Proper Resolutions and Auslander-Type Conditions of Modules

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Dec 8, 2010 - It is well known that commutative Gorenstein rings are fundamental and ... commutative Noetherian ring R is a Gorenstein ring (that is, the ...
arXiv:1012.1703v1 [math.RA] 8 Dec 2010

Proper Resolutions and Auslander-Type Conditions of Modules ∗† Zhaoyong Huang‡ Department of Mathematics, Nanjing University, Nanjing 210093, Jiangsu Province, P. R. China

Abstract We obtain some methods to construct a (strongly) proper resolution (resp. coproper coresolution) of one end term in a short exact sequence from that of the other two terms. By using this method, we prove that for a left and right Noetherian ring R, R R satisfies the Auslander condition if and only if so does every flat left R-module, if and only if the injective dimension of the ith term in a minimal flat resolution of any injective left R-module is at most i − 1 for any i ≥ 1, if and only if the flat (resp. injective) dimension of the ith term in a minimal injective (resp. flat) resolution of any left R-module M is at most the flat (resp. injective) dimension of M plus i − 1 for any i ≥ 1, if and only if the flat (resp. injective) dimension of the injective envelope (resp. flat cover) of any left R-module M is at most the flat (resp. injective) dimension of M , and if and only if any of the opposite versions of the above conditions hold true. Furthermore, we prove that for an Artinian algebra R satisfying the Auslander condition, R is Gorenstein if and only if the subcategory consisting of finitely generated modules satisfying the Auslander condition is contravariantly finite. As applications, we get some equivalent characterizations of Auslander-Gorenstein rings and Auslander-regular rings.

1. Introduction It is well known that commutative Gorenstein rings are fundamental and important research objects in commutative algebra and algebraic geometry. Bass proved in [B2] that a commutative Noetherian ring R is a Gorenstein ring (that is, the self-injective dimension of R is finite) if and only if the flat dimension of the ith term in a minimal injective coresolution of R as an R-module is at most i − 1 for any i ≥ 1. In non-commutative case, Auslander proved that this condition is left-right symmetric ([FGR, Theorem 3.7]). In this case, R is said to satisfy the Auslander condition. Motivated by this philosophy, Huang and Iyama introduced the notion of certain Auslander-type conditions as follows. For any m, n ≥ 0, a left and right Noetherian ring is said to be Gn (m) if the flat dimension of the ith term ∗

2000 Mathematics Subject Classification: 16E65, 16G10, 16E10, 16E30. Keywords: (Strongly) proper resolutions, (Strongly) coproper coresolutions, Auslander-type conditions, Flat dimension, Injective dimension, Minimal flat resolutions, Minimal injective coresolutions, Gorenstein algebras, Contravariantly finite. ‡ E-mail address: [email protected]

1

in a minimal injective coresolution of RR is at most m + i − 1 for any 1 ≤ i ≤ n. The Auslander-type conditions are non-commutative analogs of commutative Gorenstein rings. Such conditions play a crucial role in homological algebra, representation theory of algebras and non-commutative algebraic geometry ([AR3], [AR4], [Bj], [EHIS], [FGR], [H], [HI], [IS], [I1], [I2], [I3], [I4], [M], [Ro], [S], [W], and so on). In particular, by constructing an injective coresolution of the last term in an exact sequence of finite length from that of the other terms, Miyachi obtained in [M] an equivalent characterization of the Auslander condition in terms of the relation between the flat dimensions of any module and its injective envelope. Then he got some properties of Auslander-Gorenstein rings and Auslander-regular rings. This paper is organized as follows. In Section 2, we give some terminology and some preliminary results. In Section 3, we introduce the notion of strongly proper (co)resolutions of modules, and then give a method to construct a (strongly) proper resolution (resp. coproper coresolution) of the first (resp. last) term in a short exact sequence from that of the other two terms. We will prove the following two theorems and their dual results. Theorem 1.1. Let C be a full subcategory of Mod R closed under finite direct sums and under kernels of epimorphisms, and let 0 → X → X0 → X1 → 0 be an exact sequence in Mod R. If · · · → Cij → · · · → C1j → C0j → X j → 0 is a (strongly) proper C -resolution of X j for j = 0, 1, then M M 1 C10 → C → X → 0 Ci0 → · · · → C21 · · · → Ci+1 is also a (strongly) proper C -resolution of X, and M C00 → C01 → 0 0 → C → C11 is exact. Theorem 1.2. Let C be a full subcategory of Mod R closed under finite direct sums, and let 0 → X1 → X0 → X → 0 be an exact sequence in Mod R and Cjn → · · · → Cj1 → Cj0 → Xj → 0 2

(1.1)

a (strongly) coproper C -coresolution of Xj for j = 0, 1. If (1.1) is (strongly) HomR (C , −)exact, then C0n

M

C1n−1 → · · · → C02

M

C11 → C01

M

C10 → C00 → X → 0

is also a (strongly) coproper C -coresolution of X. Let R be a left Noetherian ring and n, k ≥ 0, and let {Mi }i∈I be a family of left Rmodules and M = − lim →Mi , where I is a directed index set. By using some techniques of i∈I

direct limits and transfinite induction, we prove in Section 4 that if the flat dimension of the (n + 1)st term in a minimal injective coresolution of Mi is at most k for any i ∈ I, then the flat dimension of the (n + 1)st term in a minimal injective coresolution of M is also at most k. For any m, n ≥ 0, we introduce in Section 5 the notion of modules satisfying the Auslander-type conditions Gn (m); in particular, a left R-module M for any ring R is said to satisfy the Auslander condition if the flat dimension of the ith term in a minimal injective coresolution of R M is at most i−1 for any i ≥ 1. By using results obtained in the former sections, we will investigate the homological behavior of modules satisfying the Auslander-type conditions in terms of the relation between the flat (resp. injective) dimensions of modules and their injective envelopes (resp. flat covers). In particular, we get the following Theorem 1.3. Let R be a left and right Noetherian ring. Then the following statements are equivalent. (1) R R satisfies the Auslander condition. (2) Every flat left R-module satisfies the Auslander condition. (3) The flat dimension of the ith term in a minimal injective coresolution of any left R-module M is at most the flat dimension of M plus i − 1 for any i ≥ 1. (4) The flat dimension of the injective envelope of any left R-module M is at most the flat dimension of M . (5) The injective dimension of the ith term in a minimal flat resolution of any injective left R-module is at most i − 1 for any i ≥ 1. (6) The injective dimension of the ith term in a minimal flat resolution of any left Rmodule M is at most the injective dimension of M plus i − 1 for any i ≥ 1. (7) The injective dimension of the flat cover of any left R-module M is at most the injective dimension of M . (i)op The opposite version of (i) (1 ≤ i ≤ 7).

3

As applications of this theorem, we obtain some equivalent characterizations of AuslanderGorenstein rings and Auslander-regular rings, respectively. Note that a commutative Noetherian ring satisfies the Auslander condition if and only if it is Gorenstein ([B2]). Auslander and Reiten conjectured in [AR3] that an Artinian algebra satisfying the Auslander condition is Gorenstein. This conjecture is situated between the well known Nakayama conjecture and the finitistic dimension conjecture. The Nakayama conjecture states that an Artinian algebra R is selfinjective if all terms in a minimal injective coresolution of

RR

are projective; and the finitistic dimension conjecture states that the

supremum of the projective dimensions of all finitely generated left R-modules with finite projective dimension for an Artinian algebra R is finite. All of these conjectures remains still open. In Section 6, we first obtain the approximation presentations of a given module relative to the subcategory of modules satisfying the Auslander condition and that of modules with finite injective dimension respectively. Then we establish the connection between the Auslander and Reiten conjecture mentioned above with the contravariant finiteness of some certain subcategories as follows. Theorem 1.4. Let R be an Artinian algebra satisfying the Auslander condition. Then the following statements are equivalent. (1) R is Gorenstein. (2) The subcategory consisting of finitely generated modules satisfying the Auslander condition is contravariantly finite. (3) The subcategory consisting of finitely generated modules which are n-syzygy for any n ≥ 1 is contravariantly finite. As a consequence, we get that an Artinian algebra is Auslander-regular if and only if the subcategory consisting of projective modules and that consisting of modules satisfying the Auslander condition coincide.

2. Preliminaries Throughout this paper, R is an associative ring with identity, Mod R is the category of left R-modules and mod R is the category of finitely generated left R-modules. We use gl.dim R to denote the global dimension of R. In this section, we give some terminology and some preliminary results. Definition 2.1. ([E]) Let C ⊆ D be full subcategories of Mod R. The homomorphism f : C → D in Mod R with C ∈ C and D ∈ D is said to be a C -precover of D if for any ′





homomorphism g : C → D in Mod R with C ∈ C , there exists a homomorphism h : C → C 4

such that the following diagram commutes: C h

C

~~

~

~

f

~



g



/D

The homomorphism f : C → D is said to be right minimal if an endomorphism h : C → C is an automorphism whenever f = f h. A C -precover f : C → D is called a C -cover if f is right minimal. Dually, the notions of a C -preenvelope, a left minimal homomorphism and a C envelope are defined. Following Auslander and Reiten’s terminology in [AR1], for a module over an Artinian algebra, a C -(pre)cover and a C -(pre)envelope are called a (minimal) right C -approximation and a (minimal) left C -approximation, respectively. If each module in D has a right (resp. left) C -approximation, then C is called contravariantly finite (resp. covariantly finite) in D. Lemma 2.2. ([X, Theorem 1.2.9]) Let C be a full subcategory of Mod R closed under direct products. If fi : Ci → Mi is a C -precover of Mi in Mod R for any i ∈ I, where I is Q Q Q Q an index set, then i∈I fi : i∈I Ci → i∈I Mi is a C -precover of i∈I Mi . We use F 0 (Mod R) and I 0 (Mod R) to denote the subcategories of Mod R consisting of flat modules and injective modules, respectively. Recall that an F 0 (Mod R)-(pre)cover and an I 0 (Mod R)-(pre)envelope are called a flat (pre)cover and an injective (pre)envelope, respectively. Bican, El Bashir and Enochs proved in [BEE, Theorem 3] that every R-module has a flat π

π

π

π

i 2 1 0 · · · −→ F1 −→ cover. For an R-module M , we call an exact sequence · · · → Fi −→ F0 −→

M → 0 a proper flat resolution of M if πi : Fi → Im πi is a flat precover of Im πi for any i ≥ 0. Furthermore, we call the following exact sequence: πi (M )

π2 (M )

π1 (M )

π0 (M )

· · · → Fi (M ) −→ · · · −→ F1 (M ) −→ F0 (M ) −→ M → 0 a minimal flat resolution of M , where πi (M ) : Fi (M ) → Im πi (M ) is a flat precover of Im πi (M ) for any i ≥ 0. It is easy to verify that the flat dimension of M is at most n if and only if Fn+1 (M ) = 0. In addition, we use 0 → M → E 0 (M ) → E 1 (M ) → · · · → E i (M ) → · · · to denote a minimal injective coresolution of M . We denote by (−)+ = HomZ (−, Q/Z), where Z is the additive group of integers and Q is the additive group of rational numbers. 5

Lemma 2.3. ([EH, Theorem 3.7]) The following statements are equivalent. (1) R is a left Noetherian ring. (2) A monomorphism f : A ֌ E in Mod R is an injective preenvelope of A if and only if f + : E + ։ A+ is a flat precover of A+ in Mod Rop . Let M ∈ Mod R. We use fdR M , pdR M and idR M to denote the flat, projective and injective dimensions of M , respectively. Lemma 2.4. (1) ([F, Theorem 2.1]) For any M ∈ Mod R, fdR M = idRop M + . (2) ([F, Theorem 2.2]) If R is a right Noetherian ring, then fdR N + = idRop N for any N ∈ Mod Rop . Recall that Fin.dim R = sup{pdR M | M ∈ Mod R with pdR M < ∞}. Observe that the first assertion in the following result was proved by Bass in [B1, Corollary 5.5] when R is a commutative Noetherian ring. Lemma 2.5. (1) For a left Noetherian ring R, idR R ≥ sup{fdR M | M ∈ Mod R with fdR M < ∞}. (2) For a left and right Noetherian ring R, idR R ≥ sup{idRop N | N ∈ Mod Rop with idRop N < ∞}. Proof. (1) Without loss of generality, assume that idR R = n < ∞. Then Fin.dim R ≤ n by [B1, Proposition 4.3]. It follows from [J1, Proposition 6] that the projective dimension of any flat left R-module is finite. So, if M ∈ Mod R with fdR M < ∞, then pdR M < ∞ and pdR M ≤ n. Thus we have fdR M (≤ pdR M ) ≤ n. (2) By [B1, Proposition 4.1], we have sup{fdR M | M ∈ Mod R with fdR M < ∞} = sup{idRop N | N ∈ Mod Rop with idRop N < ∞}. So the assertion follows from (1).



3. The constructions of (strongly) proper resolutions and coproper coresolutions In this section, we introduce the notion of strongly (co)proper (co)resolutions of modules. Then we give a method to construct a (strongly) proper resolution (resp. coproper coresolution) of the first (resp. last) term in a short exact sequence from that of the other two terms, as well as give a method to construct a (strongly) proper resolution (resp. coproper coresolution) of the last (resp. first) term in a short exact sequence from that of the other two terms. We first give the following easy observation, which is a generalization of the horseshoe lemma. 6

f

g



′′

Lemma 3.1. Let 0 → A −→ A −→ A → 0 be an exact sequence in Mod R. ′′

′′

′′

′′



(1) If there exist homomorphisms α : C → A, α : C → A and h : C → A in Mod R ′′

such that α = gh, then we have the following commutative diagram with exact rows: 0

′′ ) (10C ) L (0,1 ′′ C / ′′ /C C C 

/C

α



0





α f

/A

/

/0

′′

α





g



A

′′

/

A

/0



where α = (f α, h). ′′

′′

′′



(2) If there exist homomorphisms β : A → D, β : A → D and k : A → D in Mod R such that β = kf , then we have the following commutative diagram with exact rows: 0

f

/A

/

β

0 ′

where β =

β 1D 0

g



A 



/ β



′′

A ′′

 ′′ ) ( ) L (0,1 ′′ D /D / ′′ D D

 /D

/0

/0

k  β ′′ g .

The following observation is useful in the rest of this section. Lemma 3.2. Let M

f1

g1

/N g



f



/Y X be a commutative diagram in Mod R and C ∈ Mod R.

(1) If this diagram is a pull-back diagram of f and g and HomR (C, g) is epic, then HomR (C, g1 ) is also epic. (2) If this diagram is a push-out diagram of f1 and g1 and HomR (g1 , C) is epic, then HomR (g, C) is also epic. Proof. Assume that the given diagram is a pull-back diagram of f and g and HomR (C, g) is epic.

Let α ∈ HomR (C, X).

HomR (C, g)(β) = gβ.

Then there exists β ∈ HomR (C, N ) such that f α =

By the universal property of a pull-back diagram, there exists

γ ∈ HomR (C, M ) such that α = g1 γ = HomR (C, g1 )(γ). So HomR (C, g1 ) is epic and the assertion (1) follows. Dually, we get the assertion (2).



Let C be a full subcategory of Mod R and M ∈ Mod R. Recall that a sequence in Mod R is called HomR (C , −)-exact exact if it is exact and remains still exact after applying the 7

functor HomR (C , −); and an exact sequence: · · · → Ci → · · · → C1 → C0 → M → 0 in Mod R with each Ci ∈ C is called a C -resolution of M . Avramov and Martsinkovsky called in [AM] the above exact sequence a proper C -resolution of M if it is a C -resolution of M and is HomR (C , −)-exact. Dually, the notions of a HomR (−, C )-exact exact sequence, a C -coresolution and a coproper C -coresolution of M are defined. We now introduce the notion of strongly (co)proper (co)resolutions of modules as follows. Definition 3.3. Let C be a full subcategory of Mod R and M ∈ Mod R. (1) A sequence: · · · → Xi → · · · → X1 → X0 → M → 0 in Mod R is called strongly HomR (C , −)-exact exact if it is exact and Ext1R (C , Ki ) = 0 for any i ≥ 1, where Ki = Im(Xi → Xi−1 ). Dually, the notion of a strongly HomR (−, C )-exact exact sequence is defined. (2) An exact sequence: · · · → Ci → · · · → C1 → C0 → M → 0 in Mod R is called a strongly proper C -resolution of M if it is a C -resolution of M and is strongly HomR (C , −)-exact. Dually, the notion of a strongly coproper C -coresolution of M is defined. It is easy to see that a strongly (co)proper C -(co)resolution is a (co)proper C -(co)resolution. But the converse does not hold true in general. For example, let C be a full subcategory of Mod R such that there exists a module M ∈ C with Ext1R (M, M ) 6= 0. Then the exact sequence: M (1M (0,1M ) 0 ) 0 → M −→ M M −→ M → 0 is both a proper C -resolution and a coproper C -coresolution of M , but it neither a strongly proper C -resolution nor a strongly coproper C -coresolution of M . The following result contains Theorem 1.1, which gives a method to construct a (strongly) proper resolution of the first term in a short exact sequence from that of the last two terms. Theorem 3.4. Let C be a full subcategory of Mod R and 0 → X → X 0 → X 1 → 0 an exact sequence in Mod R. Let · · · → Ci0 → · · · → C10 → C00 → X 0 → 0 8

(3.1)

be a C -resolution of X 0 , and let · · · → Ci1 → · · · → C11 → C01 → X 1 → 0

(3.2)

be a HomR (C , −)-exact exact sequence in Mod R. Then (1) We get the following exact sequences: 1 · · · → Ci+1

M

Ci0 → · · · → C21

M

C10 → C → X → 0

(3.3)

and 0 → C → C11

M

C00 → C01 → 0

(3.4)

Assume that C is closed under finite direct sums and under kernels of epimorphisms. Then we have (2) If the exact sequence (3.2) is a C -resolution of X 1 , then the exact sequence (3.3) is a C -resolution of X. (3) If both the exact sequences (3.1) and (3.2) are strongly proper C -resolutions of X 0 and X 1 respectively, then the exact sequence (3.3) is a strongly proper C -resolution of X. (4) If both the exact sequences (3.1) and (3.2) are proper C -resolutions of X 0 and X 1 respectively, then the exact sequence (3.3) is a proper C -resolution of X. 0 ) and K 1 = Im(C 1 → C 1 ) for any i ≥ 1. Consider Proof. (1) Put Ki0 = Im(Ci0 → Ci−1 i i i−1

the following pull-back diagram: 0

0

 

K11

K11

0

/X

 /M

 / C1 0

/0

0

/X

 / X0

 / X1

/0

 

0

0

Because the third column in the above diagram is HomR (C , −)-exact exact, so is the middle column by Lemma 3.2(1). Thus by Lemma 3.1(1) we get the following commutative diagram

9

with exact columns and rows and the middle row splitting: 0

0

0

 

 _ _ _ _/ W1 _ _ _ _/ K 0 _ _ _/ 0 1      L 0 / C0 / C1 / C1 /0 C 0 1 1  0      /M /0 / K1 / X0 1       

0 _ _ _/

0

0

K21

0 where W1 = Ker(C11

L

0

0

C00 → M ). It is easy to verify the upper row in the above diagram is

HomR (C , −)-exact exact. On the one hand, we have the following pull-back diagram: 0

0 



W1

W1

0

 /C

0

 /X

/ C1 1

 L

C00

 /M





0

0

/ C1 0

/0

/ C1 0

/0

On the other hand, again by Lemma 3.1(1) we get the following commutative diagram with exact columns and rows and the middle row splitting:

10

0

0

0 

   1 _ _ _ / _ _ _ _ / W 2 _ _ _ _/ K20 _ _ _/ 0 K3 0     L 1 1 / /0 / C0 / C2 C2 C10 0 1       1 / K0 / / /0 W 1 K2 0 1     

0 where W2 =

Ker(C21

L

C10

0

0

→ W1 ) and the upper row in the above diagram is HomR (C , −)-

exact exact. Continuing this process, we get the desired exact sequences (3.3) and (3.4) with L L 0 L 0 1 ) for any i ≥ 2 and W1 = Im(C21 C10 → C). Ci → Ci1 Ci−1 Wi = Im(Ci+1 (2) It follows from the assumption and the assertion (1). (3) If both the exact sequences (3.1) and (3.2) are strongly proper C -resolutions of X 0 and X 1 respectively, then Ext1R (C , Kij ) = 0 for any i ≥ 1 and j = 0, 1. By the proof of (1), we have an exact sequence: 1 0 → Ki+1 → Wi → Ki0 → 0

for any i ≥ 1. So Ext1R (C , Wi ) = 0 for any i ≥ 1, and hence the exact sequence (3.3) is a strongly proper C -resolution of X. (4) Assume that both the exact sequences (3.1) and (3.2) are proper C -resolutions of X0

and X 1 respectively. Then by the proof of (1) and [EJ, Lemma 8.2.1], we have that

both the middle column in the second diagram and the first column in the third diagram are HomR (C , −)-exact exact; and in particular we have a HomR (C , −)-exact exact sequence: 1 · · · → Ci+1

M

Ci0 → · · · → C21

M

C10 → W1 → 0.

Thus we get the desired proper C -resolution of X.



Based on Theorem 3.4, by using induction on n it is not difficult to get the following Corollary 3.5. Let C be a full subcategory of Mod R closed under finite direct sums and under kernels of epimorphisms, and let 0 → X → X 0 → X 1 → · · · → X n → 0 be an exact sequence in Mod R. If · · · → Cij → · · · → C1j → C0j → X j → 0 11

is a (strongly) proper C -resolution of X j for any 0 ≤ j ≤ n, then ··· →

n M

i Ci+3 →

n M

i Ci+2 →

i Ci+1 →C→X →0

i=0

i=0

i=0

n M

is a (strongly) proper C -resolution of X, and there exists an exact sequence: 0→C→

n M

Cii →

n M

i Ci−1 →

i Ci−2 → · · · → C0n−1

M

C1n → C0n → 0.

i=2

i=1

i=0

n M

Remark. 3.6. By Wakamatsu’s lemma (see [X, Lemma 2.1.1]), if the full subcategory C is closed under extensions, then a minimal proper C -resolution of a module M is a strongly proper C -resolution of M . Note that any projective resolution is just a strongly proper P 0 (Mod R)-resolution, where P 0 (Mod R) = {projective left R-modules}. So putting C = P 0 (Mod R) in Corollary 3.5, we get the following Corollary 3.7. Let 0 → X → X 0 → X 1 → · · · → X n → 0 be an exact sequence in Mod R. If · · · → Pij → · · · → P1j → P0j → X j → 0 is a projective resolution of X j for any 0 ≤ j ≤ n, then ··· →

n M

i Pi+3 →

n M

i Pi+2 →

i Pi+1 →C→X →0

i=0

i=0

i=0

n M

is a projective resolution of X, and there exists an exact and split sequence: 0→C→

n M i=0

Pii



n M i=1

i Pi−1



n M

i Pi−2 → · · · → P0n−1

M

P1n → P0n → 0.

i=2

The following 3.8–3.11 are dual to 3.4–3.7 respectively. The following result gives a method to construct a (strongly) coproper coresolution of the last term in a short exact sequence from that of the first two terms. Theorem 3.8. Let C be a full subcategory of Mod R and 0 → Y1 → Y0 → Y → 0 an exact sequence in Mod R. Let 0 → Y0 → C00 → C01 → · · · → C0i → · · ·

12

(3.5)

be a C -coresolution of Y0 , and let 0 → Y1 → C10 → C11 → · · · → C1i → · · ·

(3.6)

be a HomR (−, C )-exact exact sequence in Mod R. (1) We get the following exact sequences: 0 → Y → C → C01

M

C12 → · · · → C0i

M

C1i+1 → · · ·

(3.7)

and 0 → C10 → C00

M

C11 → C → 0

(3.8)

Assume that C is closed under finite direct sums and under cokernels of monomorphisms. Then we have (2) If the exact sequence (3.6) is a C -coresolution of Y1 , then the exact sequence (3.7) is a C -coresolution of Y . (3) If both the exact sequences (3.5) and (3.6) are strongly coproper C -coresolutions of Y0 and Y1 respectively, then the exact sequence (3.7) is a strongly coproper C -coresolution of Y. (4) If both the exact sequences (3.5) and (3.6) are coproper C -coresolutions of Y0 and Y1 respectively, then the exact sequence (3.7) is a coproper C -coresolution of Y . Proof. It is dual to the proof of Theorem 3.4, we give the proof here for the sake of completeness. Put K0i = Im(C0i−1 → C0i ) and K1i = Im(C1i−1 → C1i ) for any i ≥ 1. Consider the following push-out diagram: 0

0

0

 / Y1

 / Y0

/Y

/0

0

 / C0 1

 /N

/Y

/0





K11

K11





0

0

Because the first column in the above diagram is HomR (−, C )-exact exact, so is the middle column by Lemma 3.2(2). Then by Lemma 3.1(2) we get the following commutative diagram 13

with exact columns and rows and the middle row splitting: 0

0

0



0

 / Y0

0

 / C0 0

 /N    / C0 0

L 

C11

 / K1 1

/0

 / C1 1

/0

 



0 _ _ _/

K01 

0 L 0

where W 1 = Coker(N → C0

  _ _ _ _/ W 1 _ _ _ _/ K 2 _ _ _/ 0 1     

0

0

C11 ). It is easy to verify that the bottom row in the above

diagram is HomR (−, C )-exact exact. On the one hand, we have the following push-out diagram:

0

/ C0 1

0

/ C0 1

/ C0 0

0

0

 /N

 /Y

/0

 /C

/0

 L

C11





W1

W1





0

0

On the other hand, again by Lemma 3.1(2) we get the following commutative diagram with

14

exact columns and rows and the middle row splitting: 0

0

0

 



0

0

/

K01

/ W1   

 / C1 0

/ C1 0

L 

C12

 / K2 1

/0

 / C2 1

/0

 



0 _ _ _/

  _ _ _ _/ W 2 _ _ _ _/ K 3 _ _ _/ 0 1     

K02 

0 L 1

where W 2 = Coker(W 1 → C0

0

0

C12 ) and the bottom row in the above diagram is HomR (−, C )-

exact exact. Continuing this process, we get the desired exact sequences (3.7) and (3.8) with L L i+1 L C1 ) for any i ≥ 2 and W 1 = Im(C → C01 C12 ). W i = Im(C0i−1 C1i → C0i (2) It follows from the assumption and the assertion (1). (3) If both the exact sequences (3.5) and (3.6) are strongly coproper C -coresolutions of Y0 and Y1 respectively, then Ext1R (Kji , C ) = 0 for any i ≥ 1 and j = 0, 1. By the proof of (1), we have an exact sequence: 0 → K0i → W i → K1i+1 → 0 for any i ≥ 1. So Ext1R (W i , C ) = 0 for any i ≥ 1, and hence the exact sequence (3.7) is a strongly coproper C -coresolution of Y . (4) Assume that both the exact sequences (3.5) and (3.6) are coproper C -coresolutions of X0 and X1 respectively. Then by the proof of (1) and the dual version of [EJ, Lemma 8.2.1], we have that both the middle column in the second diagram and the first column in the third diagram are HomR (−, C )-exact exact; and in particular we have a HomR (−, C )-exact exact sequence: 0 → W 1 → C01

M

C12 → · · · → C0i

M

C1i+1 → · · · .

Thus we get the desired coproper C -coresolution of X.



Based on Theorem 3.8, by using induction on n it is not difficult to get the following Corollary 3.9. Let C be a full subcategory of Mod R closed under finite direct sums and under cokernels of monomorphisms and let 0 → Yn → · · · → Y1 → Y0 → Y → 0 be an exact

15

sequence in Mod R. If 0 → Yj → Cj0 → Cj1 → · · · Cji → · · · is a (strongly) coproper C -coresolution of Yj for any 0 ≤ j ≤ n, then 0→Y →C →

n M

n M

Cii+1 →

n M

Cii+2 →

i=0

i=0

i=0

Cii+3 → · · ·

is a (strongly) coproper C -coresolution of Y , and there exists an exact sequence: 0→

Cn0



0 Cn−1

M

Cn1

→ ··· →

n M

Cii−2



n M

Cii−1



Cii → C → 0.

i=0

i=1

i=2

n M

Remark. 3.10. By Wakamatsu’s lemma (see [X, Lemma 2.1.1]), if the full subcategory C is closed under extensions, then a minimal coproper C -coresolution of a module M is a strongly coproper C -coresolution of M . Note that any injective coresolution is just a strongly coproper I 0 (Mod R)-coresolution. So putting C = I 0 (Mod R) in Corollary 3.9, we get the following Corollary 3.11. ([M, Corollary 1.3]) Let 0 → Yn → · · · → Y1 → Y0 → Y → 0 be an exact sequence in Mod R. If 0 → Yj → Ij0 → Ij1 → · · · Iji → · · · is an injective coresolution of Yj for any 0 ≤ j ≤ n, then 0→Y →C→

n M

Iii+1



n M

Iii+2



n M i=0

i=0

i=0

Iii+3 → · · ·

is an injective coresolution of Y , and there exists an exact and split sequence: 0 0 → In0 → In−1

M

In1 → · · · →

n M

Iii−2 →

i=2

n M

Iii−1 →

i=1

n M

Iii → C → 0.

i=0

The following result contains Theorem 1.2, which gives a method to construct a (strongly) proper resolution of the last term in a short exact sequence from that of the first two terms. Theorem 3.12. Let C be a full subcategory of Mod R and 0 → X1 → X0 → X → 0

16

(3.9)

an exact sequence in Mod R. Let C0n → · · · → C01 → C00 → X0 → 0

(3.10)

be a HomR (C , −)-exact exact sequence, and C1n−1 → · · · → C11 → C10 → X1 → 0

(3.11)

a C -resolution of X1 in Mod R. (1) We get the following exact sequences: M M M C10 → C00 → X → 0 C11 → C01 C1n−1 → · · · → C02 C0n

(3.12)

Assume that C is closed under finite direct sums. Then we have (2) If the exact sequence (3.10) is a C -resolution of X0 , then the exact sequence (3.12) is a C -resolution of X. (3) If the exact sequence (3.9) is strongly HomR (C , −)-exact and both the exact sequences (3.10) and (3.11) are strongly proper C -resolutions of X0 and X1 respectively, then the exact sequence (3.12) is a strongly proper C -resolution of X. (4) If the exact sequence (3.9) is HomR (C , −)-exact and both the exact sequences (3.10) and (3.11) are proper C -resolutions of X0 and X1 respectively, then the exact sequence (3.12) is a proper C -resolution of X. Proof. (1) Put Kji = Im(Cji → Cji−1 ) for any 1 ≤ i ≤ n − j and j = 0, 1. Consider the following pull-back diagram: 0

0

 

K01

K01

0

 / W1

 / C0 0

/X

/0

0

 / X1

 / X0

/X

/0





0

0

Note that the middle column in the above diagram is HomR (C , −)-exact exact. So by Lemma 3.2(1), the first column is also HomR (C , −)-exact exact. Then by Lemma 3.1(1) we get the following commutative diagram with exact columns and rows and the middle row splitting: 17

0

0

0 

   2 _ _ _ / _ _ _ _ / W 2 _ _ _ _/ K11 _ _ _/ 0 K0 0     L 1 1 / /0 / C0 / C0 C0 C10 0 1       1 / X1 / / /0 W 1 K0 0     

0 where W2 =

Ker(C01

L

C10

0

0

→ W1 ). It is easy to check that the upper row in the above

diagram is HomR (C , −)-exact exact. Then by using Lemma 3.1(1) iteratively we get the L L i−1 exact sequence (3.12) with Wi = Im(C0i C1 → C0i−1 C1i−2 ) for any 2 ≤ i ≤ n and L W1 = Im(C01 C10 → C00 ). (2) It follows from the assumption and the assertion (1). (3) If the exact sequence (3.9) is strongly HomR (C , −)-exact and both the exact sequences (3.10) and (3.11) are strongly proper C -resolutions of X0 and X1 respectively, then Ext1R (C , Kji ) = 0 for any 1 ≤ i ≤ n − j and j = 0, 1. By the proof of (1), we have an exact sequence: 0 → K0i → Wi → K1i−1 → 0 for any 1 ≤ i ≤ n (where K10 = X1 ). So Ext1R (C , Wi ) = 0 for any 1 ≤ i ≤ n, and hence the exact sequence (3.12) is a strongly proper C -resolution of X. (4) If the exact sequence (3.9) is HomR (C , −)-exact and both the exact sequences (3.10) and (3.11) are proper C -resolutions of X0 and X1 respectively, then the middle row in the first diagram is HomR (C , −)-exact exact by Lemma 3.2(1). Thus by [EJ, Lemma 8.2.1], the exact sequence (3.12) is a proper C -resolution of X.



Based on Theorem 3.12, by using induction on n it is not difficult to get the following Corollary 3.13. Let C be a full subcategory of Mod R closed under finite direct sums, and let Xn → · · · → X1 → X0 → X → 0

(3.13)

and Cjn−j → · · · → Cj1 → Cj0 → Xj → 0 18

(3.14(j))

be exact sequences in Mod R for any 0 ≤ j ≤ n. (1) If the exact sequence (3.13) is strongly HomR (C , −)-exact, and if (3.14(j)) is a strongly proper C -resolution of Xj for any 0 ≤ j ≤ n, then n M

Cin−i



n−1 M

(n−1)−i

Ci

→ · · · → C01

M

C10 → C00 → X → 0

(3.15)

i=0

i=0

is a strongly proper C -resolution of X. (2) If the exact sequence (3.13) is HomR (C , −)-exact, and if (3.14(j)) is a proper C resolution of Xj for any 0 ≤ j ≤ n, then (3.15) is a proper C -resolution of X. The following corollary is an immediate consequence of Corollary 3.13. Corollary 3.14. Let Xn → · · · → X1 → X0 → X → 0 be an exact sequence in Mod R. If Pjn−j → · · · → Pj1 → Pj0 → Xj → 0 is a projective resolution of Xj for any 0 ≤ j ≤ n, then n M i=0

Pin−i →

n−1 M

(n−1)−i

Pi

→ · · · → P01

M

P10 → P00 → X → 0

i=0

is a projective resolution of X. The following 3.15–3.17 are dual to 3.12–3.14 respectively. The following result gives a method to construct a strongly coproper coresolution of the first term in a short exact sequence from that of the last two terms. Theorem 3.15. Let C be a full subcategory of Mod R and 0→Y →Y0 →Y1 →0

(3.16)

0 → Y 0 → C00 → C10 → · · · → Cn0

(3.17)

an exact sequence in Mod R. Let

be a HomR (−, C )-exact exact sequence, and 1 0 → Y 1 → C01 → C11 → · · · → Cn−1

(3.18)

a C -coresolution of Y 1 in Mod R. (1) We get the following exact sequences: 0 → Y → C00 → C01

M

C10 → C11 19

M

1 C20 → · · · → Cn−1

M

Cn0

(3.19)

Assume that C is closed under finite direct sums. Then we have (2) If the exact sequence (3.17) is a C -coresolution of Y 0 , then the exact sequence (3.19) is a C -coresolution of X. (3) If the exact sequence (3.16) is strongly HomR (−, C )-exact and both the exact sequences (3.17) and (3.18) are strongly coproper C -coresolutions of Y 0 and Y 1 respectively, then the exact sequence (3.19) is a strongly coproper C -coresolution of Y . (4) If the exact sequence (3.16) is HomR (−, C )-exact and both the exact sequences (3.17) and (3.18) are coproper C -coresolutions of Y 0 and Y 1 respectively, then the exact sequence (3.19) is a coproper C -coresolution of Y . Proof. It is dual to the proof of Theorem 3.12, we give the proof here for the sake of j → Cij ) for any 1 ≤ i ≤ n − j and j = 0, 1. Consider the completeness. Put Kij = Im(Ci−1

following push-out diagram: 0

0

0

/Y

 /Y0

 /Y1

/0

0

/Y

 / C0 0

 / W1

/0





K10

K10

 

0

0

Note that the middle column in the above diagram is HomR (−, C )-exact exact by assumption. So the third column is also HomR (−, C )-exact exact by Lemma 3.2(2). Then by Lemma 3.1(2) we get the following commutative diagram with exact columns and rows and

20

the middle row splitting: 0

0

0



0

 /Y1

0

 / C1 0

  / W1    / C1 0

L 

C10

 / K0 1

/0

 / C0 1

/0

   1 _ _ _ _/ 0 _ _ _/ _ _ _ / _ _ _ _ / 2 K K 0 0 W 1 2     

0 L 1

where W 2 = Coker(W 1 → C0

0

0

C10 ). It is easy to verify that the bottom row in the above

diagram is HomR (−, C )-exact exact. Then by using Lemma 3.1(2) iteratively we get the L 0 L 0 1 1 Ci ) for any 2 ≤ i ≤ n and Ci−1 → Ci−1 exact sequence (3.19) with W i = Im(Ci−2 L 0 1 0 1 C1 ). W = Im(C0 → C0 (2) It follows from the assumption and the assertion (1). (3) If the exact sequence (3.16) is strongly HomR (−, C )-exact and both the exact sequences (3.17) and (3.18) are strongly coproper C -coresolutions of Y 0 and Y 1 respectively, then Ext1R (Kij , C ) = 0 for any 1 ≤ i ≤ n − j and j = 0, 1. By the proof of (1), we have an exact sequence: 1 0 → Ki−1 → W i → Ki0 → 0

for any 1 ≤ i ≤ n (where K01 = Y 1 ). So Ext1R (W i , C ) = 0 for any 1 ≤ i ≤ n, and hence the exact sequence (3.19) is a strongly coproper C -coresolution of Y . (4) If the exact sequence (3.16) is HomR (−, C )-exact and both the exact sequences (3.17) and (3.18) are coproper C -coresolutions of Y 0 and Y 1 respectively, then the middle row in the first diagram is HomR (−, C )-exact exact by Lemma 3.2(2). Thus by the dual version of [EJ, Lemma 8.2.1], the exact sequence (3.19) is a coproper C -coresolution of Y .



Based on Theorem 3.15, by using induction on n it is not difficult to get the following Corollary 3.16. Let C be a full subcategory of Mod R closed under finite direct sums, and let 0 → Y → Y 0 → Y 1 → ··· → Y n

(3.20)

j 0 → Y j → C0j → C1j → · · · Cn−j

(3.21(j))

and

21

be exact sequences in Mod R for any 0 ≤ j ≤ n. (1) If the exact sequence (3.20) is strongly HomR (−, C )-exact, and if (3.21(j)) is a strongly coproper C -coresolution of Y j for any 0 ≤ j ≤ n, then 0→Y →

C00



C10

M

C01 · · ·



n−1 M

i C(n−1)−i



n M

i Cn−i

(3.22)

i=0

i=0

is a strongly coproper C -coresolution of Y . (2) If the exact sequence (3.20) is HomR (−, C )-exact, and if (3.21(j)) is a coproper C -coresolution of Y j for any 0 ≤ j ≤ n, then (3.22) is a coproper C -coresolution of Y . The following corollary is an immediate consequence of Corollary 3.16. Corollary 3.17. Let 0 → Y → Y 0 → Y 1 → · · · → Y n be an exact sequence in Mod R. If j 0 → Y j → I0j → I1j → · · · In−j

is an injective coresolution of Y j for any 0 ≤ j ≤ n, then 0 → Y → I00 → I10

M

I01 · · · →

n−1 M i=0

i I(n−1)−i →

n M

i In−i

i=0

is an injective coresolution of Y .

4. Flat dimension of En of direct limits In this section, R is a left Noetherian ring. The aim of this section is to prove the following Theorem 4.1. Let n, k ≥ 0 and let {Mi }i∈I be a family of left R-modules, where I is a n n directed index set. If M = lim −→Mi and fdR E (Mi ) ≤ k for any i ∈ I, then fdR E (M ) ≤ k. i∈I

By [R, Theorem 5.40], every flat left R-module is a direct limit (over a directed index set) of finitely generated free left R-modules. So by Theorem 4.1, we have the following Corollary 4.2. fdR E n (R R) = sup{fdR E n (F ) | F ∈ Mod R is flat} for any n ≥ 0. Before giving the proof of Theorem 4.1, we need some preliminaries. Definition 4.3. ([J2]) Let β be an ordinal number. A set S is called a continuous union of a family of subsets indexed by ordinals α with α < β if for each such α we have a subset ′

Sα ⊂ S such that if α ≤ α then Sα ⊂ Sα′ , and such that if γ < β is a limit ordinal then S Sγ = α