Cofiniteness with respect to extension of Serre subcategories at small dimensions

Let $R$ be a commutative noetherian ring, $\frak a$ be an ideal of $R$, $\cS$ be an arbitrary Serre subcategory of $R$-modules and let $\cN$ be the subcategory of finitely generated $R$-modules. In this paper, we study $\cN\cS$-$\frak a$-cofinite modules with respect to the extension subcategory $\cN\cS$ when $\dim R/\frak a\leq 2$. We also study $\frak a$-cofiniteness with respect to a new dimension.


Introduction
Throughout this paper R is a commutative noetherian ring, a is an ideal of R. Given a Serre subcategory S of R-modules, an R-module M is said to be S-a-cofinite if Supp M ⊆ V (a) and Ext i R (R/a, M ) ∈ S for all integers i ≥ 0. Let N be the subcategory of finitely generated Rmodules.
The extension subcategory induced by N and S is denoted by N S, consisting of those R-modules M for which there exist an exact sequence 0 −→ N −→ M −→ S −→ 0 such that N ∈ N and S ∈ S. It has been proved by [Y] that N S is Serre. A well-known example for these subcategories is N A, the subcategory of minimax modules studied by [Z] where A is the subcategory of artinian modules. Another example is N F, the subcategory of FSF modules introduced by [Q], where F consists of all modules of finite support. When S = 0, an N S-a-cofinite module was known as an a-cofinite module which is defined for the first time by Hartshorne [H], giving a negative answer to a question of [G,Expos XIII,Conjecture 1.1]. Many people [BNS,M1,M2,NS] studied a-cofiniteness in various cases.
The main aim of this paper is to extend the fundamental results about a-cofinite modules at small dimensions to N S-a cofinite modules. We recall that a Serre subcategory S satisfies the condition C a if for every R-module M , the following implication holds. Assume that M is an R-module such that Supp R M ⊆ V (a). Melkersson [M1,Theorem 2.3] showed that if dim R/a = 1, then M is a-cofinite if and only if Hom R (R/a, M ) and Ext 1 R (R/a, M ) are finitely generated. The above theorem generalizes this result when R is a local ring and S(p) satisfies the condition pR p for each p ∈ V (a). Indeed we deduce that if dim R/a = 1, then M is N S-a-cofinite if and only if Hom R (R/a, M ) and Ext 1 R (R/a, M ) ∈ N S. Moreover, if R is a local ring of dimension 2 such that S(p) satisfies the condition C pRp for every prime ideal p of R with dim R/p ≤ 1, then M is N S-a-cofinite if and only if Hom R (R/a, M ) and Ext 1 R (R/a, M ) ∈ N S. Bahmanpour et all [BNS,Theorem 3.5] showed that if R is a local ring such that dim R/a = 2, then M is a-cofinite if and only if Ext i R (R/a, M ) are finitely generated for i = 0, 1, 2. As another conclusion, we generalize this result when S(p) satisfies the condition pR p for each p ∈ V (a). To be more prcise, we deduce that if dim R/a = 2, then M is N S-a-cofinite if and only if Ext i R (R/a, M ) ∈ N S for i = 0, 1, 2. Moreover, if R is a local ring of dimension 3 such that S(p) satisfies the condition pR p for every prime ideal p with dim R/p ≤ 2, then M is N S-a-cofinite if and only if Ext i R (R/a, M ) ∈ N S for i = 0, 1, 2. We prove the following result about local cohomology which generalizes [NS,Theorem 3.7]. For the basic properties and unexplained terminology of local cohomology, we refer the reader to the textbook by Brodmann and Sharp [BS].
Theorem 1.2. Let (R, m) be a local ring, let a be an ideal of R such that dim R/a = 2 and let S(p) satisfy the condition pR p for each p ∈ V (a). If n is a non-negative integer such that Ext i R (R/a, M ) ∈ N S for all i ≤ n + 1, then the following conditions are equivalent.
In Section 3, we define a new dimension of modules which is an upper bound of the dimension mentioned in Section 3. For every R-module M , we define dimM = dim R/ Ann R M . For every non-negative integer n, we denote by D ≤n , the subcategory of all R-modules M satisfying dimM ≤ n. It is proved that D ≤n is Serre for each n ≥ 0. We show that the subcategory a-cofinite modules in D ≤1 is Serre, the subcategory of a-cofinite modules in D ≤2 is abelian. Finally we show that the kernel and cokernel of a homomorphism f : M −→ N of a-cofinite modules in D ≤3 is a-cofinite if and only if (0 : Coker f a) is finitely generated.

Cofiniteness with respect to Extension subcategories
Let C be an abelian category and S be a subcategory of C. We denote by S sub (S quot ), the smallest subcategory of C containing S which is closed under subobjects (quotients). These subcategories can be specified as follows: (S) sub = {N ∈ C| N is a subobject object of an object of S}; (S) quot = {M ∈ C| M is a quotient object of an object of S.
Let T be another subcategory of C. We denote by ST , the extension subcategory of S and T which is: For any n ∈ N 0 , we set S 0 = {0} and S n = SS n−1 . In the case where S 2 = S, we say that S is closed under extension. We also define (S) ext = n≥0 S n the smallest subcategory of C containing S which is closed under extension.
A full subcategory S of C is called Serre if it is closed under taking subobjects, quotients and extensions. For any subcategory X of C, we denote by (X ) Serre , the smallest Serre subcategory of A containing X . Lemma 2.1. Let S be a subcategory of C. Then we have the following conditions. [K,Proposition 2.4]. (ii) Given N ∈ (S) ext , there exists a positive integer n such that N ∈ S n . We prove by induction on n that there exists a filtration 0 ⊂ N 1 ⊂ N 2 ⊂ · · · ⊂ N n = N of subobjects of N of length n such that N i /N i−1 ∈ S for each 1 ≤ i ≤ n. If n = 1, there is nothing to prove. Since N ∈ S n , there exists an exact sequence 0 −→ N 1 −→ N −→ N/N 1 −→ 0 such that N 1 ∈ S and N/N 1 ∈ S n−1 . By the induction hypothesis, there exists a finite filtration In the rest of this paper, a is an ideal of R, S is a Serre subcategory of R-modules and N is the subcategory of finitely generated modules in its corresponding module category (such as in the category of R-modules or in the category of R p -modules for some prime ideal p of R). We have the following lemma. An R-module M is said to be S-a-cofinite if Supp M ⊆ V (a) and Ext i R (R/a, M ) ∈ S for all integers i ≥ 0. An N -a-cofinite module is called a-cofinite. We recall that S satisfies the condition C a if for every R-module M , the following implication holds. Proof. See [SR,Proposition 2.4].
Lemma 2.4. Let N be a finitely generated R-module and M be an arbitrary R-module such that for a non-negative integer n, we have Ext i R (N, M ) ∈ S for each i ≤ n. Then Ext i R (L, M ) ∈ S for each finitely generated R-module L with Supp R L ⊆ Supp R N and each i ≤ n.
Proof. See [AS,Lemma 2.1] In this section for every R-module M , dim M means dimension of Supp R M which is the length of the longest chain of prime ideals in Supp R M .
for all i ≥ 0 and all finitely generated R-modules N with Supp R N ⊆ V (a) and dim N ≤ 1 such that S and satisfying the condition C Ann R N .
such that N has finite length and S ∈ S. Since MaxM ⊆ Supp S, we deduce that N ∈ S so that Hom R (R/b, M ) ∈ S and since S satisfies the condition C b , we have M ∈ S. This implies that , using a similar argument as mentioned above, Ext i R (R/b, M ) ∈ S for all i ≤ d; and hence it follows from [AM,Theorem 2.9] that H i b (M ) ∈ S for all i ≥ 0; and hence the assertion is clear in this case. If dim R/b = 1, it follows from [AS,Theorem 3.5 It thus follows from Lemma 2.2 and Lemma 2.4 that For every Serre subcategory S of R-modules and every p ∈ Spec R, we denote by S(p) the smallest Serre subcategory of R p -modules containing S p = {M p | M ∈ S}. We have the following lemma.
Lemma 2.6. Let p be an ideal of R. Then S p is closed under subobjects and quotients. In The second assertion follows from Lemma 2.1.
By virtue of Lemma 2.6, if (R, m) is a local ring, then we have S(m) = S. Lemma 2.7. Let p be a prime ideal of R and X ∈ S p . Then there exists an R-module S ∈ S such that S is an essential R-submodule of S p and X = S p .
Proof. Since X ∈ S p , there exists T ∈ S such that X = T p . If ϕ : T −→ T p is the canonical homomorphism, then S = T / ker ϕ is the desired module.
Proof. If dim N = 0, then Ann R N is m-primary. It follows from the assumption and Lemma 2.3 that N satisfies the condition C AnnR N ; and hence Proposition 2.5 implies that Ext i . We show that L ∈ N S. In view of the previous argument there exists a finitely generated submodule K of L such that L p /K p ∈ S(p). Consider the canonical morphism ϕ : which yields the following exact sequence We observe that p + xR is m-primary and according to the assumption and Lemma 2.3, S satisfies the condition C p+xR , and hence Proposition 2.5 implies that Ext i R (R/p + xR, M ) ∈ N S. Thus (0 : L x) ∈ N S. Since (L/K) p ∈ S(p), using Lemma 2.1, there exists a positive integer t such that (L/K) p = S t p . Without loss of generality, we may assume that t = 2, the other cases are similar. Then there exists an exact sequence of R p -modules Using Lemma 2.7 we may assume that S ′ and S ′′ are R-submodule of S ′ p and S ′′ p , respectively. Since L/L 1 is an essential R-submodule of (L/L 1 , the module F has finite length and hence F ∈ S as R/m ∈ S. This implies that Hom R (R/xR, L 1 /K) ∈ S and hence Hom R (R/m, L 1 /K) ∈ S. Since S satisfies the condition C m , we deduce that L 1 /K ∈ S so that L 1 ∈ N S and consequently L 2 ∈ N S. Consider the canonical homomorphism ϕ 1 : From the induced essential monomorphism L/L 3 −→ (L/L 2 ) p = (L/L 3 ) p and using a similar argument as mentioned above, we may assume that S ′′ is a submodule of L/L 3 and so there exists a submodule L 4 of L containing L 3 such that S ′′ = L 4 /L 3 and (L/L 4 ) p = 0. Therefore we have Supp R L/L 4 ⊆ V (m). The exact sequence 0 −→ L 2 −→ L 3 −→ L 3 /L 2 −→ 0, Lemma 2.2 and the fact that L 2 ∈ N S imply that Hom R (R/xR, L 3 /L 2 ) ∈ N S and so using a similar argument as mentioned above, we deduce that L 3 /L 2 ∈ S so that L 3 ∈ N S and hence L 4 ∈ N S. Now applying Hom R (R/xR, −) to the exact sequence 0 −→ L 4 −→ L −→ L/L 4 −→ 0 and using again a similar argument as mentioned before, we deduce that L ∈ N S.
The following corollary generalizes a result due to Melkersson [M2,Theorem 2.3].
Corollary 2.9. Let (R, m) be a local ring, let dim R/a ≤ 1 and let S(p) satisfy the condition pR p for each p ∈ V (a). If M is an R-module such that Supp R M ⊆ V (a) and Hom R (R/a, M ), Ext 1 R (R/a, M ) ∈ N S, then M is N S-a-cofinite. Proof. The assertion follows immediately from Theorem 2.8. Theorem 2.10. Let (R, m) be a local ring, let S(p) satisfy the condition pR p for each p ∈ V (a).
Proof. Similar to the proof of Theorem 2.8, we may assume that N = R/p for some p ∈ V (a) with dim R/p = 2 and Ext i Rp (R p /pR p , M p ) ∈ N S(p) for each i ≥ 0. Assume that i ≥ 0 and L = Ext i R (R/p, M ) and we show that L ∈ N S. There exists a finitely generated submodule K of L such that L p /K p ∈ S(p). Consider the canonical morphism ϕ L/K : L/K −→ (L/K) p with Ker ϕ L/K = L 1 /K where L 1 is a submodule of L. Clearly (L 1 /K) p = 0 and since p ⊆ Ann R L 1 /K and dim R/p = 2, every q ∈ Supp R L 1 /K with dim R/q = 1 is a minimal prime ideal of Ann R L 1 /K. Then the set T = {q ∈ Supp R L 1 /K| dim R/q = 1} is finite. Assume that T = {q 1 , . . . , q n }. Then p ∩ n j=1 q j and so there x ∈ ∩ n i=1 q i \ p. Then there is an exact sequence 0 −→ R/p which yields the following exact sequence −→ Ext i R (R/p, M ). Since dim R/p + xR ≤ 1, it follows from Theorem 2.8 that Ext i R (R/p + xR, M ) ∈ N S; and hence (0 : L x) ∈ N S. Now assume that L 2 /K = Ker ϕ L1/K where ϕ L1/K : L 1 /K −→ (L 1 /K) q1 . Then we have (L 2 /K) q1 = 0. Continuing this way, assume that . . q n , m} for each 1 ≤ i ≤ n. Since (0 : L x) ∈ N S, we have (0 : Ln+1 x) ∈ N S. Now applying the functor Hom R (R/xR, −) to the exact sequence 0 −→ K −→ L n+1 → L n+1 /K −→ 0, we deduce that Hom R (R/xR, L n+1 /K) ∈ N S and so there is an exact sequence of R-modules such that F is finitely generated and S ∈ S. Using the same argument as mentioned in the proof of Theorem 2.8, we deduce that L n+1 /K ∈ S so that L n+1 ∈ N S. Since Hom R (R/xR, L n ) ∈ N S, Hom R (R/xR, L n ) qn ∈ N S(q n ) so that Hom R (R/xR, L n /K) qn ∈ N S(q) and a similar proof as mentioned above gives (L n /K) qn = (L n /L n+1 ) qn ∈ S(q n ). To be convenient, set q = q n . in view of Lemma 2.6, we may assume that (L n /L n+1 ) q = S 2 q and the other cases are similar. Then there exists an exact sequence Using Lemma 2.7, we may assume that S ′ and S ′′ are R-submodules of S ′ p and S ′′ p , respectively. Since L n /L n+1 is an essential submodule of (L n /L n+1 ) q , the submodule S ′ ∩ L n /L n+1 is nonzero and (S ′ ∩ L n /L n+1 ) p = S ′ p ; and hence replacing S ′ by S ′ ∩ L n /L n+1 , we may assume that S ′ is a submodule of L n /L n+1 . Assume that S ′ = L ′ n /L n+1 for some submodule L ′ n of L n and so L ′ n ∈ N S. Thus (L n /L ′ n ) q = S ′′ q . Consider the canonical exact sequence 0 −→ L ′′ n /L ′ n −→ L n /L ′ n −→ (L n /L ′ n ) q which forces Supp R L ′′ n /L ′ n ⊆ V (m). Since Hom R (R/xR, L ′′ n ) ∈ N S and L ′ n ∈ N S, using a similar argument as above, we deduce that L ′′ n ∈ N S. The essential monomorphism L n /L ′′ n −→ (L n /L ′ n ) q and a similar argument as above imply that S ′′ is a submodule of L n /L ′′ n and so S ′′ = L ′′ /L ′′ n for some submodule L ′′ of L n . The fact that L ′′ n ∈ N S implies that L ′′ ∈ N S. Since (L n /L ′′ ) q = 0, we deduce that Supp R L n /L ′′ ⊆ V (m). A similar argument as above implies that L n /L ′′ ∈ S, and hence L n ∈ N S. Continuing this manner we deduce that L = L 1 ∈ N S. Therefore, we may assume that ϕ L/K is an essential monomorphism. According to Lemma 2.6, there exists a positive integer t such that (L/K) p ∈ (S p ) t . Set A = L/K. Without loss of generality, we may assume that t = 2 and so by a similar argument as mentioned above, there exists an exact sequence of R p -modules 0 −→ S ′ p −→ A p −→ S ′′ p −→ 0 such S ′ is a submodule of L/K and S ′′ is an R-submodule of A/S ′ and so ((A/S ′ )/S ′′ ) p = 0; and hence p / ∈ Supp R (A/S ′ )/S ′′ . Since p ⊆ Ann R (A/S ′ )/S ′′ and dim R/p = 2, every q ∈ Supp R (A/S ′ )/S ′′ with dim R/q = 1 is a minimal prime ideal of Ann R (A/S ′ )/S ′′ ; and hence the set is finite. Assume that U = {p 1 , . . . p m } and set D = (A/S ′ )/S ′′ . We notice that Supp R D = {p 1 , . . . , p m , m}. It follows from Proposition 2.5 that D pj ∈ N S(p j ) for each 1 ≤ j ≤ m. Then for each j, there exists an exact sequence such that N j is a finitely generated R-module and (D/N ) pj ∈ (S pj ) ext . In view of the preceding arguments, we may assume that S ∈ S pj so that there exists S j ∈ S such that S = S jp j . Furthermore, we may assume that S j is a submodule of D/N j and ((D/N j )/S j ) pj = 0. Then there exists a submodule X j of D such thst S j = X j /N j and so this implies that X j ∈ N S for each 1 ≤ j ≤ m. Now, set X = X 1 + · · ·+ X m and so clearly X ∈ N S and Supp D/X ⊆ V (m). We notice that there exists a submodule A ′ of A such that S ′′ = A ′ /S ′ and so A ′ ∈ S. On the other hand, there exists a submodule L ′ of L such that A ′ = L ′ /K which implies that L ′ ∈ N S and D = L/L ′ . Moreover, since X is a submodule of D, there exists a submodule L 1 of L containing L ′ such that X = L 1 /L ′ and hence D/X = L/L 1 . We observe by Lemma 2.2 that L 1 ∈ N SN S ⊆ N S. Considering the following exact sequence 0 −→ L 1 −→ L −→ D/X → 0, we have Hom R (R/xR, D/X) ∈ N S and using a similar proof as mentioned before, we deduce that L/L 1 = D/X ∈ S. Now the fact that L 1 ∈ N S forces L ∈ N SS ⊆ N S.
Corollary 2.11. Let (R, m) be a local ring, let dim R/a ≤ 2 and let S(p) satisfy the condition pR p for each p ∈ V (a). If M is an R-module such that Supp R M ⊆ V (a) and Ext i R (R/a, M ) ∈ N S for i = 0, 1, 2, then M is N S-a-cofinite.
Proof. The assertion follows immediately from Theorem 2.10.
The subcategory of all R-modules of finite support is denoted by F . It is clear that F is Serre and it satisfies the condtion C a for every ideal a of R.

Corollary 2.12. Let (R, m) be a local ring, let dim R/a = 2 and let M be an R-module such that
It is clear that F (p) satisfies the condition C pRp for every prime ideal p of R. Therefore, the assertion follows immediately from Theorem 2.10.
Theorem 2.13. Let (R, m) be a local ring, let dim R/a = 2 and let S(p) satisfy the condition pR p for each p ∈ V (a). If n is a non-negative integer such that Ext i R (R/a, M ) ∈ N S for all i ≤ n + 1, then the following conditions are equivalent.
If n = 1, the exact sequence ( † 1 ) and Corollary 2.11 imply that (i) and (ii) are equivalent. For n > 1, since Γ a (M ) ∈ N S, it follows from ( † i ) and the previous isomorphisms that Ext j R (R/a, Q) ∈ N S for all i ≤ n so that (i) and (ii) are equivalent for Q and non-negative integer n − 1. Now, using again the previous isomorphisms, the conditions (i) and (ii) are equivalent for M and non-negative integer n.
Corollary 2.14. Let R be an local ring with dim R/a ≤ 2, let n be a non-negative integer such that Ext i R (R/a, M ) ∈ N F for all i ≤ n + 1. Then the following conditions are equivalent.
M )) ∈ N F for all i ≤ n. Proof. Since F (p) satisfies the condition C pRp for every prime ideal p of R, the results is obtained by Theorem 2.13.
Given an arbitrary Serre subcategory S of R-modules, we say that R admits the condition P S n (a) if for every R-module M , the following implication holds: Theorem 2.15. Let R be a ring of dimension d ≥ 1 admitting the condition P S d−1 (a) for all ideals a of dimension d − 1 (i.e. dim R/a ≤ d − 1), then R admits P S d−1 (a) for all ideals a of R. Proof. Let M be an R-module and a be an arbitrary ideal of R such that Supp R M ⊆ V (a) and Ext i R (R/a, M ) ∈ N S for all i ≤ d− 1. If there exists some positive integer n such that a n = 0, then M = (0 : M a n ). On the other hand, if a = (a 1 , . . . , a t ), there is an exact sequence of R-modules a 1 x, . . . , a t x) . It is clear that a i (0 : M a n ) is a submodule of (0 : M a n−1 ) for each i and hence, since N S is Serre, an easy induction on n implies that (0 : M a n ) ∈ N S. Since V (a) = Spec R, the module M is N S-cofinite. Now, assume that a is not nilpotent and so there is a positive integer n such that Γ a (R) = (0 : R a n ). Taking R = R/Γ a (R) and M = M/(0 : M a n ), it is clear that M is an R-module and since Γ a (R) = 0, a contains an R-regular element so that dim R/a + Γ a (R) ≤ d− 1. We observe that Supp R M ⊆ V (a + Γ a (R)) and it follows from Lemma 2.4 that Ext i R (R/a + Γ a (R), M ) ∈ N S for all i ≤ d − 1. Thus the assumption implies that M is N S-a + Γ a (R)-cofinite. We now show that Ext i R (R/aR, M ) ∈ N S for each i ≥ 0. Consider the Grothendieck spectral sequence . Now, assume that i > 0 and the result has been proved for all values smaller than i. Then , M ); and hence it is in N S. Therefore Ext i R (R/aR, M ) ∈ N S for all i ≥ 0. Now consider the Grothendieck spectral sequence E p,q 2 := Ext p R (Tor R q (R, R/a), M ) ⇒ H p+q = Ext p+q R (R/a, M). Using again Lemma 2.4, we deduce E p,q 2 ∈ N S for all p, q ≥ 0. For any r > 2, the R-module E p,q r is a subquotient of E p,q r−1 and so an easy induction yields that E p,q r ∈ N S for all r ≥ 2 so that E p,q ∞ ∈ N S for all p, q ≥ 0 . For any 0 ≤ t ≤ n, there is a finite filtration ∈ N S for all 0 ≤ p ≤ t and t ≥ 0, we deduce that H t ∈ N S for all t ≥ 0; and hence M is N S-a-cofinite. On the other hand, since (0 : M a n ) ∈ N S, we conclude that M is N S-a-cofinite.
Corollary 2.16. Let R be a local ring of dimension 2 such that S satisfies the condition C a for every ideal a of R with dim R/a ≤ 1. Then R admits the condition P N S 1 (a) for every ideal a of R.
Proof. It follows from [AS,Theorem 3.2] and Corollary 2.9 that R admits the condition P N S 1 (a) for all ideals with dim R/a ≤ 1. Now, the result follows from Theorem 2.15.
Corollary 2.17. Let R be a local ring of dimension 2 such that S(p) satisfies the condition pR p for every prime ideal p with dim R/p ≤ 1. Then R admits the condition P N S 2 (a) for every ideal a of R.
Proof. It follows from Corollary 2.9 that R admits the condition P N S 1 (a) for all ideals a with dim R/a ≤ 1. Now, the result follows from Theorem 2.15.
Corollary 2.18. Let R be a local ring of dimension 3 such that S(p) satisfies the condition pR p for every prime ideal p with dim R/p ≤ 2. Then R admits the condition P N S 2 (a) for every ideal a of R.
Proof. It follows from Corollary 2.11 that R admits the condition P N S 2 (a) for all ideals with dim R/a ≤ 2. Now, the result follows from Theorem 2.15.
Corollary 2.19. Let R be a local ring of dimension 3. Then R admits the condition P N F 2 (a).
Proof. It follows from Corollary 2.12 that R admits the condition P N F 2 (a) for all ideals with dim R/a = 2. Now, the result follows from Theorem 2.15.

Cofiniteness with respect to a new dimension
For every R-module M , it is clear that Supp R M ⊆ V (Ann R M ) and for the case where M is finitely generated, they are equal. We define the upper dimension of M and we denote it by dimM which is dimM = dim R/ Ann R M . Clearly dim M ≤ dimM . We first recall some results which are needed in this section. For every non-negative integer n, we denote by D ≤n , the subcategory of all R-modules M such that dimM ≤ n. We also denote by G, the subcategory of all R-modules F such that V (Ann R F ) is a finite set.
Then there exists r ∈ Ann R N \ p and so for every x ∈ Ann R M/N , we have rx ∈ Ann R M ⊆ p which implies that x ∈ p. Consequently, p ∈ V (Ann R M/N ).
Corollary 3.5. The following conditions hold.
(i) For every non-negative integer n, the suncategory D ≤n is Serre.
(ii) The subcategory G is Serre.
Proof. The proof is is straightforward by Lemma 3.4. Proposition 3.8. The subcategory of a-cofinite modules in D ≤1 is Serre.
Proof. If dimM = 0, then dim M = 0 and so the module (0 : M a) has finite length. Thus it follows from Lemma 3.3 that M is a-cofinite. Now, assume that dimM = 1 and so dim R = 1 where R = R/ Ann R M . It is clear that (0 : M aR) = (0 : M a) is finitely generated and Supp R M ⊆ V (a). It follows from [M1,Proposition 4.5] that M is a-cofinite. Finally, using Lemma 3.1, M is a-cofinite. The second assertion is straightforward. Proof. Since M ∈ N F, there exists an exact sequence of R-modules 0 −→ N −→ M −→ F −→ 0 such that N is finitely generated and F has finite support. We notice that (0 : F a) is finite, and it suffices to show that F is a-cofinite and so we may assume that V (Ann R M ) is a finite set so that dimM ≤ 1. It follows from [M1,Proposition 4.5] that M is a-cofinite where R = R/ Ann R M and a = aR. Now Lemma 3.1 implies that M is a-cofinite.
Proposition 3.10. The subcategory of a-cofinite modules in D ≤2 is abelian.
Proof. Assume that f : M −→ N be a morphism of a-cofinite modules and assume that K = Ker f, I = Im f and C = Coker f . The assumption implies that (0 : I a) = (0 : I aR) is finitely generated R-module where R = R/ Ann R M . If dim R = 0, the module (0 : I a) has finite length and so I is Artinian. Now, Lemma 3.3 implies that I is a-cofinite. If dim R = 1, it follows from Proposition 3.8 that I is aR-cofinite as M is aR-cofinite; and hence I is a-cofinite using Lemma 3.1. If dim R = 2, it follows from [NS,Corollary 2.6] that I is aR-cofinite and so is a-cofinite by using Lemma 3.1. Now, using the exact sequences of R-modules it is straightforward to show that K and C are a-cofinite modules. Proof. By the assumption we have dim R/ Ann R M ≤ 3 and also dim R/ Ann R N ≤ 3. If we put b = Ann R M ∩ Ann R N and R = R/b, we have dim R ≤ 3 and further M and N are R/b-module. Clearly (0 : Coker f aR) is a finitely generated R-module. It follows from [NS,Theorem 2.8] that ker f and Coker f are aR-cofinite and so using Lemma 3.1, they are a-cofinite.