aa r X i v : . [ m a t h . A C ] O c t Another version of cosupport for complexes
Xiaoyan YangDepartment of Mathematics, Northwest Normal University, Lanzhou 730070, ChinaE-mail: [email protected]
Abstract
The goal of the article is to develop a theory dual to that of support in the derivedcategory D( R ), this is done by introducing another versions of the “big” and“small” cosupport for complexes that are differ from the cosupport in [J. ReineAngew. Math. 673 (2012) 161–207]. We provide some properties for cosupportthat are similar–or rather dual–to those of support for complexes, study somerelations between the “big” and “small” cosupport and give some computationsand comparisons of the “small” support and cosupport. Finally, we investigatethe dual notion of associated primes of complexes. Key Words: complex; support; cosupport; coassociated prime
Introduction
Support is a fundamental concept in commutative algebras, which provides a geometricapproach for studying various algebraic structures. Based on certain localization functors oncompactly generated triangulated categories, Benson, Iyengar and Krause [3, 4, 6] developedthe theories of support and cosupport. Suitably specialized their approach recovers thesupport theory of Foxby [10] and Neeman [13, 14] for commutative noetherian rings, thetheory of Avramov and Buchweitz for complete intersection local rings [1, 2]. Their worksalso play a pivotal role on a classification theorem for the thick subcategories of modulesand the localizing subcategories of the stable module category (see [11] and [5, 7]).Despite the many ways in which cosupport is dual to the notion of support, cosupportseems to be more mysterious, even in the setting of a commutative noetherian ring. Ingeneral the theory of cosupport is not completely satisfactory because this construction isnot as well understood as support. Richardson [15] introduced the concept of cosupport ofmodules and proved that the cosupport have properties dual to those of support.One purpose of this paper is to extend the concept of cosupport in [15] to unboundedcomplexes. We focus on the duality functor D R ( − ) = Hom R ( − , L E ( R/ m )) the sum runningover all maximal ideals m of R , where E ( R/ m ) is the injective envelope of R/ m . For an R -complex M , the co-localization of M relative to a prime ideal p is the R p -complex M = Hom R p ( D R ( M ) p , E R p ( k ( p ))) ≃ Hom R ( D R ( M ) , E R ( R/ p )).In Section 2, we define the set coSupp R M of “big” cosupport of M to be the set of primeideals p such that p M
0. One of the main results of this work is that coSupp R M can bedetected by the big cosupport of the homology of M . We show that Theorem A.
For any R -complex M , one has that coSupp R M = S i ∈ Z coSupp R H i ( M ) .In particular, M if and only if coSupp R M = ∅ . We provide the following (partial) duality between the big cosupport and support.
Theorem B.
Let M be an R -complex. (1) p ∈ coSupp R M if and only if p ∈ Supp R D R ( M ) . (2) If p ∈ Supp R M , then p ∈ coSupp R D R ( M ) . The converse holds when M ∈ D n ( R ) ( i.e.each H i ( M ) is noetherian ) . By an example we show that the above notion is not the same as the one in [17].Section 3 investigates the “small” cosupport of complexescosupp R M := { p ∈ Spec R | RHom R ( R/ p , p M ) } ,and proves some properties for “small” cosupport that are similar to those of “small” supportand “big” cosupport in Section 2.In Section 4, we study some relations between the “big” and “small” cosupport, and showcosupp R M ⊆ coSupp R M . By an example we show that the inclusion may be strict.Section 5 is devoted to provide some computations of the “small” support and “small”cosupport, and study the relation between cosupp R M and cosupp R H( M ). As an application,we give the comparison of the support and cosupport.The concept of coassociated primes of complexes is introduced in the last section, and anextension of Nakayama lemma is given.1. Preliminaries
Unless stated to the contrary we assume throughout this paper that R is a commutativenoetherian ring which is not necessarily local.This section is devoted to recalling some notions and basic consequences for use throughoutthis paper. For terminology we shall follow [9] and [17]. Complexes.
The category of chain R -complexes is denoted by C( R ). The derived cate-gory of R -complexes is denoted by D( R ).Let M be an object in C( R ) and n ∈ Z . The soft right-truncation, σ > n ( M ), of M at n and the soft left-truncation, σ n ( M ), of M at n are given by σ > n ( M ) : · · · −→ M n +2 d n +2 −→ M n +1 d n +1 −→ Ker d n −→ n ( M ) : 0 −→ Coker d n +1 d n −→ M n − d n − −→ M n − −→ · · · .The differential d n is the induced morphism on residue classes.An R -complex M is called bounded above if H n ( M ) = 0 for all n ≫
0, bounded below ifH n ( M ) = 0 for all n ≪
0, and bounded if it both bounded above and bounded below. Thefull triangulated subcategories consisting of bounded above, bounded below and bounded R -complexes are denoted by D − ( R ) , D + ( R ) and D b ( R ). We denote by D n ( R ) the full trian-gulated subcategory of D( R ) consisting of R -complexes M such that H i ( M ) are noetherian R -modules for all i , and denote by D a ( R ) the full triangulated subcategory of D( R ) consistingof R -complexes M such that H i ( M ) are artinian R -modules for all i . For M ∈ D( R ),inf M := inf { n ∈ Z | H n ( M ) = 0 } , sup M := sup { n ∈ Z | H n ( M ) = 0 } .We write Spec R for the set of prime ideals of R and Max R for the set of maximal idealsof R . For an ideal a in R and p ∈ Spec R , we setU( p ) = { q ∈ Spec R | q ⊆ p } and V( a ) = { q ∈ Spec R | a ⊆ q } .Denote D R ( − ) = Hom R ( − , L m ∈ Max R E ( R/ m )) and D m ( − ) = Hom R ( − , E ( R/ m )) for m ∈ Max R . Let S be a multiplicatively closed subset of R . For an R -complex M , the co-localization of M relative to S is the S − R -module S − M = D S − R ( S − D R ( M )). If S = R − p for some p ∈ Spec R , we write p M for S − M . We also set M ∼ = Q m ∈ Max R D m ( D m ( M )). Support and cosupport.
The “small” support of an R -complex M is the setsupp R M = { p ∈ Spec R | k ( p ) ⊗ L R M } ,where k ( p ) = R p / p R p . The “big” support of M is the setSupp R M = { p ∈ Spec R | M p } .It follows from [9, 6.4.2.1, 6.1.3.2] that supp R M ⊆ Supp R M = S i ∈ Z Supp R H i ( M ).The “small” cosupport of an R -complex M is the setco-supp R M = { p ∈ Spec R | RHom R ( k ( p ) , M ) } .The “big” cosupport of M is the setCo-supp R M = { p ∈ Spec R | RHom R ( R p , M ) } .It follows from [17, Corollary 4.6] that co-supp R M ⊆ Co-supp R M .Richardson [15] defined the cosupport of an R -module K , coSupp R K , as the setcoSupp R K := { p ∈ Spec R | p K = 0 } .Yassemi [16] introduced the cocyclic modules and another cosupport of modules. An R -module L is cocyclic if L is a submodule of E ( R/ m ) for some m ∈ Max R . The cosupport of K is defined as the set of prime ideals p such that there is a cocyclic homomorphic image L of K such that p ⊆ Ann R L , the annihilator of L , and denoted this set by Cosupp R K . Lemma 1.1.
Let K be an R -module and p a point in Spec R . If p ∈ Cosupp R K , then p ∈ coSupp R K . (2) If R is a semi-local ring or K is a finitely generated R -module, then p ∈ coSupp R K ifand only if p ∈ Cosupp R K .Proof. (1) The exact sequence 0 → E ( R/ m ) → L m ∈ Max R E ( R/ m ) → L m = m ′ ∈ Max R E ( R/ m ′ ) → → Hom R (Hom R ( K, L m = m ′ ∈ Max R E ( R/ m ′ )) , E ( R/ p )) → Hom R ( D R ( K ) , E ( R/ p )) → Hom R ( D m ( K ) , E ( R/ p )) → R ( L m ∈ Max R D m ( K ) , E ( R/ p )) = 0, thenHom R ( D m ( K ) , E ( R/ p )) = 0 for some m , and hence p K = 0 and p ∈ coSupp R K .(2) If R is semi-local or K is finitely generated, then p K ∼ = Hom R ( L m ∈ Max R D m ( K ) , E ( R/ p )).Hence the equivalence follows from the remark after [16, Theorem 3.8]. (cid:3) Lemma 1.2.
Let K be an R -module and p a point in Spec R . If p ∈ Supp R K , then p ∈ coSupp R D R ( K ) . The converse holds when K is finitely generated.Proof. Since p ∈ Supp R K , p ∈ Cosupp R D m ( K ) for some m ∈ Max R ∩ V( p ) by [16, Lemma2.8], and hence p ∈ Cosupp R D R ( K ). Consequently, p ∈ coSupp R D R ( K ) by Lemma 1.1.Conversely, if p ∈ coSupp R D R ( K ) then p D R ( K ) = 0, which implies that K p = 0 since K isfinitely generated. Therefore, p ∈ Supp R K . (cid:3) Another version of big cosupport
This section introduces the set coSupp R M of the “big” cosupport of an R -complex M ,which is differ from the “big” cosupport in [17]. We show that coSupp R M is completelyrelated to coSupp R H i ( M ), and give a (partial) duality between coSupp R M and Supp R M . Definition 2.1.
Let M be an R -complex. The “big” cosupport of M is defined ascoSupp R M := { p ∈ Spec R | p M } .The next theorem establishes the fact that the big cosupport for an R -complex is com-pletely related to the big cosupport of the homology modules of complexes, which bring ananalogue of the big support (see [9, 6.1.3.2]). Theorem 2.2.
Let M be an R -complex. One has an equality coSupp R M = S i ∈ Z coSupp R H i ( M ) .Proof. One has the following equivalences ∈ coSupp R M ⇐⇒ H i ( p M ) = 0 for some i ⇐⇒ Hom R ( D R (H i ( M )) , E ( R/ p )) = 0 for some i ⇐⇒ p H i ( M ) = 0 for some i ⇐⇒ p ∈ [ i ∈ Z coSupp R H i ( M ) , where the second equivalence is because E ( R/ m ) and E ( R/ p ) are injective. (cid:3) Corollary 2.3.
For an R -complex M , one has M if and only if coSupp R M = ∅ .Proof. One has that coSupp R M = ∅ if and only if coSupp R H i ( M ) = ∅ for some i if and onlyif H i ( M ) = 0 for some i if and only if M
0, where the first equivalence is by Theorem 2.2,the second one is by [15, Theorem 2.7]. (cid:3)
If 0 M ∈ D nb ( R ), then Supp R M = V(Ann R M ). The next corollary is dual to this. Corollary 2.4.
For any M ∈ D ab ( R ) , one has that coSupp R M = V(Ann R M ) = Supp R ( R/ Ann R M ) .Proof. Set i = inf M and s = sup M . We havecoSupp R M = s [ j = i coSupp R H j ( M )= s [ j = i V(Ann R H j ( M ))= V( s \ j = i Ann R H j ( M ))= V(Ann R M ) , where the second equality is by [15, Theorem 2.7]. (cid:3) The following result play an important role in the rest of the paper.
Theorem 2.5.
Let M be an R -complex. The following are equivalent: (1) p ∈ coSupp R M ; (2) p ∈ Supp R D R ( M ) .If in addition R is semi-local, then (1) and (2) are equivalent to (3) p ∈ Supp R D m ( M ) for some m ∈ Max R ∩ V( p ) ; (4) RHom R ( R p , M ∼ ) .Proof. (1) ⇔ (2) One has the following equivalences ∈ coSupp R M ⇐⇒ p ∈ coSupp R H i ( M ) for some i ⇐⇒ p ∈ Supp R D R (H i ( M )) for some i ⇐⇒ p ∈ Supp R H − i ( D R ( M )) for some i ⇐⇒ p ∈ Supp R D R ( M ) , where the first one is by Theorem 2.2, the second one is by [15, Theorem 2.7], the third oneis since L m ∈ Max R E ( R/ m ) is injective.Next assume that R is semi-local.(2) ⇔ (3) One has the following equivalences p ∈ Supp R D m ( M ) ⇐⇒ p ∈ Supp R D m (H i ( M )) for some i ⇐⇒ p ∈ coSupp R H i ( M ) for some i ⇐⇒ p ∈ Supp R D R (H i ( M )) for some i ⇐⇒ p ∈ Supp R D R ( M ) , where the second equivalence is by [16, Lemma 2.5] and Lemma 1.1, the third one is by [15,Theorem 2.7].(1) ⇔ (4) One has the following equivalencesRHom R ( R p , M ∼ ) ⇐⇒ Y m ∈ Max R H i ( D m ( D m ( M ) p )) = 0 for some i ⇐⇒ Y m ∈ Max R D m ( D m (H i ( M )) p ) = 0 for some i ⇐⇒ Y m ∈ Max R Hom R ( R p , D m ( D m (H i ( M )))) = 0 for some i ⇐⇒ Hom R ( R p , H i ( M ) ∼ ) = 0 for some i, where the second and the third equivalences are because E ( R/ m ) is injective and R p is flat.Hence Theorem 2.2 and [16, Theorem 2.15] imply the desired equivalence. (cid:3) Let U be a subset of Spec R . The specialization closure of U is the setcl U = { p ∈ Spec R | there is q ∈ U with q ⊆ p } .The subset U is specialization closed if cl U = U . Remark 2.6. (1) For any R -complex M , one has that coSupp R M = coSupp R Σ M .(2) For an exact triangle L → M → N in D( R ), we havecoSupp R M ⊆ coSupp R L ∪ coSupp R N .(3) For any R -complex M , the set coSupp R M is specialization closed.(4) H( p M ) ∼ = p H( M ) for any p ∈ Spec R .(5) Let M ∈ D nb ( R ) and N ∈ D( R ). One has two isomorphisms ( M ⊗ L R N ) ≃ M p ⊗ L R p p N and p RHom R ( M, N ) ≃ RHom R p ( M p , p N ),in D( R ), which implies thatcoSupp R ( M ⊗ L R N ) ⊆ Supp R M ∩ coSupp R N ,coSupp R RHom R ( M, N ) ⊆ Supp R M ∩ coSupp R N .(6) By Lemma 1.1, one has that RHom R ( R p , M ∼ ) p M R -complex M is not the same as the one in[17]. For example, let M = R = k [ x ] for any field k . Then Co-supp R M = Spec R . ButcoSupp R M = Max R = Spec R by Theorem 2.5.(8) Let M be an R -complex and p ∈ Spec R . If each H i ( M ) is a Matlis reflexive R -module (i.e. H i ( M ) ∼ = D R ( D R (H i ( M )))), then M ≃ D R ( D R ( M )), and so p D R ( M ) ≃ Hom R p ( D R ( D R ( M )) p , E ( k ( p ))) ≃ D R p ( M p ). Consequently, p ∈ Supp R M ⇐⇒ p ∈ coSupp R D R ( M ).In general, we have the following result. Proposition 2.7.
Let M be an R -complex. (1) If p ∈ Supp R M , then p ∈ coSupp R D R ( M ) . (2) If M ∈ D n ( R ) , then p ∈ Supp R M if and only if p ∈ coSupp R D R ( M ) .Proof. (1) Since p ∈ Supp R M , p ∈ Supp R H i ( M ) for some i , and so p ∈ coSupp R H − i ( D R ( M ))by Lemma 1.2. Therefore, p ∈ coSupp R D R ( M ) by Theorem 2.2.(2) “Only if” part by (1). “If” part. Since p ∈ coSupp R D R ( M ), p ∈ coSupp R H i ( D R ( M ))for some i by Theorem 2.2, i.e., p ∈ coSupp R D R (H − i ( M )). Hence Lemma 1.2 implies that p ∈ Supp R H − i ( M ). Consequently, p ∈ Supp R M . (cid:3) The following example shows that the reverse of (1) in the above proposition does nothold in general.
Example 2.8. ([16]) Let ( R, m , k ) be a local domain with dim R >
0. Consider the complex M = 0 → L n> R/ m n →
0. Then (0) ∈ Supp R D R ( D R ( M )) and so (0) ∈ coSupp R D R ( M )by Theorem 2.5. However, (0) Supp R M .3. Another version of small cosupport
This section introduces the set cosupp R M of “small” cosupport of an R -complex M , andprovide a duality between the “small” cosupport and support as Section 2. Definition 3.1.
Let M be an R -complex. The “small” cosupport of M is defined ascosupp R M := { p ∈ Spec R | RHom R ( R/ p , p M ) } .Next we bring an analogue of Theorem 2.5. heorem 3.2. Let M be an R -complex. The following are equivalent: (1) p ∈ cosupp R M ; (2) RHom R ( D R ( M ) , k ( p )) ; (3) p ∈ supp R D R ( M ) ; (4) k ( p ) ⊗ L R p p M ; (5) p R p ∈ cosupp R p p M .If in addition R is semi-local, then (1) – (5) are equivalent to (6) RHom R ( k ( p ) , M ∼ ) ; (7) Hom R ( ` m ∈ Max R D m ( M ) , k ( p )) ; (8) p ∈ supp R D m ( M ) for some m ∈ Max R ∩ V( p ) ; (9) k ( p ) ⊗ L R p RHom R ( R p , M ∼ ) .Proof. One has the following isomorphisms in D( R ): p RHom R ( R/ p , M ) ≃ RHom R ( D R ( M ) , k ( p )) ≃ RHom R ( R/ p , p M ), D R (RHom R ( R/ p , M )) p ≃ ( R/ p ⊗ L R D R ( M )) p ≃ k ( p ) ⊗ L R D R ( M ).Hence Theorem 2.5 implies the equivalences of (1)–(3).(1) ⇔ (4) This follows from [17, Fact 3.5] and the isomorphism RHom R ( R/ p , p M ) ∼ =RHom R p ( k ( p ) , p M ) in D( R ).(1) ⇔ (5) Since RHom R p ( R p / p R p , p R p ( p M )) ≃ D R p ( D R p (RHom R ( R/ p , p M ))), it followsthat RHom R p ( R p / p R p , p R p ( p M )) R ( R/ p , p M )
0, as desired.One has the following isomorphisms in D( R ):RHom R ( k ( p ) , M ∼ ) ≃ RHom R ( R p , RHom R ( R/ p , M ) ∼ ),Hom R ( ` m ∈ Max R D m ( M ) , k ( p )) ≃ Hom R ( ` m ∈ Max R D m (RHom R ( R/ p , M )) , E ( R/ p )), D m (RHom R ( R/ p , M )) p ≃ ( R/ p ⊗ L R D m ( M )) p ≃ k ( p ) ⊗ L R D m ( M ),Hence Theorem 2.5 implies the equivalences of (1) ⇔ (6) ⇔ (7) ⇔ (8).(6) ⇔ (9) This follows from [17, Fact 3.5] and the isomorphism RHom R ( k ( p ) , M ∼ ) ≃ RHom R p ( k ( p ) , RHom R ( R p , M ∼ )) in D( R ). (cid:3) Corollary 3.3.
Let M be an R -complex. Then M if and only if cosupp R M = ∅ .Proof. M D R ( M ) R D R ( M ) = ∅ if and only ifcosupp R M = ∅ by Theorem 3.2. (cid:3) Corollary 3.4.
Let M be an R -complex. One has that cosupp R M = min(cosupp R H( M )) .Proof. One has the following equivalences ∈ cosupp R M ⇐⇒ p ∈ supp R D R ( M ) ⇐⇒ p ∈ min(supp R H( D R ( M ))) ⇐⇒ p ∈ min(supp R D R (H( M ))) ⇐⇒ p ∈ min(cosupp R H( M )) , where the first and the last equivalences are by Theorem 3.2, the second one is by [3, Theorem5.2] and the third one is as L m ∈ Max R E ( R/ m ) is injective. (cid:3) Remark 3.5. (1) For any R -complex M , one has p RHom R ( R/ p , M ) ≃ RHom R ( R/ p , p M ).Hence p ∈ cosupp R M ⇐⇒ p ∈ coSupp R RHom R ( R/ p , M ).(2) If M is an R -module, then cosupp R M = { p ∈ Spec R | p Ext iR ( R/ p , M ) = 0 for some i } .(3) Let V be a specialization closed subset of Spec R . For each R -module M , one hascosupp R M ⊆ V ⇐⇒ p M = 0 for each p ∈ Spec R \ V.(4) For each R -module M , one has inclusionscosupp R M ⊆ cl(cosupp R M ) = coSupp R M ⊆ V(Ann R M ). Proposition 3.6. (1)
Let M ∈ D fb ( R ) and N ∈ D( R ) . One has that cosupp R RHom R ( M, N ) = supp R M ∩ cosupp R N . (2) Let M ∈ D f+ ( R ) and N ∈ D + ( R ) or M ∈ D fb ( R ) and N ∈ D( R ) . One has that cosupp R ( M ⊗ L R N ) = supp R M ∩ cosupp R N .In particular, for any ideal a of R and an arbitrary R -complex M , we have cosupp R RHom R ( R/ a , M ) = cosupp R M ∩ V( a ) = cosupp R ( R/ a ⊗ L R M ) .Proof. (1) One has the following equivalences p ∈ cosupp R RHom R ( M, N ) ⇐⇒ p ∈ supp R D R (RHom R ( M, N )) ⇐⇒ p ∈ supp R ( M ⊗ L R D R ( N )) ⇐⇒ p ∈ supp R M ∩ supp R D R ( N ) ⇐⇒ p ∈ supp R M ∩ cosupp R N, where the first and the fourth equivalences are by Theorem 3.2, the second one is by [9,Theorem 2.5.6] and the third one is by [17, Proposition 3.12].(2) One has the following equivalences p ∈ cosupp R ( M ⊗ L R N ) ⇐⇒ p ∈ supp R D R ( M ⊗ L R N ) ⇐⇒ p ∈ supp R RHom R ( M, D R ( N )) ⇐⇒ p ∈ supp R M ∩ supp R D R ( N ) ⇐⇒ p ∈ supp R M ∩ cosupp R N, here the first and the fourth equivalences are by Theorem 3.2, the third one is by [17,Proposition 3.16]. (cid:3) The following proposition is an analogue of Proposition 2.7.
Proposition 3.7.
Let M be an R -complex. (1) If p ∈ supp R M , then p ∈ cosupp R D R ( M ) . (2) If M ∈ D n ( R ) , then p ∈ supp R M if and only if p ∈ cosupp R D R ( M ) .Proof. (1) Let p ∈ supp R M . Then p ∈ Supp R ( R/ p ⊗ L R M ), and so p ∈ coSupp R D R ( R/ p ⊗ L R M )by Proposition 2.7(1). But D R ( R/ p ⊗ L R M ) ≃ RHom R ( R/ p , D R ( M )), it follows from Remark3.5(1) that p ∈ cosupp R D R ( M ).(2) This follows from Proposition 2.7(2) since R/ p ⊗ L R M ∈ D n ( R ). (cid:3) Relations between big and small cosupport
We devote this section to some relations between “big” and “small” cosupport. We showthat cosupp R M ⊆ coSupp R M and the inclusion may be strict. Proposition 4.1.
Let M be an R -complex. The sets supp R M and cosupp R M have thesame maximal elements with respect to containment, i.e., max(supp R M ) = max(cosupp R M ) .Moreover, max(cosupp R M ) = max(co - supp R M ) .Proof. We prove that max(supp R M ) ⊆ cosupp R M and max(cosupp R M ) ⊆ supp R M .If p ∈ max(supp R M ), then co-supp R ( R/ p ⊗ L R D R ( M )) = { p } by [17, Proposition 4.10].Hence RHom R ( D R ( M ) , k ( p )) ≃ RHom R ( R/ p ⊗ L R D R ( M ) , E ( R/ p )) p ∈ cosupp R M by Theorem 3.2. If p ∈ max(cosupp R M ), then cosupp R ( R/ p ⊗ L R M ) = { p } , so p ∈ max(supp R D R ( R/ p ⊗ L R M )). Thus [17, Proposition 4.7(b)] implies that p ∈ co-supp R RHom R ( R/ p , D R ( M )). Consequently, p ∈ supp R M by [17, Proposition 4.10].The second statement follows from [6, Theorem 4.13]. (cid:3) Proposition 4.2.
For every R -complex M , one has an inclusion cosupp R M ⊆ coSupp R M ;equality holds if R is a semi-local complete ring and M ∈ D a − ( R ) .Proof. The inclusion follows from Theorems 2.5 and 3.2 since supp R D R ( M ) ⊆ Supp R D R ( M ).Now let M ∈ D a − ( R ) and p ∈ coSupp R M , i = inf p M . Then p M ∈ D a − ( R p ) by [15, Theorem2.3], and so H i (RHom R p ( k ( p ) , p M )) ∼ = Hom R p ( k ( p ) , H i ( p M )) = 0 by [16, Theorem 4.3].Consequently, RHom R ( R/ p , p M ) p ∈ cosupp R M , as claimed. (cid:3) The next example shows that the inclusion in the proposition 4.2 may be strict: xample 4.3. ([3, Example 9.4]) Let k be a field and R = k [[ x, y ]] the power series ringin indeterminates x, y , and set m = ( x, y ) the maximal ideal of R . The minimal injectiveresolution of R has the form: · · · → → Q → ` ht p =1 E ( R/ p ) → E ( R/ m ) → → · · · ,where Q denotes the fraction field of R . Let M denote the truncated complex · · · → → Q → ` ht p =1 E ( R/ p ) → → · · · .One has that coSupp R D R ( M ) = Spec R since Spec R = Supp R M ⊆ coSupp R D R ( M ). But m cosupp R D R ( M ). In fact, if m ∈ cosupp R D R ( M ) then m ∈ supp R D R ( M ) by Proposition4.1, and hence m ∈ cosupp R M by Theorem 3.2. Consequently, m ∈ supp R M by Proposition4.1 again, which is a contradiction since supp R M = Spec R \{ m } . Corollary 4.4.
Let M be an R -complex. The sets cosupp R M and coSupp R M have the sameminimal elements with respect to containment, i.e. min(cosupp R M ) = min(coSupp R M ) .Proof. This follows from Theorems 2.5, 3.2 and [17, Proposition 3.14]. (cid:3)
Corollary 4.5.
Let M be an R -complex. (1) For an ideal a of R , coSupp R M ⊆ V( a ) if and only if cosupp R M ⊆ V( a ) . (2) The Zariski closures of coSupp R M and cosupp R M are equal.Proof. This follows from Theorems 2.5, 3.2 and [17, Proposition 3.15]. (cid:3)
Proposition 4.6. (1) If M is in D n ( R ) , then cosupp R M ⊆ co - supp R M and coSupp R M ⊆ Co - supp R M . (2) Assume that R is a semi-local ring and M ∈ D( R ) . If each H i ( M ) is a Matlis reflexive R -module, then co - supp R M = cosupp R M and Co - supp R M = coSupp R M .Proof. (1) Since M ∈ D n ( R ), it follows that cosupp R M ⊆ coSupp R M ⊆ Max R . HenceProposition 4.1 implies that cosupp R M ⊆ co-supp R M . Note that cosupp R M = coSupp R M and co-supp R M ⊆ Co-supp R M , so coSupp R M ⊆ Co-supp R M .(2) Since H i ( M ) ∼ = D R ( D R (H i ( M ))) for all i , it follows that M ≃ D R ( D R ( M )). Henceco-supp R M = co-supp R D R ( D R ( M )) = supp R D R ( M ) = cosupp R M and Co-supp R M =Co-supp R D R ( D R ( M )) = Supp R D R ( M ) = coSupp R M . (cid:3) Corollary 4.7.
Assume that R is a semi-local complete ring. If M ∈ D n ( R ) or M ∈ D a ( R ) ,then co - supp R M = cosupp R M and Co - supp R M = coSupp R M . The example in Remark 2.6(7) shows that the inclusion in Proposition 4.6 may be strict. . Computations of cosupport and support
This section puts emphasis on computing the “small” support and “small” cosupport, andstudying the relation between cosupp R M and cosupp R H( M ). As an application, we give thecomparison of the “small” support and cosupport. Proposition 5.1.
Let p be a point in Spec R . One has that (1) cosupp R R = Max R and supp R R = Spec R . (2) cosupp R k ( p ) = { p } = supp R k ( p ) . (3) supp R E ( R/ p ) = { p } and cosupp R E ( R/ p ) = U( p ) .Proof. (1) It follows from Theorem 3.2 and [17, Proposition 3.11] thatcosupp R R = supp R D R ( R ) = supp R L m ∈ Max R E ( R/ m ) = Max R .It follows from Proposition 3.7 thatsupp R R = cosupp R D R ( R ) = cosupp R L m ∈ Max R E ( R/ m ) = Spec R .(1) Since supp R k ( p ) = { p } , it follows from Proposition 4.1 that cosupp R k ( p ) ⊆ U( p ). Onthe other hand, cosupp R k ( p ) = cosupp R ( R/ p ⊗ R R p ) ⊆ V( p ) by Proposition 3.6. Conse-quently, cosupp R k ( p ) = { p } .(2) This follows from [16, Corollary 2.18] and [17, Proposition 6.3]. (cid:3) Remark 5.2. (i) Example 4.3 shows that supp R M and supp R H( M ) need not coincide andcosupp R M and cosupp R H( M ) need not coincide.(ii) For any R -complex M , cosupp R M may be not a specialization closed subset.The next results study relations between cosupp R M (resp. supp R M )and cosupp R H( M )(resp. supp R H( M )). Proposition 5.3. (1)
For each M ∈ D n+ ( R ) , one has supp R M = S i ∈ Z supp R H i ( M ) . (2) If R is semi-local complete, then for M ∈ D a − ( R ) , cosupp R M = S i ∈ Z cosupp R H i ( M ) .Proof. We just prove one of the statements since the other is dual.By Proposition 4.2, cosupp R M = coSupp R M . But coSupp R M = S i ∈ Z coSupp R H i ( M ) = S i ∈ Z cosupp R H i ( M ) by Theorem 2.2, as desired. (cid:3) Proposition 5.4. (1)
For each M ∈ D − ( R ) , one has supp R M ⊆ S i ∈ Z supp R H i ( M ) . (2) For each M ∈ D + ( R ) , one has cosupp R M ⊆ S i ∈ Z cosupp R H i ( M ) .Proof. We just prove (1) since (2) follows by duality,First let M ∈ D b ( R ). If inf M = sup M = r , then M ≃ Σ r H r ( M ) and supp R M ⊆ supp R H r ( M ). Assume that sup M − inf M >
0. The exact triangle σ > inf M +1 ( M ) → M → Σ inf M H inf M ( M ) yields that upp R M ⊆ supp R σ > inf M +1 ( M ) ∪ supp R H inf M ( M ).But supp R σ > inf M +1 ( M ) ⊆ S i ∈ Z supp R H i ( σ > inf M +1 ( M )) = S i > inf M +1 supp R H i ( M ) by induc-tion, so supp R M ⊆ S i ∈ Z supp R H i ( M ). Now let M ∈ D − ( R ). Then M = lim −→ σ > n ( M ).Since supp R M ⊆ S n supp R σ > n ( M ) and supp R σ > n ( M ) ⊆ S i > n supp R H i ( M ), it followsthat supp R M ⊆ S i ∈ Z supp R H i ( M ). (cid:3) The following corollary is a generalization of [17, Theorem 6.7].
Corollary 5.5. (1)
For each M ∈ D n+ ( R ) , one has that cosupp R M ⊆ supp R M . (2) If R is a semi-local complete ring, then for M ∈ D a − ( R ) , supp R M ⊆ cosupp R M .Proof. We just prove (1) since (2) follows by duality.By Proposition 5.4 (2), cosupp R M ⊆ S i ∈ Z cosupp R H i ( M ). But H i ( M ) is noetherian andcoSupp R H i ( M ) ⊆ Max R , it follows from Propositions 4.1 and 5.3 that S i ∈ Z cosupp R H i ( M ) ⊆ S i ∈ Z supp R H i ( M ) = supp R M , as claimed. (cid:3) Remark 5.6. (i) The assumption M ∈ D n+ ( R ) in (1) and M ∈ D a − ( R ) in (2) in Corollary5.5 are essential. For example, assume that ( R, m ) is local and not artinian. One hassupp R E ( R/ m ) = { m } ( Spec R = cosupp R E ( R/ m ),cosupp R R = { m } ( Spec R = supp R R .(ii) Proposition 5.1 (1) and (3) show that one can has proper containment or equality inthe above corollary. 6. Coassociated prime for complexes
The aim of this section is to develop a theory dual that of associated primes of complexesintroduced by Christensen in [8], and find an extension of Nakayama lemma.Let ( R, m , k ) be a local ring and M an R -complex. The depth of M isdepth R M = − supRHom R ( k, M ).Following [8], we say that p ∈ Spec R is a associated prime ideal for M ∈ D − ( R ) ifdepth R p M p = − sup M p < ∞ . For M − ( R ), we setass R M = Ass R H sup M ( M ) and z R M = z R H sup M ( M ) and Z R M = S p ∈ Ass R M p .Let K be an R -module. A prime ideal p of R is called a coassoczated prime of K if thereexists a cocyclic homomorphic image L of K such that p = Ann R L . The set of coassociatedprime ideals of K is denoted by Coass R K .For an R -module K the subset w R K of R is defined byw R K = { r ∈ R | K r · −→ K is not surjective } . y [16, Theorem 1.13], w R K = S p ∈ Coass R K p .Let ( R, m , k ) be a local ring and M an R -complex. The width of M iswidth R M = inf( k ⊗ L R M ). Definition 6.1. (1) We say that p ∈ Spec R is a coassociated prime ideal for M ∈ D + ( R ) ifwidth R p p M = inf p M > −∞ , that is, Coass R M = { p ∈ coSupp R M | width R p p M = inf p M } .(2) For an R -complex M + ( R ), we setcoass R M = Coass R H inf M ( M ) and w R M = w R H inf M ( M ) and W R M = S p ∈ Coass R M p ,and for M ≃ R M = ∅ and w R M = ∅ . Theorem 6.2.
Let M ∈ D + ( R ) . Then p ∈ Coass R M if and only if p ∈ Ass R D R ( M ) . Inparticular, M if and only if Coass R M = ∅ .Proof. Since p M = Hom R p ( D R ( M ) p , E R p ( k ( p ))), it follows that − sup D R ( M ) p = inf p M = i is finite. One has the following equivalences p ∈ Coass R M ⇐⇒ inf( k ( p ) ⊗ L R p p M ) = inf p M = i ⇐⇒ k ( p ) ⊗ R p H i ( p M ) = 0 ⇐⇒ k ( p ) ⊗ R p Hom R p ( D R (H i ( M )) p , E ( k ( p )) = 0 ⇐⇒ Hom R p (Hom R p ( k ( p ) , D R (H i ( M )) p ) , E ( k ( p )) = 0 ⇐⇒ H − i (RHom R p ( k ( p ) , D R ( M ) p ) = Hom R p ( k ( p ) , H − i ( D R ( M ) p )) = 0 ⇐⇒ p R p ∈ Ass R p H − i ( D R ( M ) p ) ⇐⇒ p ∈ Ass R D R ( M ) , where the second one is by [9, Lemma 2.4.14], the third one is since E ( R/ m ) and E ( k ( p ) areinjective and R p is flat, the fourth one is by [9, Theorem 2.5.6], the fifth one is since E ( k ( p ))is faithful injective and the last one is by [8, Observations 2.4]. (cid:3) Remark 6.3. (1) Let K be an R -module. By Theorem 6.2, p ∈ Coass R K if and onlyif p ∈ Ass R D R ( K ) if and only if p R p ∈ Ass R p D R ( K ) p if and only if p R p ∈ Coass R p p K since the morphism k ( p ) → D R ( K ) p is injective if and only if the morphism p K → k ( p ) ∼ =Hom R p ( k ( p ) , E R p ( k ( p ))) is surjective.(2) Let M ∈ D + ( R ) and p ∈ coSupp R M and set inf p M = i . Then p ∈ Coass R M ⇐⇒ p ∈ Ass R D R ( M ) ⇐⇒ p R p ∈ Ass R p D R (H i ( M )) p ⇐⇒ p R p ∈ Coass R p H i ( p M ) ⇐⇒ p R p ∈ coass R p p M ⇐⇒ p ∈ Coass R H i ( M ) . n particular there exists the following inclusioncoass R M ⊆ Coass R M .Also w R M = z R D R ( M ) ⊆ Z R D R ( M ) = W R M .(3) Let M ∈ D + ( R ). Then every minimal prime ideal in coSupp R M belongs to Coass R M ,and Coass R M ⊆ cosupp R M for M ∈ D b ( R ) by [8, Proposition 2.6].(4) If M ∈ D ab ( R ) then the set of minimal prime ideals in coSupp R M is finite.(5) If M is an R -module, then coass R M = Coass R M and w R M = W R M . Proposition 6.4. (1)
Let M ∈ D fb ( R ) and N ∈ D − ( R ) . One has that Ass R RHom R ( M, N ) = Supp R M ∩ Ass R N . (2) Let M ∈ D fb ( R ) and N ∈ D + ( R ) . One has that Coass R ( M ⊗ L R N ) = Supp R M ∩ Coass R N .Proof. (1) There exists a series of equivalences p ∈ Ass R RHom R ( M, N ) ⇐⇒ − supRHom R ( M, N ) p = depth R p RHom R ( M, N ) p ⇐⇒ − supRHom R ( M, N ) p = depth R p N p + inf M p ⇐⇒ depth R p N p = − sup N p and M p . Hence we obtain the desired equality.(2) Using (1) and the isomorphism D R ( M ⊗ L R N ) ≃ RHom R ( M, D R ( N )) in D( R ). (cid:3) Corollary 6.5. (1)
Let M be in D − ( R ) . One has that Ass R M ∩ Max R = ∅ if and only if RHom R ( R/ m , M ) . (2) Let M be in D + ( R ) . Then Coass R M ∩ Max R = ∅ if and only if R/ m ⊗ L R M . The following result is an extension of Nakayama lemma.
Proposition 6.6.
Let a be an ideal of R such that a ⊆ J ( R ) the Jacobson radical of R . (1) If M is in D − ( R ) such that Ass R M ∩ Max R = ∅ , then RHom R ( R/ a , M ) . (2) If M is in D + ( R ) such that Coass R M ∩ Max R = ∅ , then R/ a ⊗ L R M .Proof. (1) Given m ∈ Ass R M ∩ Max R and set s = sup M m . Then H s (RHom R m ( k ( m ) , M m )) ∼ =Hom R m ( k ( m ) , H s ( M m )) = 0. So H s (RHom R m (( R/ a ) m , M m )) ∼ = Hom R m (( R/ a ) m , H s ( M m )) = 0since the map ( R/ a ) m ։ ( R/ m ) m is surjective, which implies that RHom R ( R/ a , M ) R D R ( M ) ∩ Max R = ∅ . Hence RHom R ( R/ a , D R ( M )) R/ a ⊗ L R M (cid:3) ACKNOWLEDGEMENTS
I am very grateful to Peder Thompson for helpful conversations. This research was par-tially supported by National Natural Science Foundation of China (11761060,11901463), mprovement of Young Teachers ′ Scientific Research Ability (NWNU-LKQN-18-30) and In-novation Ability Enhancement Project of Gansu Higher Education Institutions (2019A-002).
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