The derived category of a locally complete intersection ring
aa r X i v : . [ m a t h . A C ] M a y THE DERIVED CATEGORY OF A LOCALLY COMPLETEINTERSECTION RING
JOSH POLLITZ
Abstract.
In this paper, we answer a question of Dwyer, Greenlees, andIyengar by proving a local ring R is a complete intersection if and only ifevery complex of R -modules with finitely generated homology is proxy small.Moreover, we establish that a commutative noetherian ring R is locally acomplete intersection if and only if every complex of R -modules with finitelygenerated homology is virtually small. Introduction
The relation of the structure of a commutative noetherian ring R and that of itscategory of modules has long been a major topic of study in commutative algebra.More recently, it has been extended to studying the relations between the structureof R and that of its derived category D ( R ). Working in this setting allows one touse ideas from algebraic topology and triangulated categories to gain insight intoproperties of R .Basic information on D ( R ) is contained in its full subcategory consisting of com-plexes with finitely generated homology, denoted D f ( R ). A complex of R -modulesis said to be perfect if it is quasi-isomorphic to a bounded complex of finitely gener-ated projective R -modules. The following homotopical characterization of regularrings is well known: a commutative noetherian ring R is regular if and only if everyobject of D f ( R ) is a perfect complex .In many respects, the local rings that are closest to being regular are completeintersections. We characterize of complete intersections in terms of how each objectof D f ( R ) relates to the perfect complexes. Moreover, this yields a homotopicalcharacterization of a locally complete intersection ring. Following [10] and [11],we say that a complex of R -modules M finitely builds a complex of R -modules N provided that N can be obtained by taking finitely many cones and retracts startingfrom M . More precisely, M finitely builds N provided N is in Thick D ( R ) M (seeSection 2.5). The main result of the paper is the following: Theorem.
A commutative noetherian ring R is locally a complete intersectionif and only if every nontrivial object of D f ( R ) finitely builds a nontrivial perfectcomplex. Date : May 22, 2018.2010
Mathematics Subject Classification.
Key words and phrases. local ring, complete intersection, derived category, DG algebra, thicksubcategory, support variety.The author was partly supported through NSF grant DMS 1103176. cknowledgements. I thank my advisors Luchezar Avramov and Mark Walker fortheir support and interesting conversations on this project. I would also like tothank Srikanth Iyengar for several useful discussions regarding this work.2.
Preliminaries
Differential Graded Algebra.
Fix a commutative noetherian ring Q . Let A = { A i } i ∈ Z denote a DG Q -algebra. We only consider left DG A -modules.2.1.1 . Let M and N be DG A -modules. We say that ϕ : M → N is a morphism ofDG A -modules provided ϕ is a morphism of the underlying complexes of Q -modulessuch that ϕ ( am ) = aϕ ( m ) for all a ∈ A and m ∈ M . We write ϕ : M ≃ −→ N when ϕ is a quasi-isomorphism.2.1.2 . Let M be a DG A -module. The differential of M is denoted by ∂ M . Foreach i ∈ Z , Σ i M is the DG A -module given by( Σ i M ) n := M n − i , ∂ Σ i M := ( − i ∂ M , and a · m := ( − | a | i am. Let H( M ) := { H i ( M ) } i ∈ Z which is a graded module over the graded Q -algebraH( A ) := { H i ( A ) } i ∈ Z . We let M ♮ denote the underlying graded Q -module. Notethat A ♮ is a graded Q -algebra and M ♮ is a graded A ♮ -module.2.1.3 . A DG A -module P is semiprojective if for every morphism of DG A -modules α : P → N and each surjective quasi-isomorphism of DG A -modules γ : M → N there exists a unique up to homotopy morphism of DG A -module β : P → M suchthat α = γβ .2.1.4 . A semiprojective resolution of a DG A -module M is a surjective quasi-isomorphism of DG A -modules ǫ : P → M where P is a semiprojective DG A -module. Semiprojective resolutions exist and any two semiprojective resolutions of M are unique up to homotopy equivalence [12, 6.6].2.1.5 . For DG A -modules M and N , defineExt ∗ A ( M, N ) := H(Hom A ( P, N ))where P is a semiprojective resolution of M over A . Since any two semiprojectiveresolutions of M are homotopy equivalent, Ext ∗ A ( M, N ) is independent of choice of M . An element [ α ] of Ext ∗ A ( M, N ) is the class of a morphism of DG A -modules α : P → Σ | α | N. Moreover, given [ α ] and [ β ] in Ext ∗ A ( M, N ), then [ α ] = [ β ] if and only if α and β are homotopic morphisms of DG A -modules.2.1.6 . Let D ( A ) denote the derived category of A (see [15] for an explicit construc-tion). Recall that D ( A ), equipped with Σ , is a triangulated category. Define D f ( A )to be the full subcategory of D ( A ) consisting of all M such that H( M ) is a finitelygenerated graded module over H( A ) . We use ≃ to denote isomorphisms in D ( A )and reserve ∼ = for isomorphisms of DG A -modules. .2. Koszul Complexes.
Fix a commutative noetherian ring Q . Let f = f , . . . , f n be a list of elements in Q . Define Kos Q ( f ) to be the DG Q -algebra with Kos Q ( f ) ♮ the exterior algebra on a free Q -module with basis ξ , . . . , ξ n of homological degree1, and differential ∂ξ i = f i . We writeKos Q ( f ) = Q h ξ , . . . , ξ n | ∂ξ i = f i i . . Let f ′ = f ′ , . . . , f ′ m be in Q . Assume there exists a ij ∈ Q such that f i = m X j =1 a ij f ′ j . There exists a unique morphism of DG Q -algebras Kos Q ( f ) → Kos Q ( f ′ ) satisfying ξ i m X j =1 a ij ξ ′ j . Therefore, Kos Q ( f ′ ) is a DG Kos Q ( f )-module where the action is given by ξ i · e ′ = m X j =1 a ij ξ ′ j e ′ for all e ′ ∈ E ′ . . Assume that ( Q, n , k ) is a commutative noetherian local ring. Define K Q tobe the Koszul complex on some minimal generating set for n . Then K Q is uniqueup to DG Q -algebra isomorphism.2.3. Map on Ext.
Let Q be a commutative noetherian ring. Fix a morphism ofDG Q -algebras ϕ : A ′ → A . Let M and N be DG A -modules, ǫ : P → M be asemiprojective resolution of M over A , and ǫ ′ : P ′ → M a semiprojective resolutionof M over A ′ . There exists a unique up to homotopy morphism of DG A ′ -modules α : P ′ → P such that ǫ ′ = ǫα . Define Hom ϕ ( α, N ) to be the compositionHom A ( P, N ) Hom ϕ ( P,N ) −−−−−−−−→ Hom A ′ ( P, N ) Hom A ′ ( α,N ) −−−−−−−−→ Hom A ′ ( P ′ , N ) . This induces a map in cohomologyExt ∗ ϕ ( M, N ) : Ext ∗ A ( M, N ) → Ext ∗ A ′ ( M, N )given by Ext ∗ ϕ ( M, N ) = H(Hom ϕ ( α, N )); it is independent of choice of α, P , and P ′ .2.3.1 . Let ϕ : A ′ → A be a morphism of DG Q -algebras and let M and N be DG A -modules. If ϕ is a quasi-isomorphism, then Ext ∗ ϕ ( M, N ) is an isomorphism [12,6.10].In the following theorem, the theory of DG Γ-algebras is used. See [2, Section6] or [13, Chapter 1] as a reference for definitions and notation.
Theorem 2.3.2.
Assume ( Q, n , k ) is a regular local ring. Let R = Q/I where I is minimally generated by f = f , . . . , f n ∈ n . Let E be the Koszul complex on f over Q . Let ϕ : E → R denote the augmentation map. The canonical map Ext ∗ ϕ ( k, k ) : Ext ∗ R ( k, k ) → Ext ∗ E ( k, k ) is surjective. roof. Write E = Q h ξ , . . . , ξ n | ∂ξ i = f i i . For an element a ∈ Q , let a denote the im-age of a in R . Let s , . . . , s e be a minimal generating set for n . Let X = { x , . . . , x e } be a set of exterior variables of homological degree 1 and Y = { y , . . . , y n } a set ofdivided power variables of homological degree 2. By [2, 7.2.10], the morphism ofDG Γ-algebras ϕ : E → R extends to a morphism of DG Γ-algebras ϕ h X i : E h X | ∂x i = s i i → R h X | ∂x i = s i i such that ϕ h X i ( x i ) = x i for each 1 ≤ i ≤ e .Since f i ∈ n , there exists a ij ∈ n such that f i = e X j =1 a ij s j . For each 1 ≤ i ≤ n , we have degree 1 cycles z i := n X j =1 a ij x j − ξ i and z i := n X j =1 a ij x j in E h X i and R h X i , respectively, where ϕ h X i ( z i ) = z i . Applying [2, 7.2.10] yieldsa morphism of DG Γ-algebras ϕ h X, Y i : E h X ih Y | ∂y i = z i i → R h X ih Y | ∂y i = z i i extending ϕ h X i such that ϕ h X, Y i ( y i ) = y i for each 1 ≤ i ≤ n .By [2, 6.3.2], E h X, Y i is an acyclic closure of k over E . In particular, E h X, Y i isa semiprojective resolution of k over E . Next, s , . . . , s e is a minimal generating setfor the maximal ideal of R . Also, since f , . . . , f n minimally generates I , it followsthat [ z ] , . . . , [ z n ] is a minimal generating set for H ( R h X i ) (see [18, Theorem 4]or [13, 1.5.4]). Thus, R h X, Y i is the second step in forming an acyclic closure of k over R . Let ι : R h X, Y i ֒ → R h X, Y, V i denote the inclusion of DG Γ-algebraswhere R h X, Y, V i is an acyclic closure of k over R and V consists of Γ-variablesof homological degree at least 3. Define α : E h X, Y i → R h X, Y, V i to be themorphism of DG Γ-algebras given by α := ι ◦ ϕ h X, Y i . The following is a commutative diagram of Γ-algebras E h X, Y i ⊗ E k R h X, Y, V i ⊗ R kk h X, Y i k h X, Y, V i ∼ = α ⊗ k ∼ = ⊆ Therefore, α ⊗ k is an injective morphism of Γ-algebras. In particular, α ⊗ k isinjective as a map of graded k -vector spaces. Also, the following is a commutativediagram of graded k -vector spacesHom k ( R h X, Y, V i ⊗ R k, k ) Hom k ( E h X, Y i ⊗ E k, k )Hom R ( R h X, Y, V i , k ) Hom E ( E h X, Y i , k ) ∼ = ( α ⊗ k ) ∗ ∼ =Hom ϕ ( α,k ) Since α ⊗ k is injective, ( α ⊗ k ) ∗ is surjective. Thus, Hom ϕ ( α, k ) is surjective.Moreover, Hom E ( E h X, Y i , k ) and Hom R ( R h X, Y, V i , k ) have trivial differential (see[2, 6.3.4]). Thus, Ext ∗ ϕ ( k, k ) = Hom ϕ ( α, k ), and so Ext ∗ ϕ ( k, k ) is surjective. (cid:3) .4. Support of a Complex of Modules.
Let R be a commutative noetherianring and Spec R denote the set of prime ideals of R . For a complex of R -modules M , define the support of M to beSupp R M := { p ∈ Spec R : M p } . . Let M be in D f ( R ) and let x generate an ideal I of R . It follows fromNakayama’s lemma thatSupp R ( M ⊗ R Kos R ( x )) = Supp R M ∩ Supp R ( R/I ) . In particular, if x generates a maximal ideal m of R with m ∈ Supp R M , thenSupp R ( M ⊗ R Kos R ( x )) = { m } . Lemma 2.4.2.
Let n be a nonzero integer and let M be in D f ( R ) . If α : M → Σ n M is a morphism in D ( R ) , then Supp R M = Supp R (cone( α )) . Proof.
Let C := cone( α ). We have an exact triangle M → Σ n M → C → in D ( R ). For each p ∈ Spec R , there is an exact triangle M p → Σ n M p → C p → in D ( R p ). It follows that Supp R C ⊆ Supp R M .If p / ∈ Supp R C, then M p ≃ Σ n M p in D ( R p ). Since M p ≃ Σ n M p , M p is in D f ( R p ),and n = 0, it follows that M p ≃ . Thus, p / ∈ Supp R M . (cid:3) Thick Subcategories.
Let T denote a triangulated category. A full subcate-gory T ′ of T is called thick if it is closed under suspension, has the two out of threeproperty on exact triangles, and is closed under direct summands. For an object X of T , define the thick closure of X in T , denoted Thick T X , to be the intersectionof all thick subcategories of T containing X . Since an intersection of thick sub-categories is a thick subcategory, Thick T X is the smallest thick subcategory of T containing X . See [5, Section 2] for an inductive construction of Thick T X and adiscussion of the related concept of levels . If Y is an object of Thick T X , then wesay that X finitely builds Y .2.5.1 . Let R be a commutative ring. Recall that a complex of R -modules M is perfect if it is quasi-isomorphic to a bounded complex of finitely generated projective R -modules. By [11, 3.7], Thick D ( R ) R consists exactly of the perfect complexes.2.5.2 . Let R be a commutative ring and let m be a maximal ideal of R . By [11, 3.10], Thick D ( R ) ( R/ m ) consists of all objects M of D f ( R ) such that Supp R M = { m } . . Let F : T → T ′ be an exact functor between triangulated categories withright adjoint exact functor G . Let ǫ : F G → id T ′ and η : id T → GF be the co-unitand unit transformations.The full subcategory of T consisting of all objects X such that the natural map η X : X → GF ( X ) is an isomorphism is a thick subcategory of T . For each X in T ,the composition F ( X ) F ( η X ) −−−−→ F GF ( X ) ǫ F ( X ) −−−−→ F ( X ) s an isomorphism. Therefore, if η X is an isomorphism in T then ǫ F ( X ) is anisomorphism in T ′ and F induces an equivalence of categories Thick T X ∼ = −→ Thick T ′ F ( X ) . Lemma 2.5.4.
Let ϕ : R → S be flat morphism of commutative rings. Suppose M is in D ( R ) and the natural map M → M ⊗ R S is an isomorphism in D ( R ) . Thenthe functor − ⊗ R S : D ( R ) → D ( S ) induces an equivalence of categories Thick D ( R ) M ∼ = −→ Thick D ( S ) ( M ⊗ R S ) . In particular, for each N in Thick D ( R ) M the natural map N → N ⊗ R S is anisomorphism in D ( R ) .Proof. The restriction of scalar functor G : D ( S ) → D ( R ) is a right adjoint to − ⊗ R S : D ( R ) → D ( S ). By assumption, the natural map M → G ( M ⊗ R S )is an isomorphism in D ( R ). Hence, (2.5.3) completes the proof. (cid:3) Support of Cohomology Graded Modules.
Let A = {A i } i ≥ be a co-homologically graded, commutative noetherian ring. Recall that Proj A denotesthe set of homogeneous prime ideals of A not containing A > := {A i } i> . Forhomogeneous elements a , . . . , a m ∈ A define V ( a , . . . , a m ) = { p ∈ Proj A : a i ∈ p for each i } . For a (cohomologically) graded A -module X , set Supp + A X := { p ∈ Proj A : X p = 0 } . The following properties of (cohomologically) graded A -modules follow easilyfrom the definition of support; see [6, 2.2] Proposition 2.6.1.
Let A = {A i } i ≥ be a cohomologically graded, commutativenoetherian ring.(1) Let X be a graded A -module and n ∈ Z . Then Supp + A X = Supp + A ( Σ n X ) . (2) Given an exact sequence of graded A -modules → X ′ → X → X ′′ → then Supp + A X = Supp + A X ′ ∪ Supp + A X ′′ . (3) If X is a finitely generated graded A -module, then Supp + A X = ∅ if and onlyif X ≫ = 0 . Cohomology Operators and Support Varieties
Fixed Notation.
Throughout this section, let Q be a commutative noether-ian ring. When Q is local, we will let n denote its maximal ideal and k its residuefield.Let I be an ideal of Q and fix a generating set f = f , . . . , f n for I . Set R := Q/I and E := Q h ξ , . . . , ξ n | ∂ξ i = f i i . The augmentation map E → R is a map of DG Q -algebras. Hence, we consider DG R -modules as DG E -modules via restriction ofscalars along E → R .Let S := Q [ χ , . . . , χ n ] be a graded polynomial ring where each χ i has cohomo-logical degree 2. When Q is local, set A := S ⊗ Q k = k [ χ , . . . , χ n ] . efine Γ to be the graded Q -linear dual of S , i.e., Γ is the graded Q -module withΓ i := Hom Q ( S i , Q ) . Let { y ( H ) } H ∈ N n be the Q -basis of Γ dual to { χ H := χ h . . . χ h n n } H ∈ N n the standard Q -basis of S . Then Γ is a graded S -module via the action χ i · y ( H ) := (cid:26) y ( h ,...,h i − ,h i − ,h i +1 ,...,h n ) h i ≥ h i = 03.2. Cohomology Operators.
Let M be a DG E -module. A semiprojective res-olution ǫ : P ≃ −→ M over Q such that P has the structure of a DG E -module and ǫ isa morphism of DG E -modules is called a Koszul resolution of M . A semiprojectiveresolution of M over E is a Koszul resolution of M , and hence Koszul resolutionsexist.Let ǫ : P ≃ −→ M be a Koszul resolution of M . Define U E ( P ) to be the DG E -module with U E ( P ) ♮ ∼ = ( E ⊗ Q Γ ⊗ Q P ) ♮ and differential given by the formula ∂ = ∂ E ⊗ ⊗ ⊗ ⊗ ∂ P + n X i =1 (1 ⊗ χ i ⊗ λ i − λ i ⊗ χ i ⊗ λ i denotes left multiplication by ξ i . By [4, 2.4], U E ( P ) → M is a semipro-jective resolution over E where the augmentation map is given by a ⊗ y ( H ) ⊗ x (cid:26) aǫ ( x ) | H | = 00 | H | > U E ( P ) has a DG S -module structure where S acts on U E ( P ) via itsaction on Γ. For a DG E -module N , Hom E ( U E ( P ) , N ) is a DG S -module andhence, Ext ∗ E ( M, N ) = H(Hom E ( U E ( P ) , N ))is a graded module over S . Remark . Let M and M ′ be DG E -modules and assume that α : M → M ′ is a morphism of DG E -modules. Let F and F ′ be semiprojective resolutions of M and M ′ over E , respectively. Since F is semiprojective over E , there exists amorphism of DG E -modules ˜ α : F → F ′ lifting α that is unique up to homotopy.Moreover, ˜ α induces a morphism of DG E -modules 1 ⊗ ⊗ ˜ α : U E ( F ) → U E ( F ′ )that is S -linear and unique up to homotopy.In particular, if F and F ′ are both semiprojective resolutions of a DG E -module M , then there exists a DG E -module homotopy equivalence U E ( F ) → U E ( F ′ )that is S -linear and unique up to homotopy. Thus, the S -module structures ofH(Hom E ( U E ( F ) , N )) and H(Hom E ( U E ( F ′ ) , N )) coincide when F and F ′ are bothsemiprojective resolutions of M over E . Proposition 3.2.2.
Let M and N be in D ( E ) . Then the S -module structure on Ext ∗ E ( M, N ) is independent of choice of Koszul resolution for M . Moreover, the S -module action on Ext ∗ E ( M, N ) is functorial in M and given an exact triangle M ′ → M → M ′′ → in D ( E ) , there exists an exact sequence of graded S -modules Σ − Ext ∗ E ( M ′ , N ) → Ext ∗ E ( M ′′ , N ) → Ext ∗ E ( M, N ) → Ext ∗ E ( M ′ , N ) . roof. Let P be a Koszul resolution of M and F a semiprojective resolution of M over E . There exists a morphism of DG E -modules ˜ α : F → P lifting the identityon M which is unique up to homotopy. This induces a DG E -module homotopyequivalence 1 ⊗ ⊗ ˜ α : U E ( F ) → U E ( P ) that is S -linear and unique up to homotopy.Thus, F and P determine the same S -module structure on Ext ∗ E ( M, N ). FromRemark 3.2.1, it follows that the S -module structure on Ext ∗ E ( M, N ) is independentof choice of Koszul resolution for M .Moreover, by Remark 3.2.1 the S -module structure on Ext ∗ E ( M, N ) is functorialin M . Thus, Ext ∗ E ( − , N ) sends exact triangles in D ( E ) to exact sequences of graded S -modules. (cid:3) . Assume that ( Q, n , k ) is a local ring and recall that A = S ⊗ Q k . Let M be in D ( E ). The S -action on Ext ∗ E ( M, k ) factors through
S → A , and hence,Ext ∗ E ( M, k ) is a graded A -module. Therefore, by Proposition 3.2.2, for any exacttriangle M ′ → M → M ′′ → in D ( E ), we get an exact sequence of graded A -modules Σ − Ext ∗ E ( M ′ , k ) → Ext ∗ E ( M ′′ , k ) → Ext ∗ E ( M, k ) → Ext ∗ E ( M ′ , k ) . Lemma 3.2.4.
Assume that ( Q, n , k ) is a local ring and M is in D ( E ) . For any x ∈ n , there exists an exact sequence of graded A -modules → Σ − Ext ∗ E ( M, k ) → Ext ∗ E ( M ⊗ Q Kos Q ( x ) , k ) → Ext ∗ E ( M, k ) → . Proof.
By (3.2.3), applying Ext ∗ E ( − , k ) to the exact triangle M → M → M ⊗ Q Kos Q ( x ) → in D ( E ) gives us an exact sequences of graded A -modules Σ − Ext ∗ E ( M, k ) → Ext ∗ E ( M ⊗ Q Kos Q ( x ) , k ) → Ext ∗ E ( M, k ) x · −→ Ext ∗ E ( M, k ) . Since x is in n , we obtain the desired result. (cid:3) Proposition 3.2.5.
Assume that ( Q, n , k ) is a regular local ring. For each M in D f ( E ) , Ext ∗ E ( M, k ) is a finitely generated graded A -module.Proof. As H( M ) is finitely generated over Q and Q is regular, there exists a Koszulresolution P ≃ −→ M such that P is a bounded complex of finitely generated free Q -modules (see [4, 2.1]). Also, we have an isomorphism of graded A -modulesHom E ( U E ( P ) , k ) ♮ ∼ = A ⊗ k Hom Q ( P, k ) ♮ . Thus, Hom E ( U E ( P ) , k ) is a noetherian graded A -module. As A is a noetheriangraded ring and Ext ∗ E ( M, k ) is a graded subquotient of Hom E ( U E ( P ) , k ), it followsthat Ext ∗ E ( M, k ) is a noetherian graded A -module. (cid:3) Remark . Suppose the local ring ( Q, n , k ) is regular. By (2.2.1), K Q is a DG E -module. Assume that I ⊆ n . Left multiplication by ξ i on K Q is zero modulo n .Thus, we have an isomorphism of DG A -modulesHom E ( U E ( K Q ) , k ) ∼ = A ⊗ k Hom Q ( K Q , k ) , where both DG A -modules have trivial differential (see (2.2.1)). Therefore, thereis an isomorphism of graded A -modulesExt ∗ E ( k, k ) ∼ = A ⊗ k Hom Q ( K Q , k ) . In particular,
Supp + A (Ext ∗ E ( k, k )) = Proj A . .3. Support Varieties.
For the rest of the section, further assume that ( Q, n , k )is a regular local ring, f minimally generates I , and I ⊆ n . Recall that A = S ⊗ Q k = k [ χ , . . . , χ n ] . By Proposition 3.2.5, Ext ∗ E ( M, k ) is a finitely generated graded A -module foreach M in D f ( E ). This leads to the following definition which recovers the supportvarieties of Avramov in [3] in the case that f is a Q -regular sequence. The varieties,defined below, are investigated and further developed in [17]. Definition 3.3.1.
Let M be in D f ( E ). Define the support variety of M over E tobe V E ( M ) := Supp + A (Ext ∗ E ( M, k )) . Theorem 3.3.2.
With the assumptions above, the following hold.(1) Let M and N be in D f ( E ) . If N is in Thick D ( E ) M , then V E ( N ) ⊆ V E ( M ) .(2) For any M in D f ( E ) , V E ( M ) = V E ( M ⊗ Q K Q ) . (3) f is a regular Q -sequence if and only if V E ( R ) = ∅ .Proof. Using (3.2.3) and Proposition 2.6.1, it follows that the full subcategory of D f ( E ) consisting of objects L such that V E ( L ) ⊆ V E ( M ) is a thick subcategory of D f ( E ). Therefore, (1) holds.Iteratively applying Lemma 3.2.4 and Proposition 2.6.1(2), establishes (2).For (3), first assume that f is a Q -regular sequence. Hence, the augmentationmap E → R is a quasi-isomorphism. Therefore, (2.3.1) yields an isomorphismExt ∗ E ( R, k ) ∼ = Ext ∗ R ( R, k ) = k. Thus, V E ( R ) = Supp + A k = ∅ . Conversely, assume that V E ( R ) = ∅ . Hence, by Proposition 3.2.5 and Proposition2.6.1(3),(1) Ext ≫ E ( R, k ) = 0 . Next, let g = g , . . . , g n be a minimal generating set for I such that g ′ = g , . . . , g c is a maximal Q -regular sequence in I for some c ≤ g . Set Q := Q/ ( g ′ ), g to be theimage of g c +1 , . . . , g n in Q , and E := Kos Q ( g ) . Since g ′ is a Q -regular sequence,we have a quasi-isomorphism of DG Q -algebras E ≃ −→ E. Hence, (2.3.1) yields anisomorphism of graded k -vector spacesExt ∗ E ( R, k ) ∼ = Ext ∗ E ( R, k ) . In particular, Ext ≫ E ( R, k ) = 0 by (1). Therefore, pd Q R < ∞ (c.f. [7, B.10]). Since R = Q/IQ where IQ contains no Q -regular element, it follows that IQ = 0 (see [9,1.4.7] ). Thus, g = g ′ , that is, I is generated by a Q -regular sequence. Therefore,by [9, 1.6.19], f is Q -regular sequence. (cid:3) Remark . In [17], a different argument is used to establish Theorem 3.3.2(c).In fact, the following is shown: f is a Q -regular sequence if and only if V E ( M ) = ∅ for some nonzero finitely generated R -module M . Theorem 3.3.4.
Assume ( Q, n , k ) is a regular local ring. Let R = Q/I where I is minimally generated by f = f , . . . , f n ∈ n . Let E be the Koszul complex on over Q and set A = k [ χ , . . . , χ n ] . For each homogeneous element g ∈ A , thereexists a complex of R -modules C ( g ) in Thick D ( R ) k such that V E ( C ( g )) = V ( g ) . Proof. As Q is regular, the Koszul complex K Q is a free resolution of k over Q .Moreover, (2.2.1) says that K Q is a Koszul resolution of k . By (3.2), there exists asemiprojective resolution ǫ : U ≃ −→ k over E where U := U E ( K Q ). Let d denote thedegree of g . Define ˜ C ( g ) := cone( U g · −→ Σ d U ) . The same proof of [6, 3.10] and Remark 3.2.6 yields(2) V E ( ˜ C ( g )) = V ( g ) . Fix a projective resolution δ : P ≃ −→ k over R . Since U is a semiprojective DG E -module there exists a morphism of DG E -modules α : U → P such that δα = ǫ. Note that α is a quasi-isomorphism.By Theorem 2.3.2 and (2.1.5), there exists a morphism of complexes of R -modules γ : P → Σ d k such that(3) U Σ d UP Σ d k ≃ α g · Σ d ǫ ≃ γ is a diagram of DG E -modules that commutes up to homotopy. Define C ( g ) := cone( γ ) . Since P ≃ k and γ is a morphism of complexes of R -modules, it follows that C ( g )is in Thick D ( R ) k . Also, as α are Σ d ǫ quasi-isomorphisms and (3) commutes up tohomotopy, we get an isomorphism C ( g ) ≃ ˜ C ( g )in D ( E ) . Therefore, Equation (2) yieldsV E ( C ( g )) = V E ( ˜ C ( g )) = V ( g ) . (cid:3) Virtually Small Complexes
Let R be a commutative noetherian ring. A complex of R -modules M is virtuallysmall if M ≃ P in Thick D ( R ) M ∩ Thick D ( R ) R. If in addition P can be chosen with Supp R M = Supp R P , we say M is proxy small .These notions were introduced by Dwyer, Greenlees, and Iyengar in [10] and [11],where the authors apply methods from commutative algebra to homotopy theoryand vice versa. Remark . In [10] and [11], the objects of
Thick D ( R ) R are called the small objectsof D ( R ). With this terminology, the nontrivial virtually small objects of D ( R ) arethe complexes that finitely build a nontrivial small object.4.2 . A nontrivial object M of D f ( R ) is virtually small if and only if there exists amaximal ideal m = ( x ) of R such that Kos R ( x ) is in Thick D ( R ) M . In particular, if R is local, a nontrivial complex M in D f ( R ) is virtually small if and only if K R isin Thick D ( R ) M . This was observed in [11, 4.5], and is a consequence of a theoremof M. Hopkins [14] and Neeman [16]. s a matter of notation, let VS ( R ) to be the full subcategory of D f ( R ) consistingof all virtually small complexes. In the following lemma, the argument for “(1)implies (2)” is abstracted from the proof of [11, 9.4]. Lemma 4.3.
Let R be a commutative noetherian ring. The following are equiva-lent:(1) Thick D ( R ) ( R/ m ) is a subcategory of VS ( R ) for each maximal ideal m of R .(2) D f ( R ) = VS ( R ) .(3) VS ( R ) is a thick subcategory of D ( R ) .Proof. (1) = ⇒ (2): Let M be a nontrivial object of D f ( R ). Since M is nontrivial,there exists a maximal ideal m in Supp R M . Let x generate m and set N := M ⊗ R Kos R ( x ) . By (2.4.1), Supp R N = { m } and hence, N is in Thick D ( R ) ( R/ m ) (see (2.5.2)). Byassumption, there exists a nontrivial object P in Thick D ( R ) N ∩ Thick D ( R ) R . Finally,since N is in Thick D ( R ) M , Thick D ( R ) N is a subcategory of Thick D ( R ) M. Thus, P is in Thick D ( R ) M . That is, M is virtually small.(2) = ⇒ (3): Whenever R is noetherian, D f ( R ) is a thick subcategory of D ( R ).(3) = ⇒ (1): Let m be a maximal ideal of R and suppose x generates m .By (2.5.2), Kos R ( x ) is in Thick D ( R ) ( R/ m ). Thus, R/ m is in VS ( R ). Since VS ( R )is a thick subcategory of D ( R ), it follows that Thick D ( R ) ( R/ m ) is contained in VS ( R ). (cid:3) Lemma 4.4.
Let ϕ : R → S be a flat morphism of commutative noetherian rings.Suppose m is a maximal ideal of R such that m S is a maximal ideal of S andthe canonical map R/ m → S/ m S is an isomorphism. Then Thick D ( R ) ( R/ m ) is asubcategory of VS ( R ) if and only if Thick D ( S ) ( S/ m S ) is a subcategory of VS ( S ) . Proof.
Set K := Kos R ( x ) where x generates m . Let x ′ denote the image of x under ϕ and set K ′ := Kos S ( x ′ ) . Hence, we have an isomorphism of DG S -algebras K ′ ∼ = K ⊗ R S .Assume Thick D ( R ) ( R/ m ) is a subcategory of VS ( R ). Let N be a nontrivial objectof Thick D ( S ) ( S/ m S ). By Lemma 2.5.4, there exists a nontrivial complex M in Thick D ( R ) ( R/ m ) such that M ⊗ R S ≃ N in D ( S ). By assumption and (4.2), K isin Thick D ( R ) M . Hence, K ⊗ R S is in Thick D ( S ) ( M ⊗ R S ) . Since K ′ ∼ = K ⊗ R S and N ≃ M ⊗ R S , we conclude that K ′ is in Thick D ( S ) N . Thus, N is in VS ( S ) . Let M be a nontrivial object of Thick D ( R ) ( R/ m ). Thus, M ⊗ R S is a nontrivialobject of Thick D ( S ) ( S/ m S ). By assumption and (4.2), K ′ is in Thick D ( S ) ( M ⊗ R S ) . Therefore,(4) K ′ ∈ Thick D ( R ) ( M ⊗ R S ) . Since the natural map R/ m → S/ m S is an isomorphism and K and M are in Thick D ( R ) ( R/ m ), by applying Lemma 2.5.4 we obtain the following isomorphismsin D ( R ) K ≃ −→ K ⊗ R S ∼ = K ′ and M ≃ −→ M ⊗ R S. These isomorphisms and (4) imply that K is in Thick D ( R ) M . That is, M is in VS ( R ). (cid:3) Proposition 4.5.
Let R be a commutative noetherian ring.
1) Then D f ( R ) = VS ( R ) if and only if D f ( R m ) = VS ( R m ) for every maximalideal m of R .(2) In addition, assume ( R, m , k ) is local and let b R denote its m -adic comple-tion. Then D f ( R ) = VS ( R ) if and only if D f ( b R ) = VS ( b R ) . Proof.
By Lemma 4.3, D f ( R ) = VS ( R ) if and only if Thick D ( R ) ( R/ m ) is a subcate-gory of VS ( R ) for each maximal ideal of m of R . By Lemma 4.4, the latter holds ifand only if Thick D ( R m ) ( κ ( m )) is a subcategory of VS ( R m ) for each maximal ideal m of R where κ ( m ) = R m / m R m . Equivalently, D f ( R m ) = VS ( R m ) for each maximalideal m of R by Lemma 4.3. Thus, (1) holds.Next, Lemma 4.4 yields that Thick D ( R ) k is a subcategory of VS ( R ) if and onlyif Thick D ( b R ) k is a subcategory of VS ( b R ). Applying Lemma 4.3, finishes the proofof (2). (cid:3) The Main Results
Let ( R, m ) be a commutative noetherian local ring and let b R denote its m -adiccompletion. The local ring R is said to be a complete intersection provided b R ∼ = Q/ ( f , . . . , f c )where Q is a regular local ring and f , . . . , f c is a Q -regular sequence. In [11, 9.4],the following was established: if R is a complete intersection every object of D f ( R ) virtually small. If in addition R is a quotient of a regular local ring, every objectof D f ( R ) is proxy small . Moreover, the authors posed the following question: Question 5.1. [11, 9.4]
If every object of D f ( R ) is virtually small, is R a completeintersection? Theorem 5.2, below, answers Question 5.1 in the affirmative. Much of the workin establishing “(1) implies (3)” is done in the proof of a theorem of Bergh [8, 3.2].The theory of support varieties developed in Section 3.3 is the key ingredient usedto prove “(2) implies (1).”
Theorem 5.2.
Let R be a commutative noetherian local ring. The following areequivalent.(1) R is a complete intersection.(2) Every object of D f ( R ) is virtually small.(3) Every object of D f ( R ) is proxy small.Proof. (1) = ⇒ (3): Let M be in D f ( R ). In the proof of [8, 3.2], it is shown thereexist positive integers n , . . . , n t and exact triangles in D ( R ) M → Σ n M → M (1) → M (1) → Σ n M (1) → M (2) → ... ... ... ... M ( t − → Σ n t M ( t − → M ( t ) → such that M ( t ) is in Thick D ( R ) R . Also, it is clear that M ( t ) is in Thick D ( R ) M .Since each n i = 0, Lemma 2.4.2 yieldsSupp R M = Supp R ( M (1)) = . . . = Supp R ( M ( t )) . hus, M is proxy small.(3) = ⇒ (2): Clear from the definitions.(2) = ⇒ (1): By Proposition 4.5(2), we may assume that R is complete. Write R = Q/I where ( Q, n , k ) is a regular local ring. Assume I is minimally generatedby f = f , . . . , f n ∈ n and let E be the Koszul complex on f .Fix 1 ≤ i ≤ n . By Theorem 3.3.4, there exists C ( i ) in Thick D ( R ) k withV E ( C ( i )) = V ( χ i ) . By assumption, each C ( i ) is virtually small. Therefore, (4.2) implies that K R is in Thick D ( R ) C ( i ). Hence, V E ( K R ) ⊆ V E ( C ( i )) = V ( χ i )by Theorem 3.3.2(1). Applying Theorem 3.3.2(2) with M = R yieldsV E ( R ) = V E ( K R ) , and hence, V E ( R ) ⊆ V ( χ i ).Therefore, V E ( R ) ⊆ V ( χ ) ∩ . . . ∩ V ( χ n ) . That is, V E ( R ) = ∅ and so by Theorem 3.3.2(3), f is a Q -regular sequence. Thus, R is a complete intersection. (cid:3) This structural characterization of a complete intersection’s derived categoryyields the following corollary which was first established by Avramov in [1].
Corollary 5.3.
Assume a commutative noetherian local ring R is a complete in-tersection. For any p ∈ Spec R , R p is is a complete intersection.Proof. For any p ∈ Spec R , the functor − ⊗ R R p : D f ( R ) → D f ( R p ) is essentiallysurjective. Also, the property of proxy smallness localizes. These observations andTheorem 5.2 complete the proof. (cid:3) Let R be a commutative noetherian ring. We say that R is locally a completeintersection if R p is a complete intersection for each p ∈ Spec R . By Corollary 5.3, R is locally a complete intersection if and only if R m is a complete intersection forevery maximal ideal m of R . We obtain the following homotopical characterizationof rings that are locally complete intersections. Theorem 5.4.
A commutative noetherian ring R is locally a complete intersectionif and only if every object of D f ( R ) is virtually small.Proof. As remarked above, R is locally a complete intersection if and only if R m isa complete intersection for each maximal ideal m of R . By Theorem 5.2, the latterholds if and only if D f ( R m ) = VS ( R m ) for each maximal ideal m of R . Equivalently, D f ( R ) = VS ( R ) by Proposition 4.5(1). (cid:3) References [1] L. L. Avramov,
Flat morphisms of complete intersections , Soviet Math. Dokl. (1975),1413–1417[2] L. L. Avramov, Infinite free resolutions , Six Lectures on Commutative Algebra (Bellaterra,1996), Progr. Math. , Birkh¨auser, Basel, 1998; pp. 1–118[3] L. L. Avramov,
Modules of finite virtual projective dimension , Inventiones mathematicae (1989), 71–101
4] L. L. Avramov, R.-O. Buchweitz,
Homological algebra modulo a regular sequence with specialattention to codimension two , Journal of Algebra (2000), 24–67[5] L. L. Avramov, R.-O. Buchweitz, S. Iyengar, C. Miller,
Homology of perfect complexes , Ad-vances in Mathematics (2010), 1731–1781[6] L. L. Avramov, S. Iyengar,
Constructing modules with prescribed cohomological support ,Illinois Journal of Mathematics (2007), 1–20[7] L. L. Avramov, S. Iyengar, S. Nasseh, S. Sather-Wagstaff,
Homology over trivial extensionsof commutative dg algebras , (2015), arXiv:1508.00748v1 [math.AC][8] P. A. Bergh,
On complexes of finite complete intersection dimension , Homology, Homotopy,and Applications (2009), 49–54[9] W. Bruns and J. Herzog,
Cohen-Macaulay rings , volume 39 of Cambridge Studies in Ad-vanced Mathematics. Cambridge University Press, Cambridge, 1993[10] W. G. Dwyer, J. P. C. Greenlees, S. Iyengar,
Duality in algebra and topology , Advances inMathematics (2006): 357–402[11] W. G. Dwyer, J. P. C. Greenlees, S. Iyengar,
Finiteness in derived categories of local rings ,Commentarii Mathematici Helvetici (2006), 383–432[12] Y. Felix, S. Halperin, J. C. Thomas,
Rational homotopy theory , Graduate Texts Math. ,Springer-Verlag, New York, 2001[13] T. H. Gulliksen, G. Levin,
Homology of Local Rings , Queen’s Papers Pure Appl. Math. ,Queen’s Univ., Kingston, ON, 1969[14] M. Hopkins, Global methods in homotopy theory, in: Homotopy theory (Durham, 1985),London Math. Soc. Lect. Note Ser. , Cambridge Univ. Press, Cambridge, (1987), 73–96[15] B. Keller,
Deriving DG categories , Ann. Sci.´Ecole Norm. Sup.(4), (1994), 63-102[16] A. Neeman, The chromatic tower of D(R) , Topology (1992), 519–532.[17] J. Pollitz, Support varieties over Koszul complexes , (in preparation)[18] J. Tate,
Homology of Noetherian rings and local rings , Illinois Journal of Mathematics (1957), 14–27 Department of Mathematics, University of Nebraska, Lincoln, NE 68588, U.S.A.
E-mail address : [email protected]@huskers.unl.edu