A partial converse ghost lemma for the derived category of a commutative noetherian ring
aa r X i v : . [ m a t h . A C ] J a n A PARTIAL CONVERSE GHOST LEMMA FOR THE DERIVEDCATEGORY OF A COMMUTATIVE NOETHERIAN RING
JIAN LIU AND JOSH POLLITZ
Abstract.
In this article a condition is given to detect the containment amongthick subcategories of the bounded derived category of a commutative noether-ian ring. More precisely, for a commutative noetherian ring R and complexesof R -modules with finitely generated homology M and N , we show N is in thethick subcategory generated by M if and only if the ghost index of N p withrespect to M p is finite for each prime p of R . To do so, we establish a “conversecoghost lemma” for the bounded derived category of a non-negatively gradedDG algebra with noetherian homology. Introduction
This article is concerned with certain numerical invariants and thick subcate-gories in the bounded derived category of a commutative noetherian ring. Let R bea commutative noetherian ring, D ( R ) will denote its derived category, and D fb ( R )will be the full subcategory of D ( R ) consisting of objects with finitely generatedhomology.An object N of D ( R ) is in the thick subcategory generated by M , denoted thick D ( R ) ( M ), provided that N can be inductively built from M using the triangu-lated structure of D ( R ) (see 2.1 for more details). There are cases where a notionof support reports on whether N is in thick D ( R ) ( M ). For example, there is thecelebrated theorem of Hopkins [13, Theorem 11] and Neeman [19, Theorem 1.5]that applies when M and N are perfect complexes. Another instance is when R islocally complete intersection by using support varieties; this was proved by Steven-son for thick subcategories containing R when R is a quotient of a regular ring [22,Corollary 10.5], and in general in [18, Theorem 3.1]. However, in general, detectingcontainment among thick subcategories can be an intractable task.In this article, we give a new criterion to determine the containment among thicksubcategories of D fb ( R ) based on certain numerical invariants being locally finite.We quickly define these below; see 2.1, 2.3, and 2.8 for precise definitions.For a triangulated category T , fix objects G and X . The level of X with respectto G counts the minimal number of cones needed to generate X , up to suspensionsand direct summand, starting from G . We denote this by level G T ( X ) and note thatthis is finite exactly when X is in thick T ( G ). The coghost index of X with respectto G , denoted cogin G T ( X ), is the minimal number n satisfying that any composition X n f n −−→ X n − → . . . f −→ X = X, Mathematics Subject Classification.
Key words and phrases. derived category, (co)ghost index, (co)ghost lemma, level, DG algebra,Koszul complex. where each Hom T ( f i , Σ j G ) = 0, must be zero in T . Switching the variance inthe definition above determines the ghost index of X with respect to G , denoted gin G T ( X ) . These invariants are of independent interest and have been studied in [1, 2, 5, 7,8, 10, 16, 17, 21]. In general, they are related by the following well-known (co)ghostlemma : max { cogin G T ( X ) , gin G T ( X ) } ≤ level G T ( X ) . Oppermann and ˇS´tov´ıˇcek proved a so-called converse coghost lemma:
Namely, cogin M D fb ( R ) ( N ) and level M D fb ( R ) ( N ) agree whenever M and N are objects of D fb ( R ),see [20, Theorem 24]. In general, it remains open whether the analogous equalityholds for objects of D fb ( R ) when cogin is replaced by gin ; furthermore, [17, 2.11]notes that the techniques used in [20, Theorem 24] cannot be suitably adapted toprove this.In this article we ask whether finiteness of certain ghost indices can determinefiniteness of level , and hence containment among thick subcategories. The mainresult in this direction is the following which is contained in Theorem 3.1. Theorem A.
Let R be a commutative noetherian ring. For M and N in D f ( R ) , the following are equivalent:(1) level M D fb ( R ) ( N ) < ∞ ;(2) gin M p D fb ( R p ) ( N p ) < ∞ for all prime ideals p of R . One of the main steps in the proof of Theorem A is establishing a conversecoghost lemma for graded-commutative, bounded below DG algebras A with H( A )a noetherian H ( A )-module (cf. Theorem 2.5). We follow the proof of [20, Theo-rem 24] closely, however, extra care is needed when working with such DG algebras.Namely, we make use of certain ascending semifree filtrations, see 1.6, as the trun-cations used by Oppermann and ˇS´tov´ıˇcek are no longer available in this setting. Acknowledgements.
We thank Srikanth Iyengar and Janina Letz for helpful dis-cussions. The second author was supported by NSF grant DMS 2002173.1.
Derived Category of a DG Algebra and Semifree DG Modules
Much of this section is devoted to setting notation and reviewing the necessarybackground regarding the topics from the title of the section. Proposition 1.8 is themain technical result of the section and will be put to use in the next section.Throughout this article objects will be graded homologically. By a DG algebrawe will implicitly assume A is non-negatively graded and graded-commutative. Forthe rest of the section fix a DG algebra A . Let D ( A ) denote the derived category of (left) DG A -modules (see, for example,[2, Sections 2 & 3] or [15, Section 4]). We use Σ to denote the suspension functoron the triangulated category D ( A ) where Σ is the autoequivalence defined by( Σ M ) i := M i − , a · ( Σ m ) := ( − | a | am, and ∂ Σ M := − ∂ M . For a DG A -module X, its homology H( M ) = { H i ( M ) } i ∈ Z is naturally a gradedH( A )-module. Also, define the infimum of X to be inf( M ) := inf { n | H n ( X ) = 0 } ;its supremum is sup( M ) := sup { n | H n ( M ) = 0 } . PARTIAL CONVERSE GHOST LEMMA 3
The following triangulated subcategories of D ( A ) will be of interest in thesequel. First, let D f ( A ) denote the full subcategory of D ( A ) consisting of DG A -modules M such that each H i ( M ) is a noetherian H ( A )-module. We let D f + ( A )be the full subcategory of objects M of D f ( A ) such that inf( M ) > −∞ . Finally, D fb ( A ) consists of those objects M of D f ( A ) satisfying H i ( M ) = 0 for all | i | ≫ A ) is noetherian as a module over H ( A ) and H ( A ) is noetherian, D fb ( A )is exactly the full subcategory of D ( A ) whose objects M are those with H( M ) beingfinitely generated as a graded H( A )-module. A DG A -module F is semifree if it admits a filtration of DG A -submodules . . . ⊆ F ( − ⊆ F (0) ⊆ F (1) ⊆ . . . where F ( i ) = 0 for i ≪ F = ∪ F ( i ) and each F ( i ) /F ( i −
1) is a direct sum ofshifts of A . The filtration above is called a semifree filtration of F . By [15, Section3], F is homotopy colimit of the F ( i ) and so there is the following exact triangle in D ( A )(1) a i ∈ Z F ( i ) − s −−→ a i ∈ Z F ( i ) → F → Σ a i ∈ Z F ( i ) , where s is induced by the canonical inclusions F ( j ) ֒ → F ( j + 1) ֒ → ` F ( i ). For the following background on semifree resolutions see [11, Chapter 6] (or[4, Section 1.3]). Let M be a DG A -module. There exists a surjective quasi-isomorphism of DG A -modules ǫ : F ≃ −→ M where F is a semifree DG A -module,see [11, Proposition 6.6]; the map ǫ is called a semifree resolution of M over A .Semifree resolutions of M are unique up to homotopy equivalence. Fix a DG A -module M with semifree resolution ǫ : F ≃ −→ M. For any DG A -module N , it is clear thatHom D ( A ) ( M, N ) = Hom D ( A ) ( F, N )and the right-hand side is computed as the degree zero homology of the DG A -module Hom A ( F, N ) . That is,(2) Hom D ( A ) ( M, − ) = H (Hom A ( F, − )) . In particular, Hom D ( A ) ( M, N ) naturally inherits an H ( A )-module structure andsince A is non-negatively graded, Hom D ( A ) ( M, N ) inherits an A -module structure.As semifree resolutions are homotopy equivalent, this H ( A )-module is independentof choice of semifree resolution. Assume each H i ( A ) is finitely generated over H ( A ) and H ( A ) is itself noe-therian. Let M be an object of D f + ( A ). By [3, Appendix B.2], there exists a semifreeresolution F ≃ −→ M with F i = 0 for all i < inf( M ) and F admits a semifree filtration { F ( i ) } i ∈ Z equipped with exact sequences of DG A -modules0 → F ( i − → F ( i ) → Σ i A β i → β i ≥ The choice to allow arbitrary indices for the start of the filtration is a non-standard one butsimplifies notation in the proof of Theorem 2.5.
JIAN LIU AND JOSH POLLITZ
Lemma 1.7.
Assume H ( A ) is noetherian and that each H i ( A ) is finitely generatedover it. Let N be an object of D ( A ) such that sup( N ) < ∞ . For an object M in D ( A ) with inf( M ) > sup( N ) , Hom D ( A ) ( M, N ) = 0 . Proof.
Fix a semifree resolution F ≃ −→ M as in 1.6. By (2) in 1.5,Hom D ( A ) ( Σ i A β i , N ) ∼ = H i ( N ) β i = 0for each i ≥ inf( M ) . Combining these isomorphisms with the exact sequences0 → F ( i − → F ( i ) → Σ i A β i → D ( A ) ( F ( i ) , N ) = 0 for all i ≥ inf( M ). Finally, (1) in1.3 implies Hom D ( A ) ( F, N ) = 0, and hence Hom D ( A ) ( M, N ) = 0 (cf. 1.5). (cid:3)
Proposition 1.8.
Assume H ( A ) is noetherian and each H i ( A ) is a finitely gen-erated H ( A ) -module. Let M be in D f + ( A ) and N be an object in D ( A ) such that sup( N ) < ∞ . Suppose F ≃ −→ M is a semifree resolution of M as in 1.6, then forall i > sup( N ) the natural map below is an isomorphism Hom D ( A ) ( M, N ) ∼ = −→ Hom D ( A ) ( F ( i ) , N ) . Proof.
For each i ≥ inf( M ), there is an exact sequence of DG A -modules(3) 0 → F ( i ) → F → F ′ → F we have that inf( F ′ ) > i . Applying Hom D ( A ) ( − , N ) to (3) andappealing to Lemma 1.7 yields the desired isomorphisms whenever i > sup( N ) . (cid:3) Levels and Coghost Index in D ( A )We begin by briefly recalling the notion of level. For more details, see [2, Section2], [8, Section 2] or [21, Section 3]. Let T be a triangulated category and C be a full subcategory of T . We say C is thick if it is closed under suspensions, retracts and cones. The smallest thicksubcategory of T containing an object X is denoted thick T ( X ); this consists of allobjects Y such that one can obtain Y from X using finitely many suspensions,retracts and cones.We set level X T ( Y ) to be smallest non-negative integer n such that Y can be builtstarting from X using finitely many suspensions, finitely many retracts and exactly n − T . If no such n exists, we set level X T ( Y ) = ∞ . Note Y is in thick T ( X )if and only if level X T ( Y ) < ∞ . Also, if C is a thick subcategory of T containing X ,then level X T ( Y ) = level X C ( Y ) . Example 2.2.
Let A be a DG algbera. A DG A -module M is perfect if M is anobject of thick D ( A ) ( A ) . In this case, M is a retract of a semifree DG A -module F with finite semifree filtration0 ⊆ F (0) ⊆ F (1) ⊆ . . . ⊆ F ( n ) = F. If n is the minimal such value, then level A D ( A ) ( M ) = n −
1, see [2, Theorem 4.2].
Let T be a triangulated category with suspension functor Σ . A morphism f : X → Y in T is called G - coghost ifHom T ( f, Σ i G ) : Hom T ( Y, Σ i G ) → Hom T ( X, Σ i G ) PARTIAL CONVERSE GHOST LEMMA 5 is zero for all i ∈ Z . Following [17, Defnition 2.4], we define the coghost index of X with respect to G in T , denoted cogin GT ( X ), to be the smallest non-negative integer n such that any composition of G -ghost maps X n f n −−→ X n − f n − −−−→ . . . → X f −→ X = X is zero in T . Let T be a triangulated category with objects G and X . In this generality, level bounds cogin from above. That is, cogin G T ( X ) ≤ level G T ( X ) , see [5, Lemma 2.2(1)] (see also [21, Lemma 4.11]). However, there are known in-stances when equality holds. For example, level G T ( − ) and cogin G T ( − ) agree providedevery object in T has an appropriate left approximation by G , see [5, Lemma 2.2(2)]for more details. Another instance is when R is a commutative noetherian ring (ormore generally, a noether algebra) cogin G D fb ( R ) ( X ) = level G D fb ( R ) ( X )for each G and X in D fb ( R ); this has been coined the converse coghost lemma (see[20, Theorem 24]).We now get to the main result of the section which generalizes a particular caseof [20, Theorem 24] mentioned above. It is worth noting that [20, Theorem 24]was proved for derived categories satisfying certain finiteness conditions; however,it does not apply directly to the case considered in the theorem below. The proofof [20, Theorem 24] is suitably adapted to the setting under consideration with themain observation being that truncations need to be replaced with the ascendingfiltrations discussed in 1.6. We have indicated the necessary changes below, whileattempting to not recast the parts of the proof of [20, Theorem 24] that carry overwith only minor changes. Theorem 2.5.
Let A be a DG algebra with H( A ) noetherian as an H ( A ) -module.If M and N are in D fb ( A ) , then cogin M D fb ( A ) ( N ) = level M D ( A ) ( N ) . Remark 2.6.
For M and N in D fb ( R ), cogin M D f + ( A ) ( N ) = level M D f + ( A ) ( N ) . Indeed, one can directly apply the argument from [20, Theorem 24] once it is notedthat, by restricting scalars along the map of commutative rings A → H ( R ),Hom D f + ( A ) ( X, Σ i N ) is finitely generated over A for X in D f + ( A ) and i ∈ Z .To see the latter holds, such an X admits a semifree filtration whose subquotientsare perfect DG A -module. Also, since N is in D fb ( A ) we can apply Proposition 1.8to get Hom D f + ( A ) ( X, Σ i N ) ∼ = Hom D f + ( A ) ( P, Σ i N )where P a perfect DG A -module with a finite semifree filtration as in Example 2.2.Therefore, induction on the length of this filtration finishes the proof of the claim,where we are again using that N is in D fb ( A ). JIAN LIU AND JOSH POLLITZ
Before beginning the proof of Theorem 2.5, we record an easy but importantlemma.
Lemma 2.7.
Let A be a DG algebra. Assume α : F → F is a morphism ofbounded below semifree DG A -modules with F ji = 0 for i < inf( F j ) and semifreefiltrations { F j ( i ) } i ∈ Z for j = 1 , satisfying → F j ( i − → F j ( i ) → a ℓ ≤ i Σ ℓ A β jℓ ( i ) → for non-negative integers β jℓ ( i ) and j = 1 , . For each i ∈ Z , α restricts to amorphism of DG A -modules α ( i ) : F ( i ) → F ( i ) . Proof.
Indeed, F ( i ) = 0 for all i < inf( F ) and so there is nothing to show for suchvalues of i . Now for i ≥ inf( F ), the DG A -module F ( i ) is generated in degreesat most i and since α is degree preserving α ( F ( i )) is generated in degrees at most i. However, the assumption on the filtration { F ( j ) } also implies α ( F ( i )) ⊆ F ( i ) . Hence, setting α ( i ) := α | F ( i ) proves the claim by induction. (cid:3) Proof of Theorem 2.5.
First, by 2.1 and Remark 2.6 level M D ( A ) ( N ) = level M D f + ( A ) ( N ) = cogin M D f + ( A ) ( N ) , while the inequality(4) cogin M D f + ( A ) ( N ) ≥ cogin M D fb ( A ) ( N )is standard. So it suffices to prove the reverse inequality of (4) holds.Set n = cogin M D fb ( A ) ( N ) and consider a composition N n f n −−→ N n − f n − −−−→ . . . f −→ N f −→ N = N, where each f i is a M -coghost map in D f + ( A ). Using the assumptions on H( A )and that each N i is in D f + ( A ), there exist semifree resolution F i ≃ −→ N i withcorresponding semifree filtrations { F i ( j ) } j ∈ Z as in 1.6. Moreover, by 1.5(2), each f i determines a morphism of DG A -modules α i : F i → F i − such that the followingdiagram commutes in D ( A )(5) F i F i − N i N i − . α i ≃ ≃ f i Furthermore, by Lemma 2.7 there are the following commutative diagrams of DG A -modules(6) F i ( j ) F i − ( j ) F i − ( j ′ ) F i F i − α i ( j ) α i whenever j ′ ≥ j. Moreover, since each F i ( j ) is a perfect DG A -module and M isin D fb ( A ), the commutativity of the diagrams in (5) and the assumption that each PARTIAL CONVERSE GHOST LEMMA 7 f i is M -coghost imply the compositions along the top of (6) are M -coghost for all j ′ ≥ j ≫
0; the same argument as in proof of [20, Theorem 24] works in this setting.Combining this with Proposition 1.8 there exists integers i j such that F n ( i n ) F n − ( i n − ) . . . F ( i ) F n F n − . . . F N n N n − . . . N β n β n − β α n ≃ α n − ≃ α ≃ f n f n − f commutes in D ( A ), the natural map(7) Hom D f + ( A ) ( F n , N ) ∼ = −→ Hom D f + ( A ) ( F n ( i n ) , N )is an isomorphism and each β i is M -coghost. Now since each β i is an M -coghostmap between perfect DG A -modules then by choice of n the composition alongthe top and then down to N , denoted β, must be zero. It is worth noting thatthe previous step needs the assumption that H( A ) is finitely generated over H ( A )since, in this case, each map in the composition defining β must be in D fb ( A ) . Finally, the isomorphism in (7) identifies β with f = f f . . . f n . Hence, f = 0and so cogin M D f + ( A ) ( N ) ≤ n = cogin M D fb ( A ) ( N ) , as needed. (cid:3) Let T be a triangulated category and fix G and X in T . The ghost index of X with respect to G in T , denoted gin GT ( X ), to be the least non-negative integer n such that any composition of G -ghost maps X = X n f n −−→ X n − f n − −−−→ . . . → X f −→ X is zero in T ; this where a map g is G - ghost provided Hom T ( Σ i G, g ) = 0 for all i ∈ Z . That is, gin G T ( X ) = cogin G T op ( X ) . In general, gin G T ( X ) ≤ level G T ( X ) and itis unknown whether equality holds when R is a commutative noetherian ring and T = D fb ( R ). The point of the next section is to provide a partial “converse.”3. A Partial Converse Ghost Lemma
In this section R is a commutative noetherian ring. As localization defines anexact functor D ( R ) → D ( R p ), level cannot increase upon localization. Hence, for M and N in D fb ( R ), if N is in thick D ( R ) ( M ), then gin M p D fb ( R p ) ( N p ) < ∞ for all p ∈ Spec R. The converse and an evident corollary are established below.
Theorem 3.1.
Let R be a commutative noetherian ring and fix M and N in D fb ( R ) .If gin M p D fb ( R p ) ( N p ) < ∞ for all p ∈ Spec R , then N is an object of thick D ( R ) ( M ) . Corollary 3.2. If gin M p D fb ( R p ) ( N p ) < ∞ for all p ∈ Spec R , then gin M D fb ( R ) ( N ) < ∞ . JIAN LIU AND JOSH POLLITZ
To prove Theorem 3.1, there are essentially two steps. We first go to derivedcategories of certain Koszul complexes where it is shown that cogin , gin and level allagree using Theorem 2.5. Second, we apply a local-to-global principle to concludethe desired result. We explain this below and give the proof of the theorem at theend of the section. Assume R is local with maximal ideal m , we let K R be the Koszul complexon a minimal generating set for m . It is regarded as a DG algebra in the usual wayand is well-defined up to an isomorphism of DG R -algebras, see [9, Section 1.6].For any p ∈ Spec R , let M be an object of D ( R ) . We set M ( p ) := M p ⊗ R p K R p which is a DG K R p -module. Restricting scalars along the morphism of DG algebras R p → K R p we may regard M ( p ) as an object of D ( R p ). In [6, Theorem 5.10], Ben-son, Iyengar and Krause proved the following local-to-global principle: For objects M and N in D fb ( R ), N is in thick D ( R ) ( M ) if and only if N ( p ) is in thick D ( R p ) ( M ( p )). Lemma 3.4.
Let R be a commutative noetherian local ring. For M and N in D fb ( K R ) , level M D ( K R ) ( N ) = cogin M D fb ( K R ) ( N ) = gin M D fb ( K R ) ( N ) . Proof.
The natural map K R → K b R is a quasi-isomorphism of DG algebras and soit induces an exact equivalence D fb ( K R ) ≡ −→ D fb ( K b R ) . Since cogin , gin and level are invariant under exact equivalences we can assume R is complete and set K = K R .As R is complete, it is well known that R admits a dualizing DG module ω ; see,for example, [14, Corollay 1.4]. Now applying [12, Theorem 2.1], Hom R ( K, ω ) isa dualizing DG K -module. In particular, setting ( − ) † := Hom K ( − , Hom R ( K, ω ))then for any M in D fb ( K ), M † is in D fb ( K ) and the natural biduality map M ≃ −→ M †† is an isomoprhism in D fb ( K ) . Hence, ( − ) † restricts to an exact auto-equivalence of D fb ( K ).Finally, as ( − ) † is an exact auto-equivalence of D fb ( K ) interchanging coghost andghost maps, from Theorem 2.5 the desired equality follows. (cid:3) Remark 3.5.
The lemma holds for any DG alegbra A satisfying the hypotheses ofTheorem 2.5 which admits a dualizing DG module as defined in [12, 1.8]. Anotherexample, generalizing the Koszul complex above, would be the DG fiber of any localring map of finite flat dimension whose target ring admits a dualizing complex [12,Theorem VI]. Lemma 3.6.
Let R be a commutative noetherian local ring and let t : D ( R ) → D ( K R ) denote − ⊗ R K R . If M and N are objects of D fb ( R ) , then gin t M D ( K R ) ( t N ) ≤ gin M D ( R ) ( N ) . Proof.
We set K = K R . For X in D ( R ) and Y in D ( K ), there is an adjunctionisomorphism(8) Hom D ( K ) ( t X, Y ) ∼ = Hom D ( R ) ( X, Y ) , PARTIAL CONVERSE GHOST LEMMA 9 which is induced by the natural map η X : X → t X given by x x ⊗ . Moreover,when f : Y → Z is a t M -ghost map in D fb ( K ), then (8) implies that f is a M -ghostmap in D fb ( R ).Assume n := gin M D fb ( R ) ( N ) < ∞ and Suppose g : t N → Y in D fb ( K ) factors as thecomposition of n maps in D fb ( K ) which are t M -ghost. Hence, by the adjunctionabove g is the composition of n maps in D fb ( R ) which are M -ghost, and thus sois g ◦ η N . Therefore, by assumption g ◦ η N = 0 and so from (8) we conclude that g = 0 in D fb ( K ), completing the proof. (cid:3) Proof of Theorem 3.1.
Let p ∈ Spec R , then by assumption gin M p D fb ( R p ) ( N p ) < ∞ .Also, gin M p D fb ( R p ) ( N p ) ≥ gin M ( p ) D fb ( K R p ) ( N ( p ))= cogin M ( p ) D fb ( K R p ) ( N ( p ))= level M ( p ) D ( K R p ) ( N ( p ))where the inequality is from Lemma 3.6 and the equalities are from Lemma 3.4.Thus level M ( p ) D ( K R p ) ( N ( p )) < ∞ , and so N ( p ) is in thick D ( K R p ) ( M ( p )) for all p ∈ Spec R . Now by restricting scalars along R p → K R p we conclude that N ( p ) is in thick D ( R p ) ( M ( p )) for all p ∈ Spec R . Finally, we apply 3.3 to obtain the desiredresult. (cid:3) References
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School of Mathematical Sciences, University of Science and Technology of China,Hefei 230026, Anhui, P.R. China.
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