The singularity category of an exact category applied to characterize Gorenstein schemes
Lars Winther Christensen, Nanqing Ding, Sergio Estrada, Jiangsheng Hu, Huanhuan Li, Peder Thompson
aa r X i v : . [ m a t h . K T ] S e p THE SINGULARITY CATEGORY OF AN EXACT CATEGORYAPPLIED TO CHARACTERIZE GORENSTEIN SCHEMES
LARS WINTHER CHRISTENSEN, NANQING DING, SERGIO ESTRADA,JIANGSHENG HU, HUANHUAN LI, AND PEDER THOMPSON
Abstract.
We study singularity categories of exact categories with a focuson those associated to a complete hereditary cotorsion pair. As an applicationwe identify a non-affine analogue of the singularity category of a Gorensteinlocal ring; with this Buchweitz’s classic equivalence of three categories overGorenstein local rings has been generalized to schemes, a project started byMurfet and Salarian more than ten years ago. As another application we usethe framework to characterize rings of finite finitistic dimension.
Introduction
The singularity category of a commutative noetherian ring A is the Verdier quo-tient of the finite bounded derived category of A by the subcategory of perfect A -complexes. A motivation for this paper comes from work of Buchweitz [7]: Incontemporary terminology, he proved that the singularity category of a Gorensteinlocal ring A is equivalent to the stable category of finitely generated Gorensteinprojective A -modules and to the homotopy category of totally acyclic complexes offinitely generated projective A -modules; in symbols( ⋄ ) D sg ( A ) ≃ StGor prj ( A ) ≃ K tac ( prj ( A )) . There are “big” versions of all three categories obtained by dropping the assump-tions of finite generation. The big singularity category of a Gorenstein ring A isby work of Beligiannis [4, Thm. 6.9] equivalent to the stable category StGor
Prj ( A )of Gorenstein projective A -modules, and the latter category is for every ring A equivalent to the homotopy category K tac ( Prj ( A )) of totally acyclic complexes ofprojective A -modules, see for example [14, Corollary 3.9].Murfet and Salarian [29] initiated the quest for non-affine analogues of the cat-egories in ( ⋄ ). As an analogue of the category K tac ( Prj ( A )) for a semi-separatednoetherian scheme X , they identified the category D F-tac ( Flat ( X )) := K F-tac ( Flat ( X )) K pac ( Flat ( X )) , Date : 2 September 2020.1991
Mathematics Subject Classification.
Primary 14F08. Secondary 16E65; 18G20.
Key words and phrases.
Cotorsion pair; defect category; finitistic dimension; global dimension;Gorenstein scheme; Iwanaga–Gorenstein ring; singularity category.L.W.C. was partly supported by Simons Foundation grant 428308. N.D. was supported byNSFC grant 11771202. S.E. was partly supported by grant MTM2016-77445-P and FEDER fundsand by grant 19880/GERM/15 from the Fundaci´on S´eneca-Agencia de Ciencia y Tecnolog´ıa de laRegi´on de Murcia. J.H. was partly supported by NSFC grant 11771212. H.L. was partly supportedby NSFC grant 11626179. Part of the work was done when J.H. visited Universidad de Murcia inFebruary 2020; the hospitality of this institution is acknowledged with gratitude.
L.W. CHRISTENSEN, N. DING, S. ESTRADA, J. HU, H. LI, AND P. THOMPSON i.e. the Verdier quotient of the homotopy category of F -totally acyclic complexes offlat quasi-coherent sheaves on X by its subcategory of pure-acyclic complexes. In-deed, for a commutative noetherian ring A of finite Krull dimension and the scheme X = Spec( A ), the categories K tac ( Prj ( A )) and D F-tac ( Flat ( X )) are equivalent by [29,Lemma 4.22]. An analogue for schemes of the stable category was identified in [13]:For a semi-separated noetherian scheme X , the category D F-tac ( Flat ( X )) is equiv-alent to the homotopy category K tac ( FlatCot ( X )) of totally acyclic complexes offlat-cotorsion sheaves on X , and that category is equivalent to the stable category StGor
FlatCot ( X ) of Gorenstein flat-cotorsion sheaves on X . The primary goal ofthis paper is to identify an analogue of the singularity category in the non-affinesetting and to show that it is equivalent to the categories K tac ( FlatCot ( X )) and StGor
FlatCot ( X ) for a Gorenstein scheme X of finite Krull dimension.We achieve this goal as an application of a more general theory that we developfor singularity categories associated to a cotorsion pair. Other applications includecharacterizations of Iwanaga–Gorenstein rings and finiteness of finitistic dimension.Here is an outline: Let A be an abelian category. To an additive full subcategory E of A that is closed under extensions and direct summands, we associate in Section 1two singularity categories relative to the projective and injective objects in E . If E is part of a complete hereditary cotorsion pair, then there are natural functorsfrom categories of Gorenstein objects into these singularity categories. This allowsus in Section 2—inspired by works of Bergh, Jorgensen, and Oppermann [6] andIyengar and Krause [20]—to define associated defect categories. Given a completehereditary cotorsion pair ( U , V ) in A , we build on the theory from [14] to developGorenstein dimensions for objects in U and V . Assuming that A is Grothendieck—as is the case for the categories of modules over a ring and quasi-coherent sheaveson a semi-separated noetherian scheme—these dimensions extend through work ofGillespie to invariants on the derived category of A ; that’s the topic of Section 3. InSection 4 we arrive at equivalences of categories akin to those in ( ⋄ )—but now for aGorenstein scheme. In Section 5 we characterize rings of finite finitistic dimensionin terms of the singularity and defect categories developed in the first sections. ∗ ∗ ∗ Throughout the paper, A denotes an abelian category; hom-sets of objects in A are denoted hom A , and that notation is also used for the induced functor from A to abelian groups. By a subcategory of A we mean a full subcategory that isclosed under isomorphisms. Let S be a subcategory of A . The category of chaincomplexes of objects from S , or S -complexes, is denoted C ( S ) and K ( S ) is theassociated homotopy category; the latter is triangulated. A complex X in C ( S ) issaid to be bounded below if X n = 0 holds for n ≪ bounded above if X n = 0holds for n ≫
0, and bounded if X n = 0 holds for | n | ≫
0. The full subcategoriesof bounded below, bounded above, and bounded complexes are denoted C + ( S ), C − ( S ), and C b ( S ), respectively. The essential images of these subcategories in thehomotopy category K ( S ) are triangulated subcategories denoted K + ( S ), K − ( S ), and K b ( S ), respectively.Given a complex X ∈ C ( A ) with differential ∂ X we write B n ( X ) and Z n ( X )for the subobjects of boundaries and cycles in X n . The cokernel of ∂ Xn +1 is thequotient object C n ( X ) := X n / B n ( X ), and the homology of X in degree n is thesubquotient object H n ( X ) := Z n ( X ) / B n ( X ). Given a second A -complex Y , wewrite Hom A ( X, Y ) for the total hom-complex in C ( A ). HE SINGULARITY CATEGORY OF AN EXACT CATEGORY 3 Singularity categories of exact categories
Let E ⊆ A be an additive subcategory closed under extensions and direct summands;it is an idempotent complete exact category with the exact structure induced from A , see B¨uhler [8, Rmk. 6.2 and Lem. 10.20]. The setup developed in this firstsection could be done in the generality of idempotent complete exact categories.We don’t need that for the applications we pursue in this paper, but we structurethe arguments in such a way as to make the generalizations straightforward for theinitiated reader.Given a subcategory S ⊆ E , a complex X in K ( S ) is called E -acyclic if(1.0.1) 0 −→ Z n ( X ) −→ X n −→ Z n − ( X ) −→ E for every n ∈ Z . The full subcategory of such complexes isdenoted K E -ac ( S ); it is isomorphism closed, see Keller [23, 4.1], and is indeed trian-gulated, see Neeman [30, Lem. 1.1]. The symbol K E -ac ( S ) is used with subscripts +, − , and b to indicate boundedness: K E -ac , b ( S ) := K E -ac ( S ) ∩ K b ( S ) etc. An A -acycliccomplex is simply referred to as acyclic , and the corresponding category is denoted K ac ( S ). A morphism in K ( S ) is called an E -quasi-isomorphism if its mapping coneis E -acyclic and simply a quasi-isomorphism if its mapping cone is acyclic.The derived category of E is defined to be the Verdier quotient D ( E ) := K ( E ) / K E -ac ( E ) ;see [30] and Keller [24]. In the case where E = A is the module category of a ring,this construction yields the usual derived category. The bounded versions D + ( E ), D − ( E ), and D b ( E ) are defined analogously: D + ( E ) := K + ( E ) / K E -ac , + ( E ) etc.A complex X in K ( S ) is called eventually E -acyclic if the sequences (1.0.1) areexact in E for all | n | ≫
0. The full subcategory of such complexes is denoted K ( E -ac) ( S ); this notation is also used with subscripts to indicate boundedness.Recall that for an A -complex X and an integer n , the hard truncation below of X at n is the complex X > n := · · · → X n +1 → X n →
0, and the soft truncationbelow of X at n is the complex X ⊇ n := · · · → X n +1 → Z n ( X ) →
0. Hard andsoft truncations above are defined similarly: X n := 0 → X n → X n − → · · · and X ⊆ n := 0 → C n ( X ) → X n − → · · · . Let E be an additive subcategory of A that is closed under ex-tensions and direct summands; let S be an additive subcategory of E . (a) The category K ( E -ac) , + ( S ) is a triangulated subcategory of K + ( S ) . (b) The category K ( E -ac) , − ( S ) is a triangulated subcategory of K − ( S ) . Proof.
We prove (a); the proof of (b) is similar. That K ( E -ac) , + ( S ) is additive andclosed under shifts is evident; it remains to show that it is closed under isomorphismsand cones.Let α : X → Y be an isomorphism in K + ( S ) with X ∈ K ( E -ac) , + ( S ). The complexCone( α ) is contractible and hence E -acyclic as E is closed under direct summands.For every n ∈ Z the short exact sequence(1) 0 −→ Y n −→ Cone( α ) n −→ (Σ X ) n −→ L.W. CHRISTENSEN, N. DING, S. ESTRADA, J. HU, H. LI, AND P. THOMPSON yields a commutative diagram(2) Cone( α ) n / / (cid:15) (cid:15) X n − (cid:15) (cid:15) / / n − (Cone( α )) / / Z n − ( X ) . For all n ≫ / / Z n (Cone( α )) / / (cid:15) (cid:15) Cone( α ) n / / (cid:15) (cid:15) Z n − (Cone( α )) / / (cid:15) (cid:15) / / Z n − ( X ) / / X n − / / Z n − ( X ) / / −→ Z n ( Y ) −→ Y n −→ Z n − ( Y ) −→ . In (2) the upper horizontal and right-hand vertical morphisms are epimorphismswith kernels in E , namely Y n and Z n − ( X ). As E is an exact category closed underdirect summands, [8, Prop. 7.6] shows that the lower horizontal morphism in (2)is an epimorphism with kernel in E . By left exactness of the cycle functor, it nowfollows from (1) that Z n − ( Y ) is in E ; thus Y belongs to K ( E -ac) , + ( S ).Finally, if α : X → Y is a morphism in K ( E -ac) , + ( S ), then for n ≫ X ⊇ n and Y ⊇ n are E -acyclic and α induces a map α ′ between them. In high degrees,the cones of α and α ′ agree, and the cone of α ′ is E -acyclic; see [8, Lem. 10.3]. (cid:3) We denote by
Prj ( E ) and Inj ( E ) the subcategories of projective and injectiveobjects in E . Let E be an additive subcategory of A that is closed under ex-tensions and direct summands. (a) If E has enough projectives, then there are triangulated equivalences of tri-angulated categories K + ( Prj ( E )) ≃ D + ( E ) and K ( E -ac) , + ( Prj ( E )) ≃ D b ( E ) . (b) If E has enough injectives, then there are triangulated equivalences of trian-gulated categories K − ( Inj ( E )) ≃ D − ( E ) and K ( E -ac) , − ( Inj ( E )) ≃ D b ( E ) . Proof.
We prove part (a); the proof of part (b) is dual. For every complex X in K + ( E ) there is a distinguished triangle in K + ( E ), p ( X ) −→ X −→ a ( X ) −→ Σ p ( X ) , with p ( X ) in K + ( Prj ( E )) and a ( X ) an E -acyclic complex; moreover the inclusion of E -acyclic complexes into K + ( E ) has a left adjoint. For a proof (of the dual result)see [23, Lem. 4.1]. It is now standard, see Krause [26, Prop. 4.9.1 and the proofof Prop. 4.13.1], that the functor G : K + ( Prj ( E )) → K + ( E ) → D + ( E ) yields theequivalence K + ( Prj ( E )) ≃ D + ( E ); see also [24, Exa. 12.2]. HE SINGULARITY CATEGORY OF AN EXACT CATEGORY 5
By Proposition 1.1 the triangulated subcategory K ( E -ac) , + ( Prj ( E )) is equivalent toits essential image under G in D + ( E ). For X in K b ( E ) the complex p ( X ) belongs to K ( E -ac) , + ( Prj ( E )), and a complex in K ( E -ac) , + ( Prj ( E )) is E -quasi-isomorphic in K + ( E )to a complex in K b ( E ). As D b ( E ) is equivalent to the subcategory of D ( E ) generatedby K b ( E ), see [24, Lem. 11.7], the restriction of G to K ( E -ac) , + ( Prj ( E )) induces thesecond equivalence. (cid:3) As K b ( Prj ( E )) evidently is a thick subcategory of K + ( Prj ( E )), it follows thatfor a category E as in Proposition 1.2 with enough projectives, the subcategory K b ( Prj ( E )) is equivalent to a thick subcategory of D b ( E ). Similarly, if E has enoughinjectives, then K b ( Inj ( E )) is equivalent to a thick subcategory of D b ( E ). Let E be an additive subcategory of A that is closed under exten-sions and direct summands.(a) Assume that E has enough projectives. The Verdier quotient D sgPrj ( E ) := D b ( E ) / K b ( Prj ( E ))is called the projective singularity category of E or Prj ( E )- singularity category. (b) Assume that E has enough injectives. The Verdier quotient D sgInj ( E ) := D b ( E ) / K b ( Inj ( E ))is called the injective singularity category of E or Inj ( E )- singularity category. With an eye towards module categories, the next result justifies the term “singu-larity category.” Resolutions of objects in exact categories are defined in the usualway, see [8, Sec. 12].
Let E be an additive subcategory of A that is closed under exten-sions and direct summands. (a) Assume that E has enough projectives. The category D sgPrj ( E ) vanishes if andonly if every object in E has a bounded resolution by objects from Prj ( E ) . (b) Assume that E has enough injectives. The category D sgInj ( E ) vanishes if andonly if every object in E has a bounded coresolution by objects from Inj ( E ) . Proof.
We prove (a); the proof of (b) is dual. The category D sgPrj ( E ) is per Proposi-tion 1.2 equivalent to K ( E -ac) , + ( Prj ( E )) / K b ( Prj ( E )). The “only if” statement is clear.For the “if” statement let P be a complex in K ( E -ac) , + ( Prj ( E )). For n ≫ n ( P ) belong to E , and the sequences 0 → Z n +1 ( P ) → P n +1 → Z n ( P ) → P is isomorphic to a complex in K b ( Prj ( E )). (cid:3) Given a ring A we write Mod ( A ) for the abelian category of A -modules. Let A be a ring. The category D sgPrj ( Mod ( A )) vanishes if and onlyif every A -module has finite projective dimension; that is, if and only if A hasfinite global dimension. In particular, for a commutative noetherian local ring A the category D sgPrj ( Mod ( A )) provides a measure of how singular—i.e. how far frombeing regular— A is. It thus serves the same purpose as the classic singularitycategory D sg ( A ) recalled in the introduction. L.W. CHRISTENSEN, N. DING, S. ESTRADA, J. HU, H. LI, AND P. THOMPSON
A variety of relative singularity categories are considered in the lit-erature. They are in certain aspects more general and/or more restrictive than ournotions. Closest to ours is, perhaps, the category considered by Chen [11]: In thecase E = A his category D Prj ( A ) ( A ) agrees with D sgPrj ( A ).Let ( U , V ) be a cotorsion pair in A ; the next result shows that the setup developedabove applies to U and V , and in the balance of the paper, we work in that setting.A cotorsion pair ( U , V ) is called complete if for each object M ∈ A there are exactsequences 0 → V → U → M → → M → V ′ → U ′ → U, U ′ ∈ U and V, V ′ ∈ V . Such sequences are known as special U -precovers and special V -envelopes, respectively. A cotorsion pair is called hereditary if Ext i A ( U, V ) = 0holds for all objects U ∈ U and V ∈ V and all i > Let ( U , V ) be a complete cotorsion pair in A . (a) The category V has enough projectives, and one has Prj ( V ) = U ∩ V . (b) The category U has enough injectives, and one has Inj ( U ) = U ∩ V . Proof.
We prove (a); the proof of (b) is dual. Let M ∈ V ; since ( U , V ) is a completecotorsion pair in A , there is an exact sequence0 −→ V −→ U −→ M −→ V ∈ V and U ∈ U . As V is closed under extensions, the object U is in U ∩ V . Itis evident that objects in U ∩ V are projective in V , and given any projective object M in V the exact sequence above shows that it is a quotient object, and hence adirect summand, of an object in U ∩ V . Thus one has Prj ( V ) = U ∩ V . (cid:3) Let ( U , V ) be a complete cotorsion pair. It follows in view of Theo-rem 1.4 that the projective singularity category of V vanishes if and only if everyobject in A has a bounded resolution by objects from U . Indeed, every object in A has a resolution by objects from U in which the first syzygy belongs to V , and D sgPrj ( V ) vanishes if and only if this object has a bounded resolution by objects from U ∩ V . Similarly, vanishing of the category D sgInj ( U ) means that every object in A has a bounded coresolution by objects from V .As U contains Prj ( A ) the singularity category D sgPrj ( U ) is defined if A has enoughprojectives, but we are not aware of any interpretation of it in this generality.For the absolute cotorsion pair ( Prj ( A ) , A ) in a category with enough projectives,the singularity category D sgPrj ( Prj ( A )), of course, vanishes. The category D sgInj ( V ) isdefined if A has enough injectives and in general it appears equally intractable. Let A be a ring and consider the cotorsion pair ( Flat ( A ) , Cot ( A )).The singularity category D sgPrj ( Cot ( A )) vanishes if and only if every A -modulehas finite flat dimension; that is, if and only if A has finite weak global dimension.The singularity category D sgInj ( Flat ( A )) vanishes if and only if every A -modulehas finite cotorsion dimension; by a result of Mao and Ding [28, Thm. 19.2.5] thisis equivalent to A being n -perfect—every flat module has projective dimension atmost n —for some n .If A has finite global dimension, then both D sgPrj ( Cot ( A )) and D sgInj ( Flat ( A )) van-ish, and the converse holds by [28, Thm. 19.2.14]. HE SINGULARITY CATEGORY OF AN EXACT CATEGORY 7 Gorenstein defect categories
In this section, ( U , V ) is a complete cotorsion pair in A . The additive categories U and V are closed under extensions and direct summands, and by Proposition 1.7the category V has enough projectives, and U has enough injectives. Thus thesingularity categories D sgPrj ( V ) and D sgInj ( U ) are defined in this context. In the caseof a complete hereditary cotorsion pair we give a more concrete interpretation ofthese singularity categories in terms of the notions of U - and V -Gorenstein objects.We recall from [14] that an acyclic complex T of objects from U is called right U -totally acyclic if and only if the cycle objects Z n ( T ) belong to V and the complexhom A ( T, W ) is acyclic for every W ∈ U ∩ V . As V is extension closed, the objects ina U -totally acyclic complex in fact belong to U ∩ V ; the symbol K R U -tac ( U ∩ V ) denotesthe homotopy category of such complexes. An object is called right U -Gorenstein if it equals Z ( T ) for some right U -totally acyclic complex T . The subcategoryof right U -Gorenstein objects in A is denoted RGor U ( A ); it is by [14, Thm. 2.11]a Frobenius category, and the associated stable category is denoted StRGor U ( A ).Dually one defines left V -totally acyclic complexes, left V -Gorenstein objects, andassociated categories K L V -tac ( U ∩ V ), LGor V ( A ), and StLGor V ( A ).It is straightforward to verify that the functors described in the next theoremare those induced by the embeddings of V and U into D b ( V ) and D b ( U ). Let ( U , V ) be a complete cotorsion pair in A . There are fully faithfultriangulated functors F Prj : StRGor U ( A ) −→ D sgPrj ( V ) and F Inj : StLGor V ( A ) −→ D sgInj ( U ) , where F Prj sends a right U -Gorenstein object to the hard truncation below at ofa defining U -totally acyclic complex, and F Inj is defined similarly.
Proof.
We prove the assertion regarding F
Prj ; the one for F
Inj is proved similarly.The functor
RGor U ( A ) → K R U -tac ( U ∩ V ) that maps a right U -Gorenstein objectto a defining right U -totally acyclic complex induces a triangulated equivalence StRGor U ( A ) → K R U -tac ( U ∩ V ); this was shown in [14, Thm. 3.8]. The arguments inthe proof of [6, Thm. 3.1] show that hard truncation below at 0 yields a fully faithfultriangulated functor K R U -tac ( U ∩ V ) → K ( U -ac) , + ( U ∩ V ) / K b ( U ∩ V ). Propositions 1.2and 1.7 now yield the desired fully faithful triangulated functor F Prj . (cid:3) Theorem 2.1 facilitates the following definition; see also [6].
Let ( U , V ) be a complete cotorsion pair in A . The Verdier quotientof the projective singularity category by the essential image of F Prj , D defPrj ( V ) := D sgPrj ( V ) / im(F Prj ) , is called the projective Gorenstein defect category of V or Prj ( V ) -Gorenstein de-fect category . Similarly, the injective Gorenstein defect category of U or Inj ( U ) -Gorenstein defect category is the Verdier quotient D defInj ( U ) := D sgInj ( U ) / im(F Inj ) . It was shown in [14, Lem. 2.10] that the categories
RGor U ( A ) and LGor V ( A ) areclosed under extensions. In the context of a complete hereditary cotorsion pair, wenow show that they are closed under direct summands; our proof relies of work ofChen, Liu, and Yang [10, 33]. L.W. CHRISTENSEN, N. DING, S. ESTRADA, J. HU, H. LI, AND P. THOMPSON
Let ( U , V ) be a complete hereditary cotorsion pair in A . The cate-gories RGor U ( A ) and LGor V ( A ) are closed under direct summands. Proof.
It follows from [14, Def. 1.1] and [33, Lem. 3.2] that the class
RGor U ( A ) isthe intersection of V and the class G ( U ) defined in [33, Def. 3.1]. (We notice that[33, 3.1–3.3] do not depend on the blanket assumption that the underlying abeliancategory is bicomplete.) It now follows from [10, Prop. 3.3] that RGor U ( A ) is closedunder direct summands. That LGor V ( A ) is closed under direct summands is provedsimilarly. (cid:3) Let ( U , V ) be a complete hereditary cotorsion pair in A . (a) Let → V ′ → V → V ′′ → be an exact sequence in V . (1) If V ′′ is in RGor U ( A ) , then V ′ is in RGor U ( A ) if and only if V is in RGor U ( A ) . (2) If V ′ and V are in RGor U ( A ) , then V ′′ ∈ RGor U ( A ) if and only if Ext A ( V ′′ , W ) = 0 holds for all W ∈ U ∩ V . (b) Let → U ′ → U → U ′′ → be an exact sequence in U . (1) If U ′ is in LGor V ( A ) , then U ′′ is in LGor V ( A ) if and only if U is in LGor V ( A ) . (2) If U and U ′′ are in LGor V ( A ) , then U ′ is in LGor V ( A ) if and only if Ext A ( W, U ′ ) = 0 holds for all W ∈ U ∩ V . Proof.
As noticed in the proof of Lemma 2.3, the class
RGor U ( A ) is the intersectionof V and the class G ( U ) defined in [33, Def. 3.1]. The assertions in part (a) nowfollow from [33, Prop. 3.3]; the proof of part (b) is similar. (cid:3) The next definition foreshadows notions of Gorenstein dimensions relative to acotorsion pair.
Let ( U , V ) be a complete hereditary cotorsion pair in A .(a) A complex V in D b ( V ) is called RGor U ( A ) -perfect if V is isomorphic in D b ( V )to a complex X with X n ∈ RGor U ( A ) for all n ∈ Z and X n = 0 for | n | ≫ U in D b ( U ) is called LGor V ( A ) -perfect if U is isomorphic in D b ( U )to a complex Y with Y n ∈ LGor V ( A ) for all n ∈ Z and Y n = 0 for | n | ≫ Let ( U , V ) be a complete hereditary cotorsion pair in A . (a) For a complex V ∈ D b ( V ) the following conditions are equivalent: ( i ) V is RGor U ( A ) -perfect. ( ii ) For every complex X ∈ K ( V -ac) , + ( U ∩ V ) isomorphic to V in D b ( V ) onehas Z n ( X ) ∈ RGor U ( A ) for n ≫ . ( iii ) There exists a complex X ∈ K ( V -ac) , + ( U ∩ V ) isomorphic to V in D b ( V ) with Z n ( X ) ∈ RGor U ( A ) for n ≫ . ( iv ) For every complex X ∈ K ( V -ac) , + ( RGor U ( A )) isomorphic to V in D b ( V ) one has Z n ( X ) ∈ RGor U ( A ) for n ≫ . (b) For a complex U ∈ D b ( U ) the following conditions are equivalent: ( i ) U is LGor V ( A ) -perfect. HE SINGULARITY CATEGORY OF AN EXACT CATEGORY 9 ( ii ) For every complex Y ∈ K ( U -ac) , − ( U ∩ V ) isomorphic to U in D b ( U ) onehas Z n ( Y ) ∈ LGor V ( A ) for n ≪ . ( iii ) There exists a complex Y ∈ K ( U -ac) , − ( U ∩ V ) isomorphic to U in D b ( U ) with Z n ( Y ) ∈ LGor V ( A ) for n ≪ . ( iv ) For every complex Y ∈ K ( U -ac) , − ( LGor V ( A )) isomorphic to U in D b ( U ) one has Z n ( Y ) ∈ LGor V ( A ) for n ≪ . Proof.
We prove (a); the proof of (b) is dual.( i ) = ⇒ ( ii ): Let X be a complex in K ( V -ac) , + ( U ∩ V ) with V ∼ = X in D b ( V ).As X is isomorphic in K ( U ∩ V ) to a bounded below complex, and as U ∩ V isclosed under direct summands, the complex X is isomorphic in K ( U ∩ V ) to asoft truncation below, so we may assume that X n = 0 holds for n ≪
0. Byassumption, there is a complex X ′ isomorphic to V in D b ( V ) with X ′ n ∈ RGor U ( A )for all n ∈ Z and X ′ n = 0 for | n | ≫
0. As X ∼ = X ′ in D b ( V ), there exists a V -quasi-isomorphism α : X → X ′ ; indeed, this follows from the dual statement toB¨uhler’s [9, Lem. 3.3.3], cf. Proposition 1.7. In particular, Cone( α ) is a V -acycliccomplex. The category RGor U ( A ) is additive, so Cone( α ) is a complex of right U -Gorenstein objects and Cone( α ) n = 0 holds for n ≪
0. It now follows fromLemma 2.4(a,1) that Z n (Cone( α )) is right U -Gorenstein for all n ∈ Z . For n ≫ n (Cone( α )) ∼ = Z n ( X ) and thus Z n ( X ) belongs to RGor U ( A ).( ii ) = ⇒ ( iii ): Trivial in view of Proposition 1.2.( iii ) = ⇒ ( iv ): Let X be a complex in K ( V -ac) , + ( RGor U ( A )) with V ∼ = X in D b ( V ).By assumption, there exists a complex X ′ in K ( V -ac) , + ( U ∩ V ) such that X ∼ = X ′ in D b ( V ) and Z n ( X ′ ) ∈ RGor U ( A ) for n ≫
0. By Lemma 2.3 the category
RGor U ( A ) is closed under direct summands, and so is U ∩ V . It follows that X and X ′ are isomorphic to soft truncations below, so without loss of generality weassume that X n = 0 = X ′ n holds for n ≪
0. As above there exists a V -quasi-isomorphism α : X ′ → X . Thus Cone( α ) is a bounded below V -acyclic complex ofright U -Gorenstein objects, so Z n (Cone( α )) is right U -Gorenstein for every n ∈ Z by Lemma 2.4(a,1). For n ≫ → X → Cone( α ) → Σ X ′ → → Z n ( X ) → Z n (Cone( α )) → Z n − ( X ′ ) →
0, andanother application of Lemma 2.4(a,1) yields Z n ( X ) ∈ RGor U ( A ).( iv ) = ⇒ ( i ): In view of Proposition 1.2 there exists a complex X in K ( V -ac) , + ( U ∩ V )with V ∼ = X in D b ( V ), and as above we can without loss of generality assume that X n = 0 holds for n ≪
0. By ( iv ) the cycle object Z n ( X ) is right U -Gorenstein forsome n ≫
0, so the complex 0 → Z n ( X ) → X n → · · · is a bounded complex thatis isomorphic to V in D b ( V ) and whose objects belong to RGor U ( A ). (cid:3) Let ( U , V ) be a complete hereditary cotorsion pair in A . We con-sider the following subcategories of D b ( V ) and D b ( U ): Perf
RGor U ( A ) ( V ) := { V ∈ D b ( V ) | V is RGor U ( A )-perfect } and Perf
LGor V ( A ) ( U ) := { U ∈ D b ( U ) | U is LGor V ( A )-perfect } . Let ( U , V ) be a complete hereditary cotorsion pair in A . (a) The category
Perf
RGor U ( A ) ( V ) is thick and it is the smallest triangulated sub-category of D b ( V ) that contains the objects from RGor U ( A ) . (b) The category
Perf
LGor V ( A ) ( U ) is thick and it is the smallest triangulated sub-category of D b ( U ) that contains the objects from LGor V ( A ) . Proof.
We prove part (a); the proof of (b) is similar. Evidently
Perf
RGor U ( A ) ( V ) isadditive and closed under shifts and isomorphisms.Let V ′ → V → V ′′ → Σ V ′ be a triangle in D b ( V ) with V ′ , V ∈ Perf
RGor U ( A ) ( V ).By Proposition 1.2, there is a triangle X ′ α −−→ X −→ Cone( α ) −→ Σ X ′ in K ( V -ac) , + ( U ∩ V ) and an isomorphism in D b ( V ) of triangles: V ′ / / ∼ = (cid:15) (cid:15) V / / ∼ = (cid:15) (cid:15) V ′′ / / ∼ = (cid:15) (cid:15) Σ V ′∼ = (cid:15) (cid:15) X ′ / / X / / Cone( α ) / / Σ X ′ . Since the category
Perf
RGor U ( A ) ( V ) is closed under isomorphisms and V ′ and V belong to Perf
RGor U ( A ) ( V ), so do X and X ′ . Consequently, there is an exact sequence0 −→ X −→ Cone( α ) −→ Σ X ′ −→ C ( V ). The sequence 0 → Z n ( X ) → Z n (Cone( α )) → Z n (Σ X ′ ) → n ≫
0, and Z n ( X ) and Z n (Σ X ′ ) belong to RGor U ( A ) by Theorem 2.6.It now follows from [14, Lem. 2.10] that Z n (Cone( α )) belongs to RGor U ( A ) for n ≫
0. Thus Cone( α ) belongs to Perf
RGor U ( A ) ( V ), again by Theorem 2.6. It followsthat Perf
RGor U ( A ) ( V ) is a triangulated subcategory of D b ( V ).The triangulated subcategory Perf
RGor U ( A ) ( V ) holds the objects from RGor U ( A ).Let h RGor U ( A ) i be the smallest triangulated subcategory of D b ( V ) that containsthe objects from RGor U ( A ), one then has h RGor U ( A ) i ⊆ Perf
RGor U ( A ) ( V ). Let V bean object in Perf
RGor U ( A ) ( V ). There exists a bounded complex X with V ∼ = X in D b ( V ) such that each X n is in RGor U ( A ). Without loss of generality we may assumethat one has X = 0 and X n = 0 for n <
0. Set s = sup { n | X n = 0 } ; we proceedby induction on s . If s = 0, clearly X ∈ h RGor U ( A ) i . For s >
0, there is a trianglein D b ( V ) X s − −→ X −→ Σ s X s −→ Σ X s − . By the induction hypothesis, both X s − and Σ s X s belong to the triangulatedcategory h RGor U ( A ) i , and hence so does X .It remains to show that Perf
RGor U ( A ) ( V ) is closed under direct summands. Let V, V ′ ∈ D b ( V ) and assume that V ⊕ V ′ is RGor U ( A )-perfect. There are V -quasi-isomorphisms X → V and X ′ → V ′ in K ( V ) with X, X ′ ∈ K ( V -ac) , + ( U ∩ V ), thisfollows from the dual to [23, Lem. 4.1]. It follows that X ⊕ X ′ → V ⊕ V ′ is a V -quasi-isomorphism in K ( V ). As V ⊕ V ′ is RGor U ( A )-perfect and X ⊕ X ′ ∈ K ( V -ac) , + ( U ∩ V ),Theorem 2.6 implies that Z n ( X ) ⊕ Z n ( X ′ ) ∼ = Z n ( X ⊕ X ′ ) belongs to RGor U ( A ) for n ≫
0. Thus Z n ( X ) and Z n ( X ′ ) belong to RGor U ( A ) for n ≫ V and V ′ belong to Perf
RGor U ( A ) ( V ). (cid:3) Given [14, Thm. 2.11], Theorem 2.6, and Proposition 2.8 one could obtain thefirst equivalence in part (a) below from a result of Iyama and Yang [19, Cor. 2.2],which elaborates on an example by Keller and Vossieck [25].
Let ( U , V ) be a complete hereditary cotorsion pair in A . (a) There are triangulated equivalences
StRGor U ( A ) ≃ Perf
RGor U ( A ) ( V ) K b ( U ∩ V ) and D defPrj ( V ) ≃ D b ( V ) Perf
RGor U ( A ) ( V ) . HE SINGULARITY CATEGORY OF AN EXACT CATEGORY 11 (b)
There are triangulated equivalences
StLGor V ( A ) ≃ Perf
LGor V ( A ) ( U ) K b ( U ∩ V ) and D defInj ( U ) ≃ D b ( U ) Perf
LGor V ( A ) ( U ) . Proof.
We prove part (a); the equivalences in part (b) have similar proofs. ByProposition 2.8 it suffices to show that the quotient
Perf
RGor U ( A ) ( V ) / K b ( U ∩ V ) isthe essential image of the functor F Prj : StRGor U ( A ) → D b ( V ) / K b ( U ∩ V ). Let V ∈ Perf
RGor U ( A ) ( V ), by Proposition 1.2 there is an isomorphism V ∼ = X in D b ( V )for some X ∈ K ( V -ac) , + ( U ∩ V ). By Theorem 2.6 there exists an integer n suchthat Z n ( X ) ∈ RGor U ( A ) and X ⊇ n is V -acyclic; indeed, the cycle objects are right U -Gorenstein by Lemma 2.4(a,1). Let e T be a right U -totally acyclic complex withZ ( e T ) = Z n ( X ). Splicing together Σ n e T and X > n +1 we obtain a right U -totallyacyclic complex T with T > n +1 = X > n +1 . By definition one has F Prj (Z ( T )) = T > ,and there are isomorphisms T > ∼ = T > n +1 ∼ = X in D sgPrj ( V ).Since the projective objects in V are those in U ∩ V , see Proposition 1.7, theequivalence D defPrj ( V ) ≃ D b ( V ) / Perf
RGor U ( A ) ( V ) follows from Definition 1.3(a) andDefinition 2.2. (cid:3) Let ( U , V ) be a complete hereditary cotorsion pair in A . (a) The following conditions are equivalent: ( i ) F Prj : StRGor U ( A ) → D sgPrj ( V ) yields a triangulated equivalence. ( ii ) One has D defPrj ( V ) = 0 . ( iii ) Every object in D b ( V ) is RGor U ( A ) -perfect. ( iv ) Every object in V is RGor U ( A ) -perfect. (b) The following conditions are equivalent: ( i ) F Inj : StLGor V ( A ) → D sgInj ( U ) yields a triangulated equivalence. ( ii ) One has D defInj ( U ) = 0 . ( iii ) Every object in D b ( U ) is LGor V ( A ) -perfect. ( iv ) Every object in U is LGor V ( A ) -perfect. Proof.
We prove part (a); the proof of part (b) is similar. The equivalence ofthe first three conditions follows immediately from Definition 2.2 and Theorem 2.9.The equivalence of ( iii ) and ( iv ) follows from Proposition 2.8, as the smallest tri-angulated subcategory of D b ( V ) that contains V is D b ( V ). (cid:3) Let ( U , V ) be a cotorsion pair in A . When one combines the pre-vious corollary with the equivalence from [14, Thm. 3.8], it transpires that if everyobject in V is RGor U ( A )-perfect, then there are triangulated equivalences D sgPrj ( V ) ≃ StRGor U ( A ) ≃ K R U -tac ( U ∩ V ) . Similarly, if every object in U is LGor V ( A )-perfect, then there are triangulated equiv-alences D sgInj ( U ) ≃ StLGor V ( A ) ≃ K L V -tac ( U ∩ V ) . In [14] right and left totally acyclic complexes and right and left Gorensteinobjects are defined for any subcategory of A . Given a cotorsion pair ( U , V ) in A ,the subcategory U ∩ V is self-orthogonal, and for such a category the right/left distinctions disappear, see [14, Def. 1.6 and 2.1], and one speaks of U ∩ V -totallyacyclic complexes and U ∩ V -Gorenstein objects. The homotopy category of U ∩ V -totally acyclic complexes is denoted K tac ( U ∩ V ) and the category of U ∩ V -Gorensteinobjects is denoted Gor U ∩ V ( A ). Let ( U , V ) be a complete hereditary cotorsion pair in A . (a) The following conditions are equivalent: ( i ) Gor U ∩ V ( A ) ⊆ V . ( ii ) Gor U ∩ V ( A ) = RGor U ( A ) .Moreover, when these conditions hold one has LGor V ( A ) = U ∩ V . (b) The following conditions are equivalent: ( i ) Gor U ∩ V ( A ) ⊆ U . ( ii ) Gor U ∩ V ( A ) = LGor V ( A ) .Moreover, when these conditions hold one has RGor U ( A ) = U ∩ V . Proof.
We prove (a); the proof of (b) is similar.( ii ) = ⇒ ( i ): The containment RGor U ( A ) ⊆ V holds by definition.( i ) = ⇒ ( ii ): The containment Gor U ∩ V ( A ) ⊇ RGor U ( A ) holds by [14, Rem. 1.8]. Ifone has Gor U ∩ V ( A ) ⊆ V , then every U ∩ V -totally acyclic complex is right U -totallyacyclic, and hence ( ii ) follows.To see that the moreover statement holds under these conditions, recall from[14, Def. 1.1 and Exa. 1.2] that there are containments U ∩ V ⊆ LGor V ( A ) ⊆ U .Further, LGor V ( A ) ⊆ Gor U ∩ V ( A ) holds by [14, Rem. 1.8], so in view of ( i ) one nowhas LGor V ( A ) ⊆ V and hence LGor V ( A ) = U ∩ V . (cid:3) Let ( U , V ) be a cotorsion pair in A . For the self-orthogonal class U ∩ V , the equivalences in Remark 2.11 simplify as follows:If every object in V is RGor U ( A )-perfect and Gor U ∩ V ( A ) ⊆ V , then there aretriangulated equivalences D sgPrj ( V ) ≃ StGor U ∩ V ( A ) ≃ K tac ( U ∩ V ) . Similarly, if every object in U is LGor V ( A )-perfect and Gor U ∩ V ( A ) ⊆ U , then thereare triangulated equivalences D sgInj ( U ) ≃ StGor U ∩ V ( A ) ≃ K tac ( U ∩ V ) . Let A be a ring and consider the cotorsion pair ( Prj ( A ) , Mod ( A )).A Prj ( A )-Gorenstein module is a Gorenstein projective module in the classic sense,see [14, Exa. 2.5]. If every A -module has finite Gorenstein projective dimension,then the categories in the first display in Remark 2.13 are “big” versions of theequivalent categories ( ⋄ ) from the introduction.3. Gorenstein dimensions for everyone
Assume that A is Grothendieck and ( U , V ) is a complete hereditary cotorsion pairin A . Under this extra, but not too restrictive, assumption on A one can extendthe notions of RGor U ( A )- and LGor V ( A )-perfection to the derived category of A : Toevery complex M of objects from A we assign two numbers, the RGor U ( A )-projectivedimension and the LGor V ( A )-injective dimension, and for objects in D b ( V ) and HE SINGULARITY CATEGORY OF AN EXACT CATEGORY 13 D b ( U ) these dimensions are finite if and only if the objects are RGor U ( A )- and LGor V ( A )-perfect, respectively.The category C ( A ) of chain complexes is also Grothendieck, and it follows fromwork of Gillespie [17, Prop. 3.6] and ˇS ’tov´ıˇcek [31, Prop. 7.14] that ( U , V ) inducestwo complete hereditary cotorsion pairs in C ( A ):(3.0.1) ( semi-U , V-ac ) and (
U-ac , semi-V ) . The complexes in
V-ac are acyclic complexes of objects in V with cycle objects in V , i.e. the V -acyclic complexes in K ( V ). The semi- U complexes, i.e. the objectsin semi-U , are complexes U of objects in U with the property that Hom A ( U, V ) isacyclic for every V -acyclic complex V . The classes U-ac and semi-V are definedsimilarly. We call a complex in semi-U ∩ semi-V a semi- U - V complex. Completeness of the cotorsion pairs in (3.0.1) yields:
For every A -complex M there are exact sequences of A -complexes,0 −→ V ′ −→ U π −−→ M −→ −→ M ι −→ V −→ U ′ −→ , where U is semi-U , V is semi-V , V ′ is V -acyclic, U ′ is U -acyclic, and the homomor-phisms π and ι are quasi-isomorphisms. See [17, Prop. 3.6] and [31, Prop. 7.14]. A complex U of objects in U with C n ( U ) ∈ U for n ≪ U complex and a complex V of objects in V with Z n ( V ) ∈ V for n ≫ V complex. This follows from [15, Props. A.1 and A.3] .In particular, a bounded below complex of objects in U is a semi- U complex anda bounded above complex of objects in V is a semi- V complex; cf. [17, Lem. 3.4]. Let M be an A -complex. A semi- U - V replacement of M is a semi- U - V complex that is isomorphic to M in the derived category D ( A ).Some technical results about the cotorsion pairs (3.0.1) and semi- U - V complexeshave been relegated to an appendix. The first step towards defining the RGor U ( A )-projective and LGor V ( A )-injective dimensions is to notice that every A -complex hasa semi- U - V replacement. Per Lemma A.4(a) every semi- V complex V has a semi- U - V replace-ment, as there is even a surjective quasi-isomorphism W ≃ −−→ V with W semi- U - V .Similarly, by Lemma A.4(b), every semi- U complex U has a semi- U - V replacementas there is even an injective quasi-isomorphism U ≃ −−→ W ′ with W ′ semi- U - V . Thesecombine to show that every A -complex M has a semi- U - V replacement:Let M be an A -complex. By Fact 3.1 there is a semi- V complex V and a quasi-isomorphism M ≃ −−→ V , and Lemma A.4(a) yields a semi- U - V complex W and aquasi-isomorphism W ≃ −−→ V . Thus W is a semi- U - V replacement of M .One could also start with a quasi-isomorphism U ≃ −−→ M as in Lemma A.4(a)and get a quasi-isomorphism U ≃ −−→ W ′ from Lemma A.4(b). Let ( U , V ) be a complete hereditary cotorsion pair in A and M an A -complex. The proofs of [15, Props. A.1 and A.3] apply to complexes of objects in any abelian category. (a) The
RGor U ( A ) -projective dimension of M is given by RGor U ( A ) -pd M =inf (cid:26) n ∈ Z (cid:12)(cid:12)(cid:12)(cid:12) There is a semi- U - V replacement W of M withH i ( W ) = 0 for all i > n and C n ( W ) in RGor U ( A ) (cid:27) with inf ∅ = ∞ by convention.(b) The LGor V ( A ) -injective dimension of M is given by LGor V ( A ) -id M =inf (cid:26) n ∈ Z (cid:12)(cid:12)(cid:12)(cid:12) There is a semi- U - V replacement W of M withH − i ( W ) = 0 for all − i > n and Z − n ( W ) in LGor V ( A ) (cid:27) with inf ∅ = ∞ by convention.Note that RGor U ( A ) -pd M = −∞ = LGor V ( A ) -id M holds if M is acyclic. Wecontinue with a caveat that compares to [12, Rmk. 5.9]: An object of RGor U ( A )-projective dimension 0 need not belong to RGor U ( A ); similarly for LGor V ( A ). For every non-zero object U ∈ U one has RGor U ( A ) -pd U = 0. In-deed, the inequality “ > ” holds as H ( U ) = 0 and the opposite inequality holds asone can construct a semi- U - V replacement of U concentrated in non-positive de-grees: Completeness of ( U , V ) yields a coresolution of U by objects in U ∩ V thatis concentrated in non-positive degrees and whose cycle objects are in U ; it is asemi- U - V complex by Example 3.2.Similarly, LGor V ( A ) -id V = 0 holds for every non-zero object V ∈ V . Let ( U , V ) be a complete hereditary cotorsion pair in A and M an A -complex with a semi- U - V replacement W . (a) For every integer n with RGor U ( A ) -pd M n one has C n ( W ) ∈ RGor U ( A ) . (b) For every integer n with LGor V ( A ) -id M n one has Z − n ( W ) ∈ LGor V ( A ) . Proof.
We prove part (a); the proof of part (b) is similar. One can assume that
RGor U ( A ) -pd M = g holds for some integer g . By assumption there exists a semi- U - V replacement W ′ of M with C g ( W ′ ) in RGor U ( A ) and H n ( W ′ ) = 0 for n > g . Byinduction it follows from Lemma 2.4(a) that C n ( W ′ ) belongs to RGor U ( A ) for every n > g . Let W be any semi- U - V replacement of M . It follows from Proposition A.7and Lemma 2.3 that C n ( W ) belongs to RGor U ( A ) for every n > g . (cid:3) Let ( U , V ) be a complete hereditary cotorsion pair in A . (a) An object V ∈ D b ( V ) is RGor U ( A ) -perfect if and only if it has finite RGor U ( A ) -projective dimension. (b) An object U ∈ D b ( U ) is LGor V ( A ) -perfect if and only if it has finite LGor V ( A ) -injective dimension. Proof.
We prove part (a); the proof of part (b) is similar. Without loss ofgenerality, let V be a bounded complex of objects in V . There exists by Theo-rem A.8(a) a bounded below semi- U - V replacement W of V . The assertion nowfollows from Theorem 2.6 and Lemma 3.7 as one has Z n ( W ) ∼ = C n +1 ( W ) for n > sup { i ∈ Z | H i ( V ) = 0 } . (cid:3) HE SINGULARITY CATEGORY OF AN EXACT CATEGORY 15
Let ( U , V ) be a complete hereditary cotorsion pair in A . Standardarguments, see [12, Thm. 4.5], based on Lemma 2.4(a,2) and 2.4(b,2) show that the RGor U ( A )-projective and LGor V ( A )-injective dimensions of an A -complex M whenfinite can be detected in terms of vanishing of cohomology: RGor U ( A ) -pd M = sup { m ∈ Z | Ext m A ( M, W ) = 0 for some W ∈ U ∩ V } and LGor V ( A ) -id M = sup { m ∈ Z | Ext m A ( W, M ) = 0 for some W ∈ U ∩ V } . Gorenstein rings and schemes
Our main interest here is schemes, but we warm up with rings.Let A be a ring. The cotorsion pair ( Flat ( A ) , Cot ( A )) in the Grothendieck cat-egory Mod ( A ) is complete and hereditary and hence gives rise to the RGor
Flat ( A )-projective dimension. By [14, Prop. 4.2] this cotorsion pair satisfies the conditionsin Proposition 2.12(a), and the RGor
Flat ( A )-projective dimension was studied in [12]under the name “Gorenstein flat-cotorsion dimension;” see also [12, Rmk. 4.8]. Inthis section we keep with that terminology as it more descriptive, and we write FlatCot ( A ) for the intersection Flat ( A ) ∩ Cot ( A ).If every A -module has finite Gorenstein flat-cotorsion dimension, then, as notedin Remark 2.13, there are triangulated equivalences D sgPrj ( Cot ( A )) ≃ StGor
FlatCot ( A ) ≃ K tac ( FlatCot ( A )) . We proceed to identify a class of rings over which these equivalences are realized.Recall that a left and right noetherian ring is called
Iwanaga–Gorenstein if ithas finite self-injective dimension on both sides. The next theorem collects charac-terizations of such rings in terms of (Gorenstein) flat–cotorsion theory that closelymirror classic characterizations in terms of (Gorenstein) projectivity. In partic-ular, ( ii ) is the analogue of Buchweitz’s equivalence between the stable categoryof finitely generated Gorenstein projective modules and the singularity category[7, Thm. 4.4.1]. Condition ( iii ) is analogous to [6, Thm. 3.6] and compares to[20, Rmk. 5.6]. Conditions ( iv ) and ( v ) compare to Auslander and Bridger’s [1,Thm. 4.20]. Let A be left and right noetherian. The following conditions areequivalent. ( i ) A is Iwanaga–Gorenstein. ( ii ) The functors F Prj : StGor
FlatCot ( A ) −→ D sgPrj ( Cot ( A )) and ˙F Prj : StGor
FlatCot ( A ◦ ) −→ D sgPrj ( Cot ( A ◦ )) yield triangulated equivalences. ( iii ) One has D defPrj ( Cot ( A )) = 0 = D defPrj ( Cot ( A ◦ )) . ( iv ) Every cotorsion A - and every cotorsion A ◦ -module has finite Gorenstein flat-cotorsion dimension. ( v ) Every A - and every A ◦ -module has finite Gorenstein flat-cotorsion dimension. ( vi ) Every A - and every A ◦ -complex with bounded above homology has finiteGorenstein flat-cotorsion dimension. Proof.
Conditions ( ii ), ( iii ), and ( iv ) are equivalent by Corollary 2.10 and Theo-rem 3.8. Conditions ( i ), ( v ), and ( vi ) are equivalent by [12, Cor. 5.10]. Evidently( v ) implies ( iv ). For the converse let M be an A -module and consider an exact se-quence 0 → M → C → F → C is cotorsion and F is flat. By [12, Cor. 5.8]an A - or A ◦ -module has finite Gorenstein flat-cotorsion dimension if and only if ithas finite Gorenstein flat dimension. It now follows from Holm [18, Thm. 3.15] thatalso M has finite Gorenstein flat-cotorsion dimension. (cid:3) Part of the allure of the category D sgPrj ( Cot ( A )) is that it does not rely on pro-jective modules and thus persists in the non-affine setting. Let X denote a schemewith structure sheaf O X . By a sheaf on X we mean a quasi-coherent sheaf, and Qcoh ( X ) denotes the category of such sheaves. We say that X is semi-separated ifit has an open affine covering with the property that the intersection of any twoopen affine sets is affine.Let X be semi-separated noetherian. The flat sheaves and cotorsion sheaves on X form a complete hereditary cotorsion pair ( Flat ( X ) , Cot ( X )) in the Grothendieckcategory Qcoh ( X ); see [13, Rmk. 2.4]. It follows from [13, Thm. 3.3] that this pairsatisfies the equivalent conditions in Proposition 2.12(a), so as above we refer tothe RGor
Flat ( X )-projective dimension as the Gorenstein flat-cotorsion dimension.
If every cotorsion sheaf on X has finite Gorenstein flat-cotorsion dimension then,as noted in Remark 2.13, there are triangulated equivalences: D sgPrj ( Cot ( X )) ≃ StGor
FlatCot ( X ) ≃ K tac ( FlatCot ( X )) . We proceed to identify a class of schemes over which these equivalences are realized.Recall that a semi-separated noetherian scheme X is Gorenstein provided that O X,x is a Gorenstein ring for every x ∈ X . If, in addition, the scheme has finite Krulldimension, then this is equivalent to saying that O X ( U ) is a Gorenstein ring foreach open affine set U in some, equivalently every, open affine covering of X . Let X be a semi-separated noetherian scheme of finite Krull di-mension. The following conditions are equivalent. ( i ) X is Gorenstein. ( ii ) The functor F Prj : StGor
FlatCot ( X ) −→ D sgPrj ( Cot ( X )) yields a triangulated equivalence. ( iii ) One has D defPrj ( Cot ( X )) = 0 . ( iv ) Every cotorsion sheaf on X has finite Gorenstein flat-cotorsion dimension. ( v ) Every sheaf on X has finite Gorenstein flat-cotorsion dimension. ( vi ) Every complex of sheaves on X with bounded above homology has finiteGorenstein flat-cotorsion dimension. Proof.
Conditions ( ii ), ( iii ), and ( iv ) are equivalent by Corollary 2.10 and The-orem 3.8. Evidently, ( vi ) implies ( v ) implies ( iv ), so it suffices to show that ( i )implies ( vi ) and ( iv ) implies ( i ). Denote by d the Krull dimension of X .( i ) = ⇒ ( vi ): Let M be a complex of sheaves on X with bounded above homologyand assume without loss of generality that H n ( M ) = 0 holds for n >
0. Let F bea semi-flat-cotorsion replacement of M . Notice that the truncated complex F > is a semi-flat-cotorsion replacement of the cokernel C = C ( F ) and consider theexact sequence 0 −→ G −→ F d − −→ · · · −→ F −→ C −→ . The truncated complex F > d yields a left resolution of G by cotorsion sheaves;splicing this with a coresolution by injective sheaves, one obtains G as a cycle inan acyclic complex of cotorsion sheaves, whence G is cotorsion by [13, Thm. 3.3].Per [13, Thm. 4.3] it now suffices to show that G is Gorenstein flat. Let U be anopen affine covering of X . For every open set U ∈ U there is an exact sequence of O X ( U )-modules,0 −→ G ( U ) −→ F d − ( U ) −→ · · · −→ F ( U ) −→ C ( U ) −→ , with F n ( U ) a flat O X ( U )-module for 0 n d −
1. As O X ( U ) is Gorensteinof Krull dimension at most d , the module G ( U ) is Gorenstein flat; see e.g. [16,Thm. 12.3.1]. From [13, Thm. 1.6] it follows that the sheaf G is Gorenstein flat.( iv ) = ⇒ ( i ): Let U be an open covering of X and fix an open set U ∈ U . The goalis to prove that the ring O X ( U ) is Gorenstein. As it has finite Krull dimension, itsuffices to show that every O X ( U )-module has finite Gorenstein flat dimension; seee.g. [16, Thm. 12.3.1]. Let M be an O X ( U )-module; denote by f M the correspondingsheaf on Spec O X ( U ) and by ( i U ) ∗ ( f M ) the sheaf on X induced by the embedding i U : Spec O X ( U ) → X . As ( Flat ( X ) , Cot ( X )) is a complete hereditary cotorsionpair in Qcoh ( X ), see [13, Rmk. 2.4], there is a flat resolution F of ( i U ) ∗ ( f M ) over X constructed by taking special flat precovers. In particular, for n > n ( F ) is cotorsion and F n is flat-cotorsion. By assumption C ( F ) has finiteGorenstein flat-cotorsion dimension, which means that C n ( F ) is Gorenstein flat-cotorsion for some n >
1. In particular, C n ( F ) is by [13, Thm. 4.3] Gorensteinflat. Thus one gets an exact sequence of O X ( U )-modules,0 −→ C n ( F )( U ) −→ F n − ( U ) −→ · · · −→ F ( U ) −→ M −→ , which shows that M has finite Gorenstein flat dimension. cf. [13, Thm. 1.6]. (cid:3) Finally we justify that D sgPrj ( Cot ( X )) is a non-affine avatar of D sgPrj ( A ): Let A be a commutative Gorenstein ring of finite Krull dimensionand set X = Spec ( A ) . There are equivalences of triangulated categories D sgPrj ( A ) ≃ StGor
Prj ( A ) ≃ K tac ( Prj ( A )) ≃ D sgPrj ( Cot ( X )) ≃ StGor
FlatCot ( X ) ≃ K tac ( FlatCot ( X )) . Proof.
A commutative noetherian ring of finite Krull dimension is Gorenstein ifand only if it is Iwanaga–Gorenstein. The horizontal equivalences thus come fromthe two theorems above combined with Remark 2.13. The vertical equivalencefollows from [14, Cor. 5.9]. (cid:3) Finitistic dimension
As discussed in Section 1, vanishing of singularity categories can capture finitenessof global dimensions. In this section we show how vanishing of a defect categorycaptures finiteness of a finitistic dimension.In this section A is a ring. The little finitistic dimension of A , findim( A ), isthe supremum of projective dimensions of finitely generated A -modules of finite projective dimension. It is conjectured that findim( A ) < ∞ holds for Artin algebras A ; see Bass [3] and [2, Conjectures]. This is the Finitistic Dimension Conjecture. Let ( U , V ) be a complete hereditary cotorsion pair in Mod ( A ). Let M be an A -module. The U ∩ V -injective dimension of M is defined as U ∩ V -id M = inf (cid:26) n ∈ Z (cid:12)(cid:12)(cid:12)(cid:12) There is a semi- U - V replacement W of M with W − i = 0 for i > n . (cid:27) . Let ( U , V ) be a complete hereditary cotorsion pair in Mod ( A ) and M an A -module. There is an inequality LGor V ( A ) -id M U ∩ V -id M and equality holds if U ∩ V -id M < ∞ . Proof.
Assume U ∩ V -id M = n holds for some integer n , otherwise the statementis trivial. Let W be a semi- U - V replacement of M with W − i = 0 for i > n . Theinequality holds as every module in U ∩ V is left V -Gorenstein; see [14, Exa. 2.2].Let m n be such that H − i ( M ) = 0 for i > m . If Z − m ( W ) is left V -Gorenstein,then Z − m ( W ) ∈ U ∩ V . Indeed, it has a finite coresolution by modules in U ∩ V which splits by Lemma 2.4(b). (cid:3) Let ( U , V ) be a complete hereditary cotorsion pair in Mod ( A ) , M an A -module in U , and n > an integer. One has LGor V ( A ) -id M n if and only ifthere exists an exact sequence of A -modules −→ M −→ G −→ L −→ with G ∈ LGor V ( A ) , L ∈ U , and U ∩ V -id L n − . Proof.
There is, by the completeness of ( U , V ), a coresolution of M consisting ofmodules in U ∩ V , concentrated in non-positive degrees and with cycle modules in U ; it is a semi- U - V complex by Example 3.2. Using this, the “only if” statement isproved the same way as [18, Thm. 2.10]. The “if” statement holds by a standardapplication of the Horseshoe Lemma, see [16, Lem. 8.2.1 and Rmk. 8.2.2], andLemma 3.7; it also follows from the second equality in Remark 3.9. (cid:3) Let ( U , V ) be a complete hereditary cotorsion pair in Mod ( A ) .For every module M in U ∩ ⊥ LGor V ( A ) one has LGor V ( A ) -id M = U ∩ V -id M . Proof.
By Lemma 5.2, it suffices to show that U ∩ V -id M LGor V ( A ) -id M holds.Set LGor V ( A ) -id M = n . If n = 0, then there is an exact sequence of A -modules0 −→ H −→ W −→ M −→ H ∈ LGor V ( A ) and W ∈ U ∩ V . Since M is in ⊥ LGor V ( A ) the sequence splits,whence M ∈ U ∩ V . If n >
1, then Lemma 5.3 yields an exact sequence of A -modules,0 → M → G → L →
0, with G ∈ LGor V ( A ), L ∈ U , and U ∩ V -id L n −
1. There isalso an exact sequence 0 → G ′ → W ′ → G → A -modules with G ′ ∈ LGor V ( A ) HE SINGULARITY CATEGORY OF AN EXACT CATEGORY 19 and W ′ ∈ U ∩ V . There is a pullback diagram with exact rows and columns:0 (cid:15) (cid:15) (cid:15) (cid:15) G ′ (cid:15) (cid:15) G ′ (cid:15) (cid:15) / / T / / (cid:15) (cid:15) W ′ / / (cid:15) (cid:15) L / / / / M / / (cid:15) (cid:15) G / / (cid:15) (cid:15) L / /
00 0 . As W ′ is in U ∩ V and U ∩ V -id L n − U ∩ V -id T n . As the left-hand column splits, one has U ∩ V -id M n . (cid:3) Let P < ∞ ( A ) denote the class of finitely generated A -modules of finite projectivedimension and ( U , V ) be the complete cotorsion pair in Mod ( A ) cogenerated by P < ∞ ; that is, V = ( P < ∞ ( A )) ⊥ and U = ⊥ V . This cotorsion pair is also hereditaryby a result of Cort´es Izurdiaga, Estrada, and Guil Asensio [21, Cor. 2.7]. Set V = ( P < ∞ ( A )) ⊥ and U = ⊥ V . The following conditions areequivalent. ( i ) One has findim( A ) < ∞ . ( ii ) Every module in U is LGor V ( A ) -perfect. Proof. ( i ) = ⇒ ( ii ): If one has findim( A ) = n for an integer n , then [21, Thm. 3.4]yields U ∩ V -id M n for every module M ∈ U ; in particular M is LGor V ( A )-perfectby Lemma 5.2 and Theorem 3.8.( ii ) = ⇒ ( i ): The projective A -module A ( A ) belongs to U , so by Theorem 3.8there is an integer n such that LGor V ( A ) -id A ( A ) = n holds. As A ( A ) also belongsto ⊥ LGor V ( A ), one has U ∩ V -id A ( A ) n by Proposition 5.4. Now apply [21,Thm. 3.4] to conclude that findim( A ) n holds. (cid:3) Set V = ( P < ∞ ( A )) ⊥ and U = ⊥ V . The following conditions areequivalent: ( i ) One has findim( A ) < ∞ . ( ii ) The functor F Inj : StLGor V ( A ) → D sgInj ( U ) yields a triangulated equivalence. ( iii ) One has D defInj ( U ) = 0 . Proof.
Combine Theorem 5.5 and Corollary 2.10. (cid:3)
Another classic conjecture for Artin algebras is the
Wakamatsu tilt-ing conjecture , which says that any Wakamatsu tilting A -module of finite projectivedimension is a tilting A -module; see Beligiannis and Reiten [5, Chapter IV]. Manteseand Reiten [27, Proposition 4.4] showed that the Wakamatsu tilting conjecture is aspecial case of the Finitistic Dimension Conjecture. A Wakamatsu tilting modulegives rise to a cotorsion pair, see Wang, Li, and Hu [32, Thm. 1.3]. If the module hasfinite projective dimension, then vanishing of an associated defect category betraysif it is tilting. We omit the details which are similar to the arguments above. Appendix: A Schanuel’s lemma for semi- U - V replacements Assume throughout this appendix that A , as in Section 3, is Grothendieck and let( U , V ) be a complete hereditary cotorsion pair in A . A.1 Lemma.
Let → M ′ → M → M ′′ → be an exact sequence in C ( A ) . (a) If M ′ is semi- V then M is semi- V if and only if M ′′ is semi- V . (b) If M ′′ is semi- U then M is semi- U if and only if M ′ is semi- U . Proof.
We prove part (a); the proof of part (b) is similar. Assume that M ′ is semi- V . It is in particular a complex of objects from V , so M is a complex of objectsfrom V if and only if M ′′ is so. Assuming that this is the case, let U be a U -acycliccomplex. As M ′ is a complex of objects from V , there is an exact sequence0 −→ Hom A ( U, M ′ ) −→ Hom A ( U, M ) −→ Hom A ( U, M ′′ ) −→ . By assumption the left-hand complex is acyclic, so the middle complex is acyclic ifand only if the right-hand complex is acyclic. (cid:3)
A.2 Proposition.
The following assertions hold: (a) An A -complex is U -acyclic if and only if it is semi- U and acyclic. (b) An A -complex is V -acyclic if and only if it is semi- V and acyclic.Moreover, an acyclic semi- U - V complex is contractible. Proof.
By [17, Lem. 3.10] a U -acyclic complex is semi- U , and a V -acyclic complexis semi- V . The category A has enough injectives, see e.g. Kashiwara and Schapira[22, Thm. 9.6.2], so it follows from [17, Thm. 3.12] that an acyclic semi- U complex is U -acyclic. As ( semi-U , V-ac ) is complete, see [17, Prop. 3.6] and [31, Prop. 7.14], itfollows from [17, Lem. 3.14(1)] that an acyclic semi- V complex is V -acyclic. Finally,an acyclic semi- U - V complex is contractible as it has cycle objects in U ∩ V . (cid:3) A.3 Corollary.
The following assertions hold. (a)
A quasi-isomorphism of semi- U complexes is a U -quasi-isomorphism. (b) A quasi-isomorphism of semi- V complexes is a V -quasi-isomorphism. (c) A quasi-isomorphism of semi- U - V complexes is a homotopy equivalence. Proof.
The mapping cone of a quasi-isomorphism of semi- U complexes is acyclicand by Lemma A.1 semi- U , so it is U -acyclic by Proposition A.2. This proves (a) andthe proof of (b) is similar. Finally, the mapping cone of a quasi-isomorphism of semi- U - V complexes is an acyclic semi- U - V complex and hence per A.2 contractible. (cid:3) A.4 Lemma.
Let M be an A -complex and consider the exact sequences from 3.1, −→ V ′ −→ U π −−→ M −→ and −→ M ι −→ V −→ U ′ −→ . (a) If M is a complex of objects in V , then π is a V -quasi-isomorphism, and if M is semi- V , then U is semi- U - V . (b) If M is a complex of objects in U , then ι is a U -quasi-isomorphism, and if M is semi- U , then V is semi- U - V . HE SINGULARITY CATEGORY OF AN EXACT CATEGORY 21
Proof.
We prove part (a); the proof of part (b) is similar. For every object U ′ in U the induced sequence0 −→ Hom A ( U ′ , V ′ ) −→ Hom A ( U ′ , U ) Hom A ( U ′ ,π ) −−−−−−−−→ Hom A ( U ′ , M ) −→ A ( U ′ , V ′ ) is acyclic. It follows that Hom A ( U ′ , π ) is a quasi-isomorphism. If M is a complex of objects from V , then so is U , whence Cone( π )is an acyclic complex of objects in V . As Hom A ( U ′ , Cone( π )) is acyclic for every U ′ ∈ U it follows that the cycles of the complex Cone( π ) belong to V . Thus π is a V -quasi-isomorphism. Finally, if M is semi- V , then by Proposition A.2 andLemma A.1 so is U . That is, U is semi- U - V . (cid:3) The lemma above provides a semi- U - V replacement of any A -complex; see Re-mark 3.4. Our next goal is a Schanuel’s lemma for semi- U - V replacements. Wemove towards it with the next result and prove it in Proposition A.7. A.5 Proposition.
Let U and U ′ be semi- U complexes and V and V ′ be a semi- V complexes. (a) Let β : U ≃ −−→ U ′ be a U -quasi-isomorphism. For every morphism α : U → V there is a morphism γ : U ′ → V with γβ ∼ α , and γ is unique up to homotopy. (b) Let β : V ′ ≃ −−→ V be a V -quasi-isomorphism. For every morphism α : U → V there is a morphism γ : U → V ′ with βγ ∼ α , and γ is unique up to homotopy. Proof.
We prove part (a); the proof of part (b) is similar. The complex Cone( β ) is U -acyclic, so the morphism Hom A ( β, V ) is a quasi-isomorphism. From this point,the proof of [12, Prop. 1.7] applies verbatim. (cid:3) A.6 Lemma.
Let W and W ′ be semi- U - V complexes. If there is an isomorphism W ≃ W ′ in D ( A ) , then there is a homotopy equivalence W → W ′ in C ( A ) . Proof.
By assumption there exists an A -complex X and a diagram in C ( A ): W ≃ ←−− X ≃ −−→ W ′ . By Fact 3.1 there is a semi- U complex U and a quasi-isomorphism U ≃ −−→ X .The composite U −→ W is by Corollary A.3(a) a U -quasi-isomorphism, so upto homotopy the composite U → W ′ lifts to a quasi-isomorphism W −→ W ′ ;see Proposition A.5. By Corollary A.3(c) this quasi-isomorphism is a homotopyequivalence. (cid:3) A.7 Proposition.
Let W and W ′ be semi- U - V complexes isomorphic in D ( A ) . Forevery n ∈ Z the following assertions hold. (a) There exist objects U and U ′ in U ∩ V such that C n ( W ) ⊕ U ∼ = C n ( W ′ ) ⊕ U ′ . (b) There exist objects V and V ′ in U ∩ V such that Z n ( W ) ⊕ V ∼ = Z n ( W ′ ) ⊕ V ′ . Proof.
We prove part (a); the proof of part (b) is similar. By Lemma A.6 thereis a homotopy equivalence α : W → W ′ . It follows from Proposition A.2 that ev-ery cycle object Z n (Cone( α )) belongs to U ∩ V . The soft truncated morphism α ⊆ n : W ⊆ n → W ′ ⊆ n is also a homotopy equivalence, so Cone( α ⊆ n ), i.e. the complex0 → C n ( W ) → C n ( W ′ ) ⊕ W n − → W ′ n − ⊕ W n − ∂ Cone( α ) n − −−−−−→ W ′ n − ⊕ W n − → · · · is contractible. Hence one has C n ( W ) ⊕ Z n − (Cone( α )) ∼ = C n ( W ′ ) ⊕ W n − . (cid:3) The point of the next result, which is crucial to our proof of Theorem 3.8, isthat one can exert some control over the boundedness of semi- U - V -replacements ofbounded complexes. A.8 Theorem.
The following assertions hold. (a)
For every bounded below semi- V complex V there exists an exact sequence −→ V ′ −→ W π −−→ V −→ such that V ′ is V -acyclic, W is semi- U - V and bounded below, and π is a V -quasi-isomorphism. (b) For every bounded above semi- U complex U there exists an exact sequence −→ U ι −→ W −→ U ′ −→ such that U ′ is U -acyclic, W is semi- U - V and bounded above, and ι is a U -quasi-isomorphism. Proof.
We prove part (a); the proof of part (b) is similar. There exists byLemma A.4(a) an exact sequence0 −→ V ′ −→ W π −−→ V −→ V ′ is V -acyclic, W is semi- U - V and π is a V -quasi-isomorphism. Since V is bounded below, the complexes V ′ and W agree in low degrees. It suffices toshow that Z n ( W ) belongs to U ∩ V for n ≪
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