is thatthe stable category of CM modules is rigid-free. Note that Λ := End R ( M ) ∈ CM R is anNCCR if and only if the stable category CM Λ is zero. Hence when passing from NCCRsto MMAs, the categories D sg (Λ) ≃ CM Λ are no longer zero, but instead rigid-free. Thismotivates § Homologically finite complexes.
In this subsection we wish to consider the gen-eral setting of commutative rings, and so in particular the Krull dimension may at times beinfinite. The following is based on some Danish handwritten notes of Hans-Bjørn Foxby.
Theorem 2.16.
Let R be a commutative ring (not necessarily of finite Krull dimension),let Λ be a module-finite R -algebra, and let M ∈ mod Λ . If proj . dim Λ p ( M p ) < ∞ for all p ∈ Spec R , then proj . dim Λ ( M ) < ∞ .Proof. Take a projective resolution . . . → P d → P d → M → , of M with finitely generated projective R -modules P i , and set K := Im d = M and K i := Im d i = Ker d i − for i >
0. Nowproj . dim Λ ( M ) < n ⇐⇒ Ext n Λ ( M, K n ) = 0 , and the right hand side is equivalent to the condition that Supp R Ext n Λ ( M, K n ) is empty.Localizing at p ∈ Spec R , we haveproj . dim Λ p ( M p ) ≥ n ⇐⇒ Ext n Λ ( M, K n ) p = Ext n Λ p ( M p , ( K n ) p ) = 0 ⇐⇒ p ∈ Supp R Ext n Λ ( M, K n ) . -FACTORIAL TERMINALIZATIONS AND MMAS Since Ext n Λ ( M, K n ) is a finitely generated R -module, we haveSupp R Ext n Λ ( M, K n ) = V (Ann R (Ext n Λ ( M, K n ))) . For ease of notation set I n = Ann R (Ext n Λ ( M, K n )), thenproj . dim Λ p ( M p ) ≥ n ⇐⇒ p ∈ V ( I n ) . (2.B)In particular, we have a decreasing sequence V ( I ) ⊇ V ( I ) ⊇ V ( I ) ⊇ . . . , so we have V ( I t ) = V ( I t +1 ) = . . . for some t since the Spec R is noetherian by our assump-tion. If p ∈ V ( I t ) = T ∞ n = t V ( I n ), then proj . dim Λ p ( M p ) = ∞ by (2.B), a contradiction.Consequently V ( I t ) = ∅ , so we have Ext t Λ ( M, K t ) = 0 and proj . dim Λ ( M ) < t . (cid:3) Corollary 2.17.
Let R be a commutative ring and Λ a module-finite R -algebra. Then (1) For every p ∈ Spec R , the algebra Λ p has only finitely many simple modules. (2) Let M ∈ mod Λ . Then proj . dim Λ ( M ) < ∞ if and only if for all X ∈ mod Λ we have Ext j Λ ( M, X ) = 0 for j ≫ .Proof. (1) Λ p is a module-finite R p -algebra. There are only finitely many simple Λ p -modules since they are annihilated by p R p and hence they are modules over the finitedimensional ( R p / p R p )-algebra Λ p / p Λ p , which has only finitely many simple modules.(2) We only have to show ‘if’ part. For any p ∈ Spec R , we know that the localizationfunctor ( − ) p : mod Λ → mod Λ p is essentially surjective, and that the completion functor d ( − ) : fl Λ p → fl b Λ p is an equivalence since the completion does not change finite lengthmodules. By (1) there exists X ∈ mod Λ such that b X p is the sum of all simple b Λ p -modules.The assumptions imply that there exists t ≥ j Λ ( M, X ) = 0 for all j > t .Thus Ext j b Λ p ( c M p , b X p ) = 0 for all j > t . This implies proj . dim b Λ p ( c M p ) ≤ t since c M p has aminimal projective resolution. Thus we have proj . dim Λ p ( M p ) ≤ t by applying [IW2, 2.26,2 ⇔
4] to the t -th syzygy of M . By 2.16, it follows that proj . dim Λ ( M ) < ∞ . (cid:3) Let T be a triangulated category with a suspension functor [1]. Recall that x ∈ T iscalled homologically finite if for all y ∈ T , Hom T ( x, y [ i ]) = 0 for all but finitely many i .The following is well-known under more restrictive hypothesis: Proposition 2.18.
Let R be a commutative ring and Λ be a module-finite R -algebra.Then the homologically finite complexes in D b (mod Λ) are precisely K b (proj Λ) .Proof. Since every object in K b (proj Λ) is homologically finite, we just need to show theconverse. If X is homologically finite, replace X by its projective resolution ( P, d ). Thetruncation of P , namely τ ≥ i P , is quasi-isomorphic to X for small enough i . Fix such an i . Thus, if we can show that Ker( d i ) has finite projective dimension, certainly it followsthat X belongs to K b (proj Λ).Now by the usual short exact sequence given by truncation, there is a triangle Q → X → Ker( d i )[ i ] → Q [1]where Q ∈ K b (proj Λ). Since the first two are homologically finite, so is Ker( d i ). Inparticular, for all Y ∈ mod Λ, we have that Ext j Λ (Ker( d i ) , Y ) = 0 for large enough j . By2.17(2), Ker( d i ) has finite projective dimension, as required. (cid:3) Singular derived categories and Q -factorial terminalizations Singular derived categories of Gorenstein schemes with isolated singulari-ties.
Recall that a scheme X is called Gorenstein if O X,x is a Gorenstein local ring for allclosed points x ∈ X . We say that a scheme X satisfies (ELF) if X is separated, noether-ian, of finite Krull dimension, such that coh X has enough locally free sheaves (i.e. for all F ∈ coh X , there exists a locally free sheaf E and a surjection E ։ F ). This is automaticin many situations, for example if X is quasi–projective over a commutative noetherianring [TT, 2.1.2(c), 2.1.3]. Definition 3.1.
We say that
F ∈
Qcoh X is locally free if X can be covered with opensets U for which each F| U is a (not necessarily finitely generated) free O X | U -module. If F ∈
Qcoh X is locally free, then F x is a free O X,x -module for all closed points x ∈ X . The converse is true if F ∈ coh X .As in [O03, §
1] we consider • the full triangulated subcategory per( X ) of D b (coh X ) consisting of objects whichare isomorphic to bounded complexes of finitely generated locally free sheaves inD b (coh( X )). • the full triangulated subcategory Lfr( X ) of D b (Qcoh X ) consisting of objectswhich are isomorphic to bounded complexes of locally free sheaves in D b (Qcoh( X )).Following Orlov [O03, O09], we denoteD sg ( X ) := D b (coh X ) / per( X ) and D SG ( X ) := D b (Qcoh X ) / Lfr( X ) . Then the natural functor D sg ( X ) → D SG ( X ) is fully faithful [O03, 1.13], and we regardD sg ( X ) as a full subcategory of D SG ( X ).The aim of this subsection is to prove the following result, which plays a crucial rolein this paper. Note that part (2) also follows from work of Orlov [O09], and is well-knownto experts [BK]. Theorem 3.2.
Suppose that X is a Gorenstein scheme of dimension d satisfying (ELF),with isolated singularities { x , . . . , x n } . (1) There is a triangle equivalence D sg ( X ) → L ni =1 D sg ( O X,x i ) = L ni =1 CM O X,x i up tosummands, given by F 7→ ( F x , . . . , F x n ) . (2) Taking the idempotent completion gives a triangle equivalence D sg ( X ) ≃ L ni =1 CM b O X,x i . Since a key role in our proof of Theorem 3.2 is played by the category D SG ( X ), weneed to deal with infinitely generated modules. Let us start with recalling results byGruson–Raynaud [GR] and Jensen [J70]. Proposition 3.3.
Let R be a noetherian ring of finite Krull dimension and M ∈ Mod R .Then (1) FPD( R ) := sup { proj . dim( M ) | M ∈ Mod R and proj . dim R ( M ) < ∞} = dim R . (2) flat . dim R ( M ) < ∞ implies proj . dim R ( M ) < ∞ .Proof. (1) is due to Bass and Gruson–Raynaud [GR, II.3.2.6] (see also [F77, 3.2]).(2) is due to Jensen [J70, Prop.6] (see also [F77, 3.3]). (cid:3) We have the following immediate consequence.
Lemma 3.4.
Let R be a noetherian ring of finite Krull dimension. Then for any M ∈ Mod R , proj . dim R m ( M m ) < ∞ for all m ∈ Max R ⇐⇒ proj . dim R ( M ) ≤ dim R .Proof. ( ⇐ ) is trivial( ⇒ ) The hypothesis implies that proj . dim R p ( M p ) < ∞ for all p ∈ Spec R . Thus by3.3(1), we have proj . dim R p ( M p ) ≤ dim R p ≤ dim R for all p ∈ Spec R . In particularflat . dim R p ( M p ) ≤ dim R for all p ∈ Spec R . Now X ∈ Mod R is zero if X p = 0 forall p ∈ Spec R . This implies flat . dim R ( M ) ≤ dim R since Tor groups localize also forinfinitely generated modules. By 3.3(2), M has finite projective dimension. Again by3.3(1), we have proj . dim R ( M ) ≤ dim R . (cid:3) The following is easy:
Lemma 3.5.
Let R be a commutative noetherian ring of finite Krull dimension, then K b (Free R ) = Lfr(Spec R ) = K b (Proj R ) .Proof. We have natural inclusions K b (Free R ) ⊆ Lfr(Spec R ) ⊆ K b (Proj R ) since everyfree module is clearly locally free, and further every locally free R -module is locally projec-tive, thus projective by [GR, II.3.1.4(3)]. We only have to show K b (Proj R ) = K b (Free R ).The proof is similar to [R89, 2.2] — by the Eilenberg swindle, if P ∈ Proj R , there ex-ists F ∈ Free R such that P ⊕ F is free. Hence from every object of K b (Proj R ) wecan get to an object of K b (Free R ) by taking the direct sum with complexes of the form0 → F → F →
0, which are zero objects in the homotopy category. (cid:3)
Corollary 3.6.
Suppose that X satisfies (ELF). If F ∈
Qcoh X satisfies proj . dim O X,x ( F x ) < ∞ for all closed points x ∈ X , then F ∈
Lfr( X ) . -FACTORIAL TERMINALIZATIONS AND MMAS Proof.
Since X satisfies (ELF), membership of Lfr( X ) can be checked locally (see [O03,1.7, proof of 1.14]). Thus we only have to show F| U ∈ Lfr( U ) for each affine open subset U = Spec R of X . Since F m is an R m -module with finite projective dimension for all m ∈ Max R , we have F| U ∈ K b (Proj R ) by 3.4. By 3.5, we have F| U ∈ Lfr( U ), so theassertion follows. (cid:3) We will also require the following well-known lemma, due to Orlov. If s is an object ina triangulated category T , we denote thick T ( s ) to be the smallest triangulated subcategoryof T which is closed under direct summands and isomorphisms and contains s . Lemma 3.7. (Orlov)
Suppose that X satisfies (ELF), and has only isolated singularities { x , . . . , x n } . Denote the corresponding skyscraper sheaves by k , . . . , k n . Then D sg ( X ) =thick D sg ( X ) ( L ni =1 k i ) .Proof. By assumption the singular locus consists of a finite number of closed points, henceis closed. It follows that D sg ( X ) = thick D sg ( X ) (coh { x ,...,x n } X ) by [C10, 1.2]. It is clearthat every coherent sheaf supported in { x , . . . , x n } belongs to thick D sg ( X ) ( L ni =1 k i ). (cid:3) Now we are ready to prove 3.2.For each x i , consider a morphism f i := (Spec O X,x i g i → Spec R i h i → X ) where Spec R i issome affine open subset of X containing x i . Let S := O X,x ⊕ . . . ⊕O X,x n and Y := Spec S .Then the collection of the f i induce a morphism Y = ` ni =1 Spec O X,x i f → X. The functor g i ∗ is just extension of scalars corresponding to a localization R i → R i m , so g i ∗ is exact and preserves quasi-coherence. Further h i is an affine morphism (since X isseparated), hence h i ∗ also is exact and preserves quasi-coherence. Hence each f i ∗ is exactand preserves quasi-coherence. Thus we have an adjoint pairQcoh X Qcoh Y = n L i =1 Qcoh O X,x i f ∗ f ∗ which are explicitly given by f ∗ F = ( F x , . . . , F x n ) and f ∗ ( G , . . . , G n ) = L ni =1 f i ∗ G i .These functors are exact, so induce an adjoint pairD b (Qcoh X ) D b (Qcoh Y ) = n L i =1 D b (Qcoh O X,x i ) f ∗ f ∗ in the obvious way.We will show f ∗ (Lfr( X )) ⊆ Lfr( Y ) and f ∗ (Lfr( Y )) ⊆ Lfr( X ), since this then inducesan adjoint pair D SG ( X ) D SG ( Y ) = L ni =1 D SG ( O X,x i ). f ∗ f ∗ by [O03, 1.2]. It is clear that f ∗ takes locally free sheaves to projective modules, and so f ∗ (Lfr( X )) ⊆ Lfr( Y ). On the other hand, each f i ∗ is an affine, flat morphism so f i ∗ O X,x i is a flat O X -module. Further, each f i ∗ preserves sums, so takes projective O X,x i -modulesto flat O X -modules. Consequently, it follows that f ∗ takes a projective S -module P toa sheaf f ∗ ( P ) for which f ∗ ( P ) x is a flat O X,x -module for all closed points x ∈ X . By3.3(2) we have proj . dim O X,x ( f ∗ ( P ) x ) < ∞ for all closed points x ∈ X , hence by 3.6 f ∗ ( P ) ∈ Lfr( X ) holds. Thus we have f ∗ (Lfr( Y )) ⊆ Lfr( X ).Now denote the skyscraper sheaves in X corresponding to the singular points x , . . . , x n by k , . . . , k n . By 3.7 we know that D sg ( X ) = thick D sg ( X ) ( L ni =1 k i ). This impliesD sg ( X ) ⊆ thick D SG ( X ) ( L ni =1 k i ).Let α : 1 → f ∗ ◦ f ∗ be the unit. Since f ∗ ◦ f ∗ ( k i ) = f ∗ (0 , . . . , , k i , , . . . ,
0) = f i ∗ k i = k i for all 1 ≤ i ≤ n , we have that α : 1 → f ∗ ◦ f ∗ is an isomorphism on thick D SG ( X ) ( L ni =1 k i ).In particular f ∗ is fully faithful on thick D SG ( X ) ( L ni =1 k i ). Since f ∗ clearly takes D sg ( X )to D sg ( Y ), and D sg ( X ) ⊆ thick D SG ( X ) ( L ni =1 k i ), we deduce thatD sg ( X ) f ∗ −→ D sg ( Y ) is fully faithful. On the other hand, since D sg ( Y ) = thick D sg ( Y ) ( f ∗ L ni =1 k i ) holds by 3.7,we have D sg ( Y ) = thick D sg ( Y ) ( f ∗ (D sg ( X ))). This immediately implies that f ∗ : D sg ( X ) → D sg ( Y ) is an equivalence up to direct summands (e.g. [N01, 2.1.39]).The first statement (1) in the theorem now follows from the well-known equivalenceD sg ( O X,x i ) ≃ CM O X,x i due to Buchweitz [B86, 4.4.1]. The second statement (2) followsfrom (1), since the idempotent completion of CM O X,x i is CM b O X,x i [KMV, A.1]. (cid:3) The first corollary of 3.2 is the following alternative proof of [O03, 1.24], which wewill use later:
Corollary 3.8.
Suppose that X is a d -dimensional Gorenstein scheme over k , satisfying(ELF), with isolated singularities. Then all Hom-sets in D sg ( X ) are finite dimensional k -vector spaces.Proof. By 3.2(1), it is enough to show that all Hom-sets in CM O X,x i are finite dimensional k -vector spaces. Since each O X,x i is isolated, all Hom-sets in CM O X,x i are finite length O X,x i -modules by 2.5(2). But each O X,x i is a localization of a finitely generated k -algebra, so by the Nullstellensatz its residue field is a finite extension of k . Thus theassertion follows. (cid:3) Rigid-freeness and Q -factoriality. In this section we give our main result whichcharacterizes the Q -factorial property in terms of the singular derived category, and thenrelate this to MMAs. Definition 3.9.
Suppose that X is a Gorenstein scheme. We say that F ∈ coh X is a Cohen–Macaulay (=CM) sheaf if F x is a CM O X,x -module for all closed points x ∈ X . Under the assumption that X is Gorenstein, F ∈ coh X is a CM sheaf if and only if E xt i ( F , O X ) x (= Ext i O X,x ( F x , O X,x )) = 0for all closed points x ∈ X and all i >
0. Thus
F ∈ coh X is a CM sheaf if and only if E xt i ( F , O X ) = 0 for all i > ⇒ (4). Proposition 3.10.
Suppose that X is a Gorenstein normal scheme satisfying (ELF), ofdimension three which has only isolated singularities { x , . . . , x n } . If D sg ( X ) is rigid-free,then X is locally factorial.Proof. We need to show that any reflexive sheaf G on X of rank one is locally free. Forall closed points x ∈ X , we have a reflexive O X,x -module G x of rank one. Since O X,x is anormal domain, we have End O X,x ( G x ) = O X,x . Since O X,x is an isolated singularity, wehave Ext O X,x ( G x , G x ) = 0 by 2.7(2).Since X satisfies (ELF), there exists an exact sequence 0 → K → V → G → V . Localizing at x , we have an exact sequence0 → K x → V x → G x → O X,x -modules with a projective O X,x -module V x . Since G x ∈ ref O X,x , we have K x ∈ CM O X,x . Since Ext O X,x ( G x , G x ) = 0, we have End O X,x ( K x ) ∈ CM O X,x by applying[IW2, 4.10] to (3.C). Thus Ext O X,x ( K x , K x ) = 0 holds, again by 2.7(2).Since K x ∈ CM O X,x , we have by 3.2(1)Hom D sg ( X ) ( K , K [1]) ∼ = n M i =1 Ext O X,xi ( K x i , K x i ) = 0 . Since D sg ( X ) is rigid-free by our assumption, we have K ∈ per( X ). Thus K x is a CM O X,x -module of finite projective dimension. By the Auslander–Buchsbaum equality, K x has to be a projective O X,x -module. By (3.C), we have proj . dim O X,x G x ≤ x ∈ X .If proj . dim O X,x G x = 1, then Ext O X,x ( G x , G x ) = 0 by [AG, 4.10], a contradiction. Thus G x is a projective O X,x -module for all x ∈ X . Hence G is locally free. (cid:3) Now we are ready to state our main result, which gives a relationship between facto-riality of schemes and rigid-freeness of their singular derived categories. -FACTORIAL TERMINALIZATIONS AND MMAS Theorem 3.11.
Suppose that X is a normal 3-dimensional Gorenstein scheme over k ,satisfying (ELF), with isolated singularities { x , . . . , x n } . (1) If O X,x i are hypersurfaces for all ≤ i ≤ n , then the following are equivalent: (a) X is locally factorial. (b) X is Q -factorial. (c) D sg ( X ) is rigid-free. (d) CM O X,x is rigid-free for all closed points x ∈ X . (2) If b O X,x i are hypersurfaces for all ≤ i ≤ n , then the following are equivalent: (a) X is complete locally factorial. (b) X is complete locally Q -factorial. (c) D sg ( X ) is rigid-free. (d) CM b O X,x is rigid-free for all closed points x ∈ X .Proof. (1) (a) ⇒ (b) is clear. (b) ⇒ (d) follows from 2.11(5) ⇒ (3). (d) ⇒ (c) follows from3.2(1). (c) ⇒ (a) follows from 3.10.(2) The proof is identical to the proof in (1). (cid:3) MMAs and NCCRs via tilting
The aim of this section is to give explicit information on rings that are derivedequivalent to certain varieties. Our main results (4.5, 4.14, 4.15 and 4.16) show a strongrelationship between the crepancy of birational morphisms and the property that therings are Cohen-Macaualy. In particular, we show that all algebras derived equivalent to Q -factorial terminalizations in dimension three are MMAs.Throughout this section, we continue to use the word variety to mean a normalintegral scheme, of finite type over a field k , satisfying (ELF). Some of the results belowremain true with weaker assumptions. If X is a variety, we denote D X := g ! k to be thedualizing complex of X [H66, Ch V § g : X → Spec k is the structure morphism.If further X is CM of dimension d , we will always use ω X to denote D X [ − d ], and refer toit as the geometric canonical . Thus, although canonical modules and dualizing complexesare not unique, throughout this section ω X has a fixed meaning. Note that crepancy isdefined with respect to this canonical, i.e. f : Y → X is called crepant if f ∗ ω X = ω Y .4.1. The endomorphism ring of a tilting complex.
We call
V ∈ per( Y ) a pretiltingcomplex if Hom D( Y ) ( V , V [ i ]) = 0 for all i = 0, and a tilting complex if it is a pretiltingcomplex satisfying thick D( Y ) ( V ) = per( Y ). If Y is derived equivalent to Λ, there exists atilting complex V such that Λ ∼ = End D( Y ) ( V ).The following two results are well-known. Lemma 4.1.
Suppose that X is a scheme over a field k satisfying (ELF). Suppose that X is derived equivalent to some ring Λ , then (1) per( X ) ≃ K b (proj Λ) . (2) D sg ( X ) ≃ D sg (Λ) .Proof. (1) Clearly the derived equivalence D b (coh X ) ≃ D b (mod Λ) induces an equivalencebetween full subcategories of homologically finite complexes. We observed in 2.18 thatK b (proj Λ) are precisely the homologically finite complexes in D b (mod Λ). On the otherhand, per( X ) are precisely the homologically finite complexes in D b (coh X ) by [O05, 1.11].Thus the assertion follows.(2) follows immediately from (1). (cid:3) Corollary 4.2.
Suppose that X is a scheme over a field k satisfying (ELF). Suppose that X is derived equivalent to some ring Λ , which is a d -sCY algebra over a normal d -sCY k -algebra. Then (1) X is smooth if and only if Λ is non-singular (in the sense of 2.4). (2) If X has only isolated singularities, then Λ has isolated singularities.Proof. The derived equivalence induces D sg ( X ) ≃ D sg (Λ) by 4.1. It is very well-knownthat X is smooth if and only if D sg ( X ) = 0, so (1) follows from 2.5(1). (2) is immediatefrom 3.8 and 2.5(2). (cid:3) In the rest of this subsection we suppose that we have a projective birational mor-phism f : Y → Spec R and that Y is derived equivalent to some ring Λ, and we investigatewhen Λ has the form End R ( M ) for some M ∈ ref R . We need the following well-knownlemma. Lemma 4.3.
Let Y → Spec R be a projective birational morphism between d -dimensionalnormal integral schemes. Let p ∈ Spec R and consider the following pullback diagram: Y ′ Y Spec R p Spec R ijg f (1) If p is a height one prime, then g is an isomorphism. (2) If V is a pretilting complex of Y with Λ := End D( Y ) ( V ) , then Λ is a module-finite R -algebra and i ∗ V is a pretilting complex of Y ′ with Λ p ∼ = End Y ′ ( i ∗ V ) .Proof. (1) Since Y and Spec R are integral schemes and f is a projective birational mor-phism, it is well-known that f is the blowup of some ideal I of R [Liu, 8.1.24]. Butblowups are preserved under flat base change, and in fact g is the blowup at the ideal I p [Liu, 8.1.14]. Now since R is normal R p is a discrete valuation ring, so every ideal isprincipal. Thus g is the blowup of a Cartier divisor, and so is an isomorphism by theuniversal property of blowing up.(2) Since coh Y is an R -linear category, so is its derived category, and hence Λ has thestructure of an R -algebra. Since V is a pretilting complex, we haveΛ = R Hom Y ( V , V ) = R f ∗ R H om Y ( V , V ) . (4.D)Now since V is perfect, R H om Y ( V , V ) is a bounded complex of coherent sheaves. Further,since f is proper, f ∗ preserves coherence and so R f ∗ : D b (coh Y ) → D b (mod R ). Therefore R f ∗ R H om Y ( V , V ) = Λ is a finitely generated R -module.Lastly, applying j ∗ to both sides of (4.D) givesΛ p = j ∗ R f ∗ R H om Y ( V , V ) = R g ∗ i ∗ R H om Y ( V , V ) = R g ∗ R H om Y ′ ( i ∗ V , i ∗ V )= R Hom Y ′ ( i ∗ V , i ∗ V ) , where the second equality is flat base change, and the third holds since i is flat and V iscoherent [H66, II.5.8]. Thus we have the assertion. (cid:3) We will also need the following result of Auslander–Goldman. A proof can be foundin [IW2, 2.11].
Proposition 4.4. [AG]
Let R be a normal domain, and let Λ be a module-finite R -algebra.Then the following conditions are equivalent: (1) There exists M ∈ ref R such that Λ ∼ = End R ( M ) as R -algebras. (2) Λ ∈ ref R and further Λ p is Morita equivalent to R p for all p ∈ Spec R with ht p = 1 . This leads to the following, which is an analogue of [IW2, 4.6(1)].
Theorem 4.5.
Let Y → Spec R be a projective birational morphism between d -dimensionalnormal integral schemes, and suppose that Y is derived equivalent to some ring Λ . Then Λ is a module-finite R -algebra. If moreover Λ ∈ ref R , then Λ ∼ = End R ( M ) for some M ∈ ref R .Proof. Since Λ ∼ = End D( Y ) ( V ) for some tilting complex V , it is a module-finite R -algebraby 4.3(2). Now consider a height one prime p ∈ Spec R , then by base change we have apullback diagram Y ′ Y Spec R p Spec R ijg f By 4.3(1) g is an isomorphism, and by 4.3(2) Λ p = R Hom Y ′ ( i ∗ V , i ∗ V ). Note that i ∗ V 6 = 0since Λ ∈ ref R , so since R is normal necessarily Λ is supported everywhere. Since R p is a -FACTORIAL TERMINALIZATIONS AND MMAS local ring, the only perfect complexes x with Hom D( R p ) ( x, x [ i ]) = 0 for all i > g ∗ i ∗ V ∼ = R a p [ b ] for some a ∈ N and b ∈ Z . HenceΛ p ∼ = End R p ( R a p ), which is Morita equivalent to R p . This holds for all height one primes,so by 4.4 the assertion follows. (cid:3) Cohen–Macaulayness and crepancy.
Let f : Y → Spec R be a projective bira-tional morphism such that Y is derived equivalent to some ring Λ. In this section we showthat f is crepant if and only if Λ ∈ CM R . To do this requires the following version ofGrothendieck duality. Theorem 4.6.
Suppose that f : Y → Spec R is a projective morphism, then R Hom Y ( F , f ! G ) = R Hom R ( R Γ F , G ) for all F ∈
D(Qcoh Y ) , and G ∈ D + (Mod R ) .Proof. The sheafified duality theorem [N96, §
6, 6.3] reads R f ∗ R H om Y ( F , f ! G ) = R H om R ( R f ∗ F , G ) . In this situation, R f ∗ = R Γ, and since Spec R is affine, global and local hom agree. (cid:3) The following is also well-known [V04a, 3.2.9].
Lemma 4.7.
Suppose that f : Y → Spec R is a projective birational morphism, where Y and R are both Gorenstein varieties of dimension d . Then (1) f ! O R is a line bundle. (2) f is crepant if and only if f ! O R = O Y .Proof. Since R is Gorenstein ω R is a line bundle and thus is a compact object in D(Mod R ).Hence by [N96, p227–228] we have f ! ω R = L f ∗ ω R ⊗ L Y f ! O R = f ∗ ω R ⊗ Y f ! O R and so ω Y = D Y [ − d ] = f ! D R [ − d ] = f ! ω R = f ∗ ω R ⊗ Y f ! O R . Since both ω Y and f ∗ ω R are line bundles, f ! O R = ( f ∗ ω R ) − ⊗ Y ω Y is a line bundle.Moreover f is crepant if and only if f ∗ ω R = ω Y if and only if f ! O R = O Y . (cid:3) The following result show that crepancy implies that Λ is Cohen-Macaulay.
Lemma 4.8.
Suppose that f : Y → Spec R is a crepant projective birational morphism,where Y and R are both Gorenstein varieties of dimension d . Then End D( Y ) ( V ) ∈ CM R for any pretilting complex V of Y .Proof. We have R H om Y ( V , V ) ∼ = R H om Y ( V , V ⊗ L Y O Y ) ∼ = R H om Y ( R H om Y ( V , V ) , O Y ) . Applying R Γ to both sides, we have R Hom Y ( V , V ) ∼ = R Hom Y ( R H om Y ( V , V ) , O Y ) ∼ = R Hom Y ( R H om Y ( V , V ) , f ! O R ) ∼ = R Hom R ( R Hom Y ( V , V ) , R ) . Since V is pretilting R Hom Y ( V , V ) = End D( Y ) ( V ), so the above isomorphism reduces toEnd D( Y ) ( V ) = R Hom R (End D( Y ) ( V ) , R ) . Hence applying H i to both sides, we obtain Ext iR (End D( Y ) ( V ) , R ) = 0 for all i >
0. ThusEnd D( Y ) ( V ) ∈ CM R , as required. (cid:3) To show the converse of 4.8, namely Λ is Cohen–Macaulay implies crepancy, willinvolve Serre functors. However, since we are in the singular setting these are somewhatmore subtle than usual. The following is based on [G06, 7.2.6].
Definition 4.9.
Suppose that Y → Spec R is a morphism where R is CM ring with acanonical module C R . We say that a functor S : per( Y ) → per( Y ) is a Serre functorrelative to C R if there are functorial isomorphisms R Hom R ( R Hom Y ( F , G ) , C R ) ∼ = R Hom Y ( G , S ( F )) in D(Mod R ) for all F , G ∈ per( Y ) . If Λ is a module-finite R -algebra, we define Serrefunctor S : K b (proj Λ) → K b (proj Λ) relative to C R in a similar way. Remark 4.10.
We remark that since we are working in the non-local CM setting, canon-ical modules are not unique. Although crepancy is defined with respect to the geometriccanonical ω R , and [G06] defines Serre functors with respect to ω R , there are benefits ofallowing the flexibility of different canonical modules. Bridgeland–King–Reid [BKR, 3.2]had technical problems when the geometric canonical is not trivial, since they first hadto work locally (where the canonical is trivial), and then extend this globally. Below, weavoid this problem by considering Serre functors with respect to the canonical module R ,using the trick in 4.13(2).The following two lemmas are standard. Lemma 4.11.
Suppose that S and T are two Serre functors relative to the same canonical C R . Then S and T are isomorphic.Proof. There are functorial isomorphisms R Hom Y ( G , S ( F )) ∼ = R Hom Y ( G , T ( F ))for all F , G ∈ per( Y ), which after applying H give functorial isomorphismsHom per( Y ) ( − , S ( − )) ∼ = Hom per( Y ) ( − , T ( − ))Since S and T take values in per( Y ), we may use Yoneda’s lemma to conclude that S and T are isomorphic. (cid:3) Lemma 4.12.
Suppose that Y → Spec R is a projective birational morphism betweenvarieties, and Y is derived equivalent to Λ . Then any Serre functor S : K b (proj Λ) → K b (proj Λ) relative to C R induces a Serre functor S ′ : per( Y ) → per( Y ) relative to C R Proof.
It is enough to show that we have a triangle equivalence F : per(Λ) → per( Y )with a functorial isomorphism R Hom Λ ( A, B ) ∼ = R Hom Y ( F A, F B ) in D( R ) for all A, B ∈ per(Λ), since then for a quasi-inverse E of F we have that S ′ := F ◦ S ◦ E : per( Y ) → per( Y )enjoys functorial isomorphisms R Hom R ( R Hom Y ( F A, F B ) , C R ) ∼ = R Hom R ( R Hom Λ ( A, B ) , C R ) ∼ = R Hom Λ ( B, S A ) ∼ = R Hom Y ( F B, F ( S A )) ∼ = R Hom Y ( F B, S ′ ( F A ))in D( R ). Let V be a tilting complex of Y , and let A be the DG endomorphism R -algebraof V .(i) Let C dg ( A ) be the DG category of DG A -modules. We denote by pretr( A ) thesmallest DG subcategory of C dg ( A ) which is closed under [ ±
1] and cones and contains A . Since the DG category per dg ( Y ) of perfect complexes of Y is pretriangulated, thereexists a fully faithful DG functor V ⊗ A − : pretr( A ) → per dg ( Y ) which induces a triangleequivalence G : per( A ) → per( Y ) [K06, Section 4.5]. In particular, we have a functorialisomorphism Hom • A ( A, B ) ∼ = Hom • Y ( V ⊗ A A, V ⊗ A B ) of DG R -modules for all A, B ∈ pretr( A ), where we denote by Hom • A and Hom • Y the Hom-sets in our DG categories.Thus we have a functorial isomorphism R Hom A ( A, B ) ∼ = R Hom Y ( GA, GB ) in D( R ) forall A, B ∈ per( A ).(ii) Let f : A → B be a quasi-isomorphism of DG R -algebras. Then the DG functor B ⊗ A − : pretr( A ) → pretr( B ) gives a triangle equivalence H : per( A ) → per( B ). For all A, B ∈ pretr( A ), we have a quasi-isomorphism Hom • A ( A, B ) → Hom • B ( B ⊗ A A, B ⊗ A B )of DG R -modules. In particular, we have a functorial isomorphism R Hom A ( A, B ) ∼ = R Hom B ( HA, HB ) in D( R ) for all A, B ∈ per( A ).(iii) Since V is a tilting complex, we have quasi-isomorphisms A ≤ → A and A ≤ → Λ -FACTORIAL TERMINALIZATIONS AND MMAS of DG R -algebras, where A ≤ is a DG sub R -algebra ( · · · → A − → Ker d → → · · · )of A . Combining (i) and (ii), we have the desired assertion. (cid:3) The following is our key observation.
Lemma 4.13.
Suppose that f : Y → Spec R is a projective birational morphism, where Y and R are both Gorenstein varieties. (1) − ⊗ Y f ! O R : per( Y ) → per( Y ) is a Serre functor relative to R . (2) f is crepant if and only if id : per( Y ) → per( Y ) is a Serre functor relative to R .Proof. (1) We know by 4.7 that f ! O R is a line bundle, so it follows that tensoring by f ! O R gives a functor − ⊗ Y f ! O R : per( Y ) → per( Y ). Further, we have functorial isomorphisms R Hom Y ( G , F⊗ Y f ! O R ) ∼ = R Hom Y ( R H om Y ( F , G ) , f ! O R ) ∼ = R Hom R ( R Hom Y ( F , G ) , O R )in D(Mod R ) for all F , G ∈ per( Y ). Thus − ⊗ Y f ! O R is a Serre functor relative to R .(2) By (1) and 4.11, id is a Serre functor relative to R if and only if − ⊗ Y f ! O R = id asfunctors per( Y ) → per( Y ). This is equivalent to f ! O R = O Y , which is equivalent to that f is crepant by 4.7. (cid:3) The following explains the geometric origin of the definition of modifying modules,and is the first main result of this subsection.
Theorem 4.14.
Let f : Y → Spec R be a projective birational morphism between d -dimensional Gorenstein varieties. Suppose that Y is derived equivalent to some ring Λ ,then the following are equivalent. (1) f is crepant. (2) Λ ∈ CM R . (3) id : per( Y ) → per( Y ) is a Serre functor relative to R .In this case Λ ∼ = End R ( M ) for some M ∈ ref R .Proof. (3) ⇔ (1) is shown in 4.13(2), and (1) ⇒ (2) is shown in 4.8.(2) ⇒ (3) Let S := R Hom R (Λ , R ) ⊗ L Λ − : D − (mod Λ) → D − (mod Λ) . By [IR, 3.5(2)(3)], there exists a functorial isomorphism R Hom Λ ( A, S ( B )) ∼ = R Hom R ( R Hom Λ ( B, A ) , R ) (4.E)in D( R ) for all A ∈ D b (mod Λ) and all B ∈ K b (proj Λ).Now suppose that Λ ∈ CM R . Since it is reflexive, Λ ∼ = End R ( M ) for some M ∈ ref R by 4.5. Then we have Λ ∼ = Hom R (Λ , R ) ∼ = R Hom R (Λ , R )in D(Λ ⊗ R Λ op ) where the first isomorphism holds by 2.8(1), and the second since Λ ∈ CM R . Thus we have S ∼ = id, and (4.E) shows that id : K b (proj Λ) → K b (proj Λ) isa Serre functor relative to R . By 4.12 it follows that id : per( Y ) → per( Y ) is a Serrefunctor relative to R . (cid:3) When further Y is smooth, we can strengthen 4.14. Corollary 4.15.
Let f : Y → Spec R be a projective birational morphism between d -dimensional Gorenstein varieties. Suppose that Y is derived equivalent to some ring Λ ,then the following are equivalent. (1) f is a crepant resolution of Spec R . (2) Λ is an NCCR of R .Proof. By 4.14 f is crepant if and only if Λ ∈ CM R . Assume that this is satisfied, thenagain by 4.14 Λ ∼ = End R ( M ) for some M ∈ ref R . By 4.2(1) Y is smooth if and only if Λis non-singular, which means that Λ is an NCCR of R . (cid:3) We are now in a position to relate maximal modification algebras and Q -factorialterminalizations in dimension three. The following is the second main result of this sub-section. Theorem 4.16.
Let f : Y → Spec R be a projective birational morphism, where Y and R are both Gorenstein varieties of dimension three. Assume that Y has (at worst) isolatedsingularities { x , . . . , x n } where each O Y,x i is a hypersurface. If Y is derived equivalentto some ring Λ , then the following are equivalent. (1) f is crepant and Y is Q -factorial. (2) Λ is an MMA of R .In this situation, all MMAs of R are derived equivalent and have isolated singularities.Proof. By 4.14 f is crepant if and only if Λ ∈ CM R . Assume that this is satisfied, thenagain by 4.14 Λ ∼ = End R ( M ) for some M ∈ ref R . Thus D sg ( Y ) ≃ CM Λ by 4.1, 2.8(2)and 2.3. Since the singularities of Y are isolated, Λ has isolated singularities by 4.2(2).Now by 3.11(1), Y is Q -factorial if and only if D sg ( Y ) ≃ CM Λ is rigid-free, which by 2.14holds if and only if Λ is an MMA. Lastly, all maximal modification algebras are derivedequivalent [IW2, 4.15], thus all are derived equivalent to Y and hence all have isolatedsingularities by 4.2(2). (cid:3) Remark 4.17.
When k = C and Y has only terminal singularities, the geometric as-sumptions in 4.16 are satisfied since terminal singularities are isolated for 3-folds [KM,5.18], and Gorenstein terminal singularities are Zariski locally hypersurfaces [R83, 0.6(I)].This shows that when k = C and Y has only terminal singularities, Y is a Q -factorialterminalization of Spec R if and only if Λ is an MMA. Remark 4.18.
We also remark that if Conjecture 1.8 is true, then the Y in 4.16 alwaysadmit a tilting complex. Note that the corresponding statement of 1.8 in dimension fouris false, see 4.20.Note that all the results in this subsection remain valid in the complete local setting.In particular 4.19 below corresponds to 4.16, and will be used in §
5. Here, a morphism f : Y → X = Spec R is called crepant if f ∗ ω X = ω Y holds (as before), where now X hasa unique canonical sheaf ω X = O R , and we choose the canonical sheaf ω Y := f ! ω X on Y . Corollary 4.19.
Suppose f : Y → Spec R is a projective birational morphism between -dimensional Gorenstein normal integral schemes, satisfying (ELF), where R is a completelocal ring containing a copy of its residue field. Suppose further that Y has (at worst)isolated singularities { x , . . . , x n } where each O Y,x i is a hypersurface. If Y is derivedequivalent to some ring Λ , then the following are equivalent. (1) Y is complete locally Q -factorial and f is crepant. (2) Y is Q -factorial and f is crepant. (3) Λ is an MMA of R .In this situation, all MMAs of R are derived equivalent and have isolated singularities.Proof. (2) ⇔ (3) is similar to 4.16 (1) ⇔ (2).(1) ⇔ (2) By 3.11(1)(2), we only have to show that D sg ( Y ) = D sg ( Y ) holds. By 4.14Λ ∼ = End R ( M ) ∈ CM R for some M ∈ ref R . We have D sg ( Y ) ≃ CM Λ by 4.1, 2.8(2) and2.3. The endomorphism ring End Λ ( X ) of any X ∈ mod Λ is again a module-finite algebraover a complete local ring R . Thus any idempotent in End Λ ( X ) is an image of someidempotent in End Λ ( X ) by [CR, 6.5, 6.7], which corresponds to some direct summandof X . Therefore the category CM Λ ≃ D sg ( Y ) is idempotent complete, and we have theassertion. (cid:3) A counterexample in dimension four.
Here we show that we cannot alwaysexpect the setup of 4.16 and 4.15 to hold in higher dimension. This puts severe limitationson any general homological theory that covers dimension four.The following is an extension of an example of Dao [D1, 3.5].
Theorem 4.20.
Let R := C [ x , x , x , x , x ] / ( x + x + x + x + x ) . This has a crepantresolution, which we denote by Y → Spec R . Then there is no algebra Λ that is derivedequivalent to Y .Proof. Note first that R is an isolated singularity, with unique singular point at the origin.A projective crepant resolution exists by [Lin, Thm. A.4]. Suppose Λ is derived equivalentto Y , then by 4.15 Λ ∼ = End R ( M ) is an NCCR of R . Completing with respect to the -FACTORIAL TERMINALIZATIONS AND MMAS maximal ideal of the origin, this implies that End b R ( c M ) is an NCCR of b R . But End b R ( c M ) ∈ CM b R with c M ∈ ref b R , so since b R is a 4-dimensional local isolated hypersurface, c M mustbe free [D1, 2.7(3)]. But this forces End b R ( c M ) ∼ = M n ( b R ), which is a contradiction since M n ( b R ) has infinite global dimension. (cid:3) cA n singularities via derived categories We now illustrate the results of the previous section to give many examples of maximalmodification algebras. The following result is due to Shepherd–Barron (unpublished).
Proposition 5.1.
Let R = C [[ u, v, x, y ]] / ( uv − f ( x, y )) be an isolated cA n singularity.Then the following are equivalent (1) There does not exist a non-trivial crepant morphism Y → Spec R . (2) f ( x, y ) is irreducible. (3) R is factorial. (4) R is Q -factorial. (5) R is an MM R -module.Proof. (1) ⇒ (2) If f ( x, y ) factors non-trivially as f = f f then blowing up the ideal ( u, f )yields a non-trivial crepant morphism (as in the calculation in § f ( x, y ) cannotfactor.(2) ⇒ (3) This is easy (see e.g. proof of [IW3, 5.9]).(3) ⇒ (4) This is clear.(4) ⇒ (5) This follows from 2.11(5) ⇒ (1).(5) ⇒ (1) Since R is an isolated cDV singularity of type A (see e.g. [BIKR, 6.1(e)]), itis a terminal singularity by [R83, 1.1]. If there exists a non-trivial crepant morphism f : Y → Spec R , then f must have one-dimensional fibres. Hence by [V04a, 3.2.10] thereexists a non-projective modifying module. Since R is a MM R -module, this contradicts2.9. (cid:3) Crepant modifications of cA n singularities. In this section we work over K , analgebraically closed field of characteristic zero. Let R := K [[ u, v, x, y ]] / ( uv − f ( x, y )) , with f ∈ m := ( x, y ) ⊆ K [[ x, y ]]. We let f = f . . . f n be a factorization into primeelements of K [[ x, y ]]. We restrict to this complete local setting since it simplifies the proofof the main theorem 5.2. A similar version of 5.2 is true when R is not complete local,though this requires a much more complicated proof [W12].We remark that when K = C , R is a cA m singularity for m := ord( f ) − R is an isolated singularity [R83, 1.1] ifand only if ( f i ) = ( f j ) for all i = j .For any subset I ⊆ { , . . . , n } we denote f I := Y i ∈ I f i and T I := ( u, f I )which is an ideal of R . For a collection of subsets ∅ ( I ( I ( ... ( I m ( { , , ..., n } , we say that F = ( I , . . . , I m ) is a flag in the set { , , . . . , n } . We say that the flag F is maximal if n = m + 1. Given a flag F = ( I , . . . , I m ), we define T F := R ⊕ m M j =1 T I j . On the other hand, geometrically, given a flag F = ( I , . . . , I m ) we define a scheme X F as follows: First, let X F → Spec R be the blowup of the ideal ( u, f I ) in R = K [[ u, v, x, y ]] / ( uv − f ). Then X F is covered by two open charts, given explicitly by R := K [[ u, v, x, y ]][ V ] v − V ff I uV − f I ! and R := K [[ u, v, x, y ]][ U ] u − f I U U v − ff I ! . The new coordinates V = f I u and U = uf I glue to give a copy of P inside X F (possiblymoving in a family), which maps to the origin of Spec R .Next, let X F → X F be the blowup of the divisor ( U , f I f I ) in the second co-ordinatechart R . Note that the zero set of the divisor ( U , f I f I ) does not intersect the first co-ordinate chart R , so R is unaffected under the second blowup. Locally above R , thecalculation to determine the structure of X F is similar to the above. Thus X F is coveredby the three affine open sets K [[ u, v, x, y ]][ V ] v − V ff I uV − f I ! and K [[ u, v, x, y ]][ U , V ] u − f I U v − ff I V U V − f I f I and K [[ u, v, x, y ]][ U ] u − f I U U v − ff I ! The U and the V coordinates again glue to produce another P (again which might movein a family) which maps to the origin of R and hence to the origin of R .Continuing by blowing up the ideal ( U , f I f I ), in this way we obtain a chain of pro-jective birational morphisms X F m → X F m − → . . . → X F → Spec R, and we define X F := X F m . See 5.5 later for a picture of this process.Note that X F , being projective over the base Spec R , automatically satisfies (ELF)[TT, 2.1.3], and by inspection of the charts, X F is a normal integral Gorenstein scheme,so we can apply the results in the previous section. Also note that if we complete each ofthe above affine open sets at the origin we obtain K [[ u, V , x, y ]]( uV − f I ) , K [[ U , V , x, y ]]( U V − f I f I ) , . . . K [[ U m − , V m , x, y ]]( U m − V m − f Im f Im − ) , K [[ U m , v, x, y ]]( U m v − ff Im ) . Theorem 5.2.
Given a flag F = ( I , . . . , I m ) , denote X F and T F as above. Then X F is derived equivalent to End R ( T F ) .Proof. In the explicit calculation for X F above, the preimage of the unique closed point n of Spec R is a chain of P ’s (some of which move in a family), in a type A m configuration.Now by [V04a, Thm. B], there is a tilting bundle on X F given as follows: let C = f − ( n ).Giving C the reduced scheme structure, write C red = S i ∈ I C i , and let L i denote the linebundle on X F such that L i · C j = δ ij . If the multiplicity of C i in C is equal to one, set M i := L i [V04a, 3.5.4], else define M i to be given by the extension0 → O r i − → M i → L i → r i − H ( X F , L − i ). Then O ⊕ ( L i ∈ I M i )is a tilting bundle on X F [V04a, 3.5.5].In our situation, let C i be the curve in X F , above the origin of Spec R , which inthe process of blowing up first appears in X F i (see 5.5 for a picture of this). We nowclaim that all the curves C i have multiplicity one. If some C i had multiplicity greaterthan one, then M i would be an indecomposable bundle with rank greater than one [V04a,3.5.4]. But this would imply, by [V04a, 3.2.9], that its global sections H ( M i ) := M i is anindecomposable CM R -module of rank greater than one, such that End R ( M i ) ∈ CM R .But this is impossible (see e.g. [IW3, 5.24]). Hence all curves have multiplicity one, andso O ⊕ ( L mi =1 L i ) is a tilting bundle on X F .Now End X F ( O ⊕ ( L mi =1 L i )) ∼ = End R ( R ⊕ ( L mi =1 H ( L i ))), and so it remains to showthat R ⊕ ( L mi =1 H ( L i )) ∼ = T F . In fact, we claim that H ( L i ) ∼ = ( u, f I i ). But it is easy -FACTORIAL TERMINALIZATIONS AND MMAS to see (for example using the ˇCech complex) that the rank one CM module H ( L i ) isgenerated as an R -module by u and f I i . (cid:3) So as to match our notation with [BIKR] and [DH], we can (and do) identify maximalflags with elements of the symmetric group S n . Hence we regard each ω ∈ S n as themaximal flag { ω (1) } ⊂ { ω (1) , ω (2) } ⊂ . . . ⊂ { ω (1) , . . . , ω ( n − } . The following is simply a special case of 5.2.
Corollary 5.3.
The scheme X ω is derived equivalent to End R ( T ω ) for all ω ∈ S n .Moreover the completions of singular points of X ω are precisely K [[ u, v, x, y ]] / ( uv − f i ) forall ≤ i ≤ n such that f i ∈ ( x, y ) . This yields the following generalization of [BIKR, 1.5] and [DH, 4.2], and provides aconceptual reason for the condition f i / ∈ m . Corollary 5.4.
Let f , . . . , f n ∈ m := ( x, y ) ⊆ K [[ x, y ]] be irreducible polynomials and R = K [[ u, v, x, y ]] / ( uv − f . . . f n ) . Then (1) Each T ω ( ω ∈ S n ) is an MM R -module which is a generator. The endomorphismrings End R ( T ω ) have isolated singularities. (2) T ω is a CT R -module for some ω ∈ S n ⇐⇒ T ω is a CT R -module for all ω ∈ S n ⇐⇒ f i / ∈ m for all ≤ i ≤ n .Proof. (1) End R ( T ω ) is derived equivalent to a scheme whose (complete local) singularitiesare listed in 5.3. Since each f i is irreducible, these are all factorial by 5.1. Hence the resultfollows from 4.19.(2) T ω is CT if and only if Γ := End R ( T ω ) is an NCCR [IW2, 5.4]. By 4.2(1) this occurs ifand only if X ω is smooth. But by 5.3, this happens if and only if each uv = f i is smooth,which is if and only if each f i / ∈ m . (cid:3) For an algebraic non-derived category proof of the above, see [IW3, § Remark 5.5.
It is useful to visualize the above in an example. X F X F X F Spec
R ϕ ϕ ϕ T I T I T I R f u f f f incinc u incx yT I R T I f f f f u inc u incx yR T I f u f f f u incyx yxRyx uv R ⊕ T I ⊕ T I ⊕ T I R ⊕ T I ⊕ T I R ⊕ T I R Consider uv = f f f f and choose maximal flag F = ( I := { } ⊂ I := { , } ⊂ I := { , , } ). Thus T I = ( u, f ), T I = ( u, f f ) and T I = ( u, f f f ). Moreover weassume, so that we can draw an accurate picture, that ( f ) = ( f ), and ( f ) = ( f ), sothat uv = f f f is non-isolated, but uv = f f is isolated.On the geometric side we have drawn the non-local picture; the complete local pictureis obtained by drawing a tubular neighbourhood around all the red curves, and restrictingto the origin in Spec R .Now on X F , the black dot is the singularity uv = f , whereas the yellow dot is uv = f f f . On X F , the latter singularity splits into uv = f (middle black dot) and uv = f f (yellow). The left hand black dot is still uv = f . On X F , the black dotsare (reading left to right) uv = f , uv = f , uv = f and uv = f . All yellow dotscorrespond to non- Q -factorial singularities, and all black dots correspond to isolated Q -factorial singularities, possibly smooth. The yellow dots with squiggles through them arenon-isolated singularities.The red curves are the P ’s which map to the origin in Spec R . The right hand curvein X F moves in a family (represented as blue lines) and ϕ contracts the whole family;consequently ϕ contracts a divisor. However both ϕ and ϕ contract a single curve butno divisor and so are flopping contractions. Note that Spec R and X F have canonicalsingularities, but X F and X F have only terminal singularities.The precise form of the quiver relies on the (easy) calculation in [IW3, 5.33], but thiscan be ignored for now. Note that the geometric picture will change depending both onthe choice of polynomials, and their ordering. Blowing up in a different order can changewhich curves move in families, and also normal bundles of the curves. Of course, thisdepends on the choice of the polynomials too. On the algebraic side, if we change thepolynomials or their order then the number of loops change, as do the relations. Changingthe ordering corresponds to mutation (see [IW3, 5.31]).On each level i the geometric space X F i is derived equivalent to the correspondingalgebra by 5.2 . The top space has only Q -factorial terminal singularities, hence the topalgebra is an MMA. Remark 5.6.
A version of 5.4 is actually true in the non-local case (see [W12]), but weexplain here why the non-local case is a much more subtle problem.For example, in the case R = C [ u, v, x, y ] / ( uv − xy ( y − M := ( u, x ), L := ( u, x ( y − L := ( u, xy ). Then we can show that both M := R ⊕ M and L := R ⊕ L ⊕ L give NCCRs by checking complete locally. Alsowe can show add M = add L by checking complete locally [IW2, 2.26]). Thus End R ( M )and End R ( L ) are Morita equivalent. Since dim R = 3 and R has an NCCR, by [V04b]we obtain a derived equivalence between these NCCRs and some crepant resolution ofSpec R .Hence our picture in 5.5 becomes R L L x ( y − uinc yu uxyinc y − u R M xu y ( y − u incy yRyx uvϕ with everything in the top row being derived equivalent. This all relies on the fact thatwe a priori know that we have an NCCR (because we can check this complete locally) andso we can use [V04b]. In a similar case uv = ( x + y ) y ( y −
2) (where no NCCR exists),we cannot play this trick, since we do not know if MMAs can be detected locally, and alsosince completion destroys Q -factoriality we cannot reduce to the complete local setting. -FACTORIAL TERMINALIZATIONS AND MMAS The other way to try to prove a non-local version of 5.4 is to try and find an explicittilting complex starting with the geometry, as in 5.2. Although a tilting bundle existsabstractly for one-dimensional fibres [V04a], to describe it explicitly is delicate, since forexample we have to deal with issues such as curves being forced to flop together. We referthe reader to [W12] for more details.
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Osamu Iyama, Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya,464-8602, Japan
E-mail address : [email protected] Michael Wemyss, School of Mathematics, James Clerk Maxwell Building, The King’sBuildings, Mayfield Road, Edinburgh, EH9 3JZ, UK.
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