Torsionfree Dimension of Modules and Self-Injective Dimension of Rings

0.15 MB·English

by  Chonghui Huang

Torsionfree Dimension of Modules and Self-Injective
Dimension of Rings∗†
Chonghui Huang1‡, Zhaoyong Huang2§
9
0 1College of Mathematics and Physics, University of South China, Hengyang 421001, P.R. China;
0 2Department of Mathematics, Nanjing University, Nanjing 210093, P.R. China.
2

n
u
J
Abstract

6
LetR be a left andrightNoetherianring. We introducethe notionofthe torsionfree

] dimension of finitely generated R-modules. For any n≥0, we prove that R is a Goren-
A
stein ring with self-injective dimension at most n if and only if every finitely generated
R leftR-moduleandeveryfinitely generatedrightR-modulehavetorsionfreedimensionat
. most n, if and only if every finitely generated left (or right) R-module has Gorenstein
h
t dimension at most n. For any n ≥ 1, we study the properties of the finitely generated
a i
R-modules M with Ext (M,R)=0 for any 1≤i≤n. Then we investigate the relation
m R
between these properties and the self-injective dimension of R.
[

1
v
1. Introduction
3
5
2 Throughoutthispaper,RisaleftandrightNoetherianring(unlessstatedotherwise)and
1
modR isthecategory offinitelygenerated leftR-modules. ForamoduleM ∈ modR, weuse
.
6
0 pdRM, fdRM, idRM to denote the projective, flat, injective dimension of M, respectively.
9
0 For any n≥ 1, we denote ⊥nRR = {M ∈ modR | ExtiR(RM,RR) = 0 for any 1≤ i ≤ n}
v: (resp. ⊥nR = {N ∈ modRop | Exti (N ,R ) = 0 for any 1 ≤ i ≤ n}), and ⊥ R =
R Rop R R R
i
X T ⊥n R (resp. ⊥R = T ⊥nR ).
n≥1 R R n≥1 R
r
a For any M ∈ modR, there exists an exact sequence:
f
P1 −→ P0 → M → 0
in modR with P0 and P1 projective. Then we get an exact sequence:
0 → M∗ → P∗ −f→∗ P∗ → TrM → 0
0 1
in modRop, where (−)∗ = Hom(−,R) and TrM = Cokerf∗ is the transpose of M ([AB]).
∗2000 Mathematics Subject Classification: 16E10, 16E05.
†Keywords: Gorenstein rings, n-torsionfree modules, torsionfree dimension, Gorenstein dimension, self-
injective dimension, thetorsionless property.
‡E-mail address: [email protected]
§E-mail address: [email protected]
1
Auslander and Bridger generalized the notions of finitely generated projective modules
and the projective dimension of finitely generated modules as follows.
Definition 1.1 ([AB]) Let M ∈modR.
(1) M is said to have Gorenstein dimension zero if M ∈ ⊥ R and TrM ∈ ⊥R .
R R
(2)Foranon-negativeintegern,theGorensteindimensionofM,denotedbyG−dim M,
R
isdefinedasinf{n≥ 0|thereexistsanexactsequence0→ Mn → ··· → M1 → M0 → M → 0
in modR with all M having Gorenstein dimension zero}. We set G−dim M infinity if no
i R
such integer exists.
Huang introduced in [Hu2] the notion of the left orthogonal dimension of modules as
follows, which is “simpler” than that of the Gorenstein dimension of modules.
Definition 1.2 ([Hu2]) For a module M ∈ modR, the left orthogonal dimension of a
module M ∈ modR, denoted by ⊥ R−dim M, is defined as inf{n ≥ 0 | there exists an
R R
exact sequence 0 → Xn → ··· → X1 → X0 → M → 0 in modR with all Xi ∈ ⊥RR}. We set
⊥ R−dim M infinity if no such integer exists.
R R
Let M ∈ modR. It is trivial that ⊥ R−dim M ≤ G−dim M. On the other hand,
R R R
by [Jo], we have that ⊥ R−dim M 6= G−dim M in general.
R R R
Recall that R is called a Gorenstein ring if idRR = idRopR < ∞. The following result
was proved by Auslander and Bridger in [AB, Theorem 4.20] when R is a commutative
Noetherian local ring. Hoshino developed in [H1] Auslander and Bridger’s arguments and
applied obtained the obtained results to Artinian algebras. ThenHuang generalized in [Hu2,
Corollary 3] Hoshino’s result with the left orthogonal dimension replacing the Gorenstein
dimension of modules.
Theorem 1.3 ([H1, Theorem] and [Hu2, Corollary 3]) The following statements are
equivalent for an Artinian algebra R.
(1) R is Gorenstein.
(2) Every module in modR has finite Gorenstein dimension.
(3) Every module in modR and every module in modRop have finite left orthogonal
dimension.
One aim of this paper is to generalize this result to left and right Noetherian rings. On
the other hand, note that the left orthogonal dimension of modules is defined by the least
length of the resolution composed of the modules in ⊥ R, which are the modules satisfying
R
one of the two conditions in the definition of modules having Gorenstein dimension zero.
2
So, a natural question is: If a new dimension of modules is defined by the least length of
the resolution composed of the modules satisfying the other condition in the definition of
modules having Gorenstein dimension zero, then can one give an equivalent characterization
of Gorenstein rings similar to the above result in terms of the new dimension of modules?
The other aim of this paper is to give a positive answer to this question. This paper is
organized as follows.
InSection2,wegivethedefinitionofn-torsionfreemodules,andinvestigatetheproperties
of such modules. We prove that a module in modR is n-torsionfree if and only if it is an
n-syzygy of a module in ⊥n R.
R
In Section 3, we introduce the notion of the torsionfree dimension of modules. Then we
give some equivalent characterizations of Gorenstein rings in terms of the properties of the
torsionfree dimension of modules. The following is the main result in this paper.
Theorem 1.4 For any n ≥ 0, the following statements are equivalent.
(1) R is a Gorenstein ring with idRR = idRopR ≤ n.
(2) Every module in modR has Gorenstein dimension at most n.
(3) Every module in modRop has Gorenstein dimension at most n.
(4) Every module in modR and every module in modRop have torsionfree dimension at
most n.
(5) Every module in modR and every module in modRop have left orthogonal dimension
at most n.
In Section 4, for any n ≥ 1, we first prove that every module in ⊥n R is torsionless
R
(in this case, ⊥nRR is said to have the torsionless property) if and only if every module in
⊥n R is ∞-torsionfree, if and only if every module in ⊥n R has torsionfree dimension at
R R
most n, if and only if every n-torsionfree module in modR is ∞-torsionfree, if and only if
every n-torsionfree module in modRop is in ⊥R , if and only if ⊥nR = ⊥R . Note that if
R R R
idRopR ≤ n, then ⊥nRR has the torsionless property. As some applications of the obtained
results, we investigate when the converse of this assertion holds true. Assume that n and
k are positive integers and ⊥n R has the torsionless property. If R is g (k) or g (k)op (see
R n n
Section 4 for the definitions), then idRopR ≤ n + k − 1. As a corollary, we have that if
idRR ≤ n, then idRR = idRopR ≤ n if and only if ⊥nRR has the torsionless property.
In view of the results obtained in this paper, we pose in Section 5 the following two
questions: (1) Is the subcategory of modR consisting of modules with torsionfree dimension
at most n closed under extensions or under kernels of epimorphisms? (2) If idRopR ≤ n,
does then every module M ∈ modR has torsionfree dimension at most n?
3
2. Preliminaries
Let M ∈ modR and n ≥ 1. Recall from [AB] that M is called n-torsionfree if TrM ∈
⊥nR ; andM iscalled ∞-torsionfree if M is n-torsionfreeforall n. We useT (modR)(resp.
R n
T(modR)) to denote the subcategory of modR consisting of all n-torsionfree modules (resp.
∞-torsionfree modules). It is well-known that M is 1-torsionfree (resp. 2-torsionfree) if
and only if M is torsionless (resp. reflexive) (see [AB]). Also recall from [AB] that M is
called an n-syzygy module (of A), denoted by Ωn(A), if there exists an exact sequence
0 → M → Pn−1 → ··· → P1 → P0 → A → 0 in modR with all Pi projective. In particular,
set Ω0(M) = M. We use Ωn(modR) to denote the subcategory of modR consisting of
all n-syzygy modules. It is easy to see that T (modR) ⊆ Ωn(modR), and in general, this
n
inclusion is strict when n ≥ 2 (see [AB]).
Jans proved in [J, Corollary 1.3] that a module in modR is 1-torsionfree if and only if it
is an 1-syzygy of a module in ⊥1 R. We generalize this result as follows.
R
Proposition 2.1 For any n ≥ 1, a module in modR is n-torsionfree if and only if it is
an n-syzygy of a module in ⊥n R.
R
Proof. Assume that M ∈ modR is an n-syzygy of a module A in ⊥n R. Then there
R
exists an exact sequence:
f
0 → M → Pn−1 → ··· → P1 −→ P0 → A→ 0
in modR with all P projective. Let
i
Pn+1 → Pn → M → 0
be a projective presentation of M in modR. Then the above two exact sequences yield the
following exact sequence:
0→ A∗ → P∗ −f→∗ ··· → P∗ → P∗ → TrM → 0.
0 n n+1
By the exactness of Pn+1 → Pn → ··· → P1 −f→ P0, we get that TrM ∈ ⊥nRR. Thus M is
n-torsionfree.
Conversely, assume that M ∈modR is n-torsionfree and
g π
P1 −→ P0 −→ M → 0
is a projective presentation of M ∈ modR. Then we get an exact sequence:
0→ M∗ −π→∗ P∗ −g→∗ P∗ → TrM → 0
0 1
4
in modRop. Let
··· h−n→+1 Qn −h→n ··· −h→1 Q0 −h→0 M∗ → 0
be a projective resolution of M∗ in modRop. Then we get a projective resolution of TrM:
··· h−n→+1 Qn −h→n ··· −h→1 Q0 π−∗→h0 P0∗ −g→∗ P1∗ → TrM → 0.
Because M is n-torsionfree, TrM ∈ ⊥nR and we get the following exact sequence:
R
0 → (TrM)∗ → P∗∗ −g→∗∗ P∗∗ h−∗0→π∗∗ Q∗ −h→∗1 ··· h−∗n→−1 Q∗ → Cokerh∗ → 0.
1 0 0 n−1 n−1
It is easy to see that M ∼= Cokerg∗∗. By the exactness of Qn−1 h−n→−1 ··· −h→1 Q0 π−∗→h0 P0∗ −g→∗
P∗, we get that Cokerh∗ ∈ ⊥n R. The proof is finished. (cid:3)
1 n−1 R
As an immediate consequence, we have the following
Corollary 2.2 For any n ≥ 1, an n-torsionfree module in modR is a 1-syzygy of an
(n − 1)-torsionfree module A in modR with A ∈ ⊥1 R. In particular, an ∞-torsionfree
R
module in modR is a 1-syzygy of an ∞-torsionfree module T in modR with T ∈ ⊥1 R.
R
We also need the following easy observation.
Lemma 2.3 For any n ≥ 1, both T (modR) and T(modR) are closed under direct
n
summands and finite direct sums.
3. Torsionfree dimension of modules
In this section, we will introduce the notion of the torsionfree dimension of modules in
modR. Then we will give some equivalent characterizations of Gorenstein rings in terms of
the properties of this dimension of modules.
We begin with the following well-known observation.
f
Lemma 3.1 ([AB, Lemma 3.9]) Let 0 → A −→ B → C → 0 be an exact sequence
in modR. Then we have exact sequences 0 → C∗ → B∗ → A∗ → Cokerf∗ → 0 and
0 → Cokerf∗ → TrC → TrB → TrA→ 0 in modRop.
The following result is useful in this section.
Proposition 3.2 Let
f
0 → M → T1 −→ T0 → A → 0
be an exact sequence in modR with both T0 and T1 in T(modR). Then there exists an exact
sequence:
0 → M → P → T → A → 0
5
in modR with P projective and T ∈ T(modR).
Proof. Let
f
0 → M → T1 −→ T0 → A → 0
be an exact sequence in modR with both T0 and T1 in T(modR). By Corollary 2.2, there
exists an exact sequence 0 → T1 → P → W → 0 in modR with P projective and W ∈
⊥1 RTT(modR). Then we have the following push-out diagram:
R
0 0
(cid:15)(cid:15) (cid:15)(cid:15)
0 //M //T1 //Imf //0
(cid:15)(cid:15) (cid:15)(cid:15)
// // // //
0 M P B 0
(cid:15)(cid:15) (cid:15)(cid:15)
W W
(cid:15)(cid:15) (cid:15)(cid:15)
0 0
Now, consider the following push-out diagram:
0 0
(cid:15)(cid:15) (cid:15)(cid:15)
0 //Imf //T0 //A //0
(cid:15)(cid:15) (cid:15)(cid:15)
// // // //
0 B T A 0
(cid:15)(cid:15) (cid:15)(cid:15)
W W
(cid:15)(cid:15) (cid:15)(cid:15)
0 0
Because W ∈ ⊥1 R, we get an exact sequence:
R
0 → TrW → TrT → TrT0 → 0
by Lemma 3.1 and the exactness of the middle column in the above diagram. Because both
W and T0 are in T(modR), both TrW and TrT0 are in ⊥RR. So TrT is also in ⊥RR and
hence T ∈ T(modR). Connecting the middle rows in the above two diagrams, then we get
the desired exact sequence. (cid:3)
6
Now we introduce the notion of the torsionfree dimension of modules as follows.
Definition 3.3 For a module M ∈ modR, the torsionfree dimension of M, denoted by
T −dimRM, is defined as inf{n≥ 0 | there exists an exact sequence 0 → Xn → ··· → X1 →
X0 → M → 0 in modR with all Xi ∈ T(modR)}. We set T −dimRM infinity if no such
integer exists.
Let M ∈ modR. It is trivial that T −dim M ≤ G−dim M. On the other hand, by
R R
[Jo], we have that T −dim M 6= G−dim M in general.
R R
Proposition 3.4 Let M ∈ modR and n ≥ 0. If T −dim M ≤ n, then there exists an
R
exact sequence 0 → H → T → M → 0 in modR with pd H ≤n−1 and T ∈ T(modR).
R
Proof. We proceed by induction on n. If n =0, then H = 0 and T = M give the desired
exact sequence. If n = 1, then there exists an exact sequence:
0 → T1 → T0 → M → 0
in modR with both T0 and T1 in T ∈ T(modR). Applying Proposition 3.2, with A = 0, we
get an exact sequence:
0→ P → T′ → M → 0
0
in modR with P projective and T′ ∈ T(modR).
0
Now suppose n ≥ 2. Then there exists an exact sequence:
0→ Tn → Tn−1 → ··· → T0 → M → 0
in modR with all Ti ∈ T(modR). Set K = Im(T1 → T0). By the induction hypothesis, we
get the following exact sequence:
0→ Pn → Pn−1 → Pn−2 → ··· → P2 → T1′ → K → 0
in modR with all Pi projective and T1′ ∈ T(modR). Set N = Im(P2 → T1′). By Proposition
3.2, we get an exact sequence:
0→ N → P1 → T → M → 0
in modR with P1 projective and T ∈ T(modR). Thus we get the desired exact sequence:
0→ Pn → Pn−1 → Pn−2 → ··· → P1 → T → M → 0
and the assertion follows. (cid:3)
7
Christensen,FrankildandHolmprovedin[CFH,Lemma2.17]thatamodulewithGoren-
stein dimensionatmostncanbeembeddedintoamodulewithprojective dimensionatmost
n, such that the cokernel is a module with Gorenstein dimension zero. The following result
extends this result.
Corollary 3.5Let M ∈ modR and n ≥ 0. If T −dim M ≤ n, then there exists an exact
R
sequences 0→ M → N → T → 0 in modR with pd N ≤ n and T ∈ ⊥1 RTT(modR).
R R
Proof. Let M ∈ modR with T −dim M ≤ n. By Proposition 3.4, there exists an exact
R
sequence 0 → H → T′ → M → 0 in modR with pd H ≤ n − 1 and T′ ∈ T(modR).
R
By Corollary 2.2, there exists an exact sequence 0 → T′ → P → T → 0 in modR with P
projective and T ∈ ⊥1 RTT(modR). Consider the following push-out diagram:
R
0 0
(cid:15)(cid:15) (cid:15)(cid:15)
0 // H //T′ //M //0
(cid:15)(cid:15) (cid:15)(cid:15)
// // // //
0 H P N 0
(cid:15)(cid:15) (cid:15)(cid:15)
T T
(cid:15)(cid:15) (cid:15)(cid:15)
0 0
Then the third column in the above diagram is as desired. (cid:3)
The following result plays a crucial role in proving the main result in this paper.
Theorem 3.6 For any n ≥ 0, if every module in modR has torsionfree dimension at
most n, then idRopR ≤ n.
To prove this theorem, we need some lemmas. We use ModR to denote the category of
left R-modules.
Lemma 3.7 ([I, Proposition 1]) idRopR = sup{fdRE | E is an injective module in
ModR}= fd Q for any injective cogenerator Q for ModR.
R
Lemma 3.8 For any n ≥ 0, idRopR ≤ n if and only if every module in modR can be
embedded into a module in ModR with flat dimension at most n.
Proof. Assume that idRopR ≤ n. Then the injective envelope of any module in modR
has flat dimension at most n by Lemma 3.7, and the necessity follows.
8
Conversely, let E be any injective module in ModR. Then by [R, Exercise 2.32], E =
limM , where {M | i ∈ I} is the set of all finitely generated submodules of M and I is a
i i
→
i∈I
directed index set. By assumption, for any i ∈ I, we have that M can be embedded into a
i
module N ∈ ModR with fd N ≤ n.
i R i
By [R, Exercise 2.32] again, M = limM , where {M | j ∈ I } is the set of all finitely
i → ij ij i
j∈Ii
generated submodules of M and I is a directed index set for any i ∈ I. Notice that each
i i
M ∈ {M |i ∈ I}, soeach correspondingN ∈ {N |i ∈ I}. PutK = ⊕ N foranyi ∈I.
ij i ij i i ij
j∈Ii
Then {K ,I} is a direct system of modules with flat dimension at most n, and there exists a
i
monomorphism 0 → M → K for any i∈ I. It follows from [R, Theorem 2.18] that we get a
i i
monomorphism 0 → E(= limM ) → limK . Because the functor Tor commutes with lim by
i i
→ → →
i∈I i∈I i∈I
[R, Theorem 8.11], fdRlimKi ≤ n. So fdRE ≤ n and hence idRopR ≤ n by Lemma 3.7. (cid:3)
→
i∈I
Proof of Theorem 3.6. By assumption and Corollary 3.5, we have that every module in
modR can beembeddedinto a modulein modR with projective dimension at most n. Then
by Lemma 3.8, we get the assertion. (cid:3)
Lemma 3.9 For any M ∈ modR and n ≥ 0, ⊥ R − dim M ≤ n if and only if
R R
Extn+i(M,R) = 0 for any i≥ 1.
R
Proof. For any M ∈ modR, consider the following exact sequence:
··· → Wn → Wn−1 → ··· → W0 → M → 0
inmodRwithallWiin⊥RR. ThenwehavethatExtiR(Im(Wn → Wn−1),R) ∼= ExtRn+i(M,R)
for any i ≥ 1. So Im(Wn → Wn−1) ∈ ⊥RR if and only if ExtRn+i(M,R) = 0 for any i ≥ 1,
and hence the assertion follows. (cid:3)
Proposition 3.10 For any n≥ 0, every module in modR has left orthogonal dimension
at most n if and only if id R ≤ n.
R
Proof. By Lemma 3.9, we have that id R ≤ n if and only if Extn+i(M,R) = 0 for any
R R
M ∈ modR and i ≥ 1, if and only if ⊥ R−dim M ≤ n for any M ∈ modR. (cid:3)
R R
Proof of Theorem 1.4. (1) ⇒ (2)+(3) follows from [HuT, Theorem 3.5].
(2) ⇒ (1)LetM beanymoduleinmodR. Thenbyassumption,wehavethatG−dim M
R
≤ n and T −dimRM ≤ n. So idRopR ≤ n by Theorem 3.6. On the other hand, because
⊥ R−dim M ≤ G−dim M, id R ≤ n by Proposition 3.10.
R R R R
Symmetrically, we get (3) ⇒ (1).
9
(4) ⇒ (1) By Theorem 3.6 and its symmetric version.
(2)+(3) ⇒ (4) Because T −dimRM ≤ G−dimRM and T −dimRopN ≤ G−dimRopN
for any M ∈ modR and N ∈ modRop, the assertion follows.
(1) ⇔ (5) By Proposition 3.10 and its symmetric version. (cid:3)
4. The torsionless property and self-injective dimension
The following result plays a crucial role in this section, which generalizes [H1, Lemma 4],
[Hu3, Lemma 2.1] and [J, Theorem 5.1].
Proposition and Definition 4.1 For any n ≥ 1, the following statements are equiva-
lent.
(1) ⊥nRR ⊆ T1(modR). In this case, ⊥nRR is said to have the torsionless property.
(2) ⊥n R ⊆ T(modR).
R
(3) Every module in ⊥n R has torsionfree dimension at most n.
R
(4) T (modR)= T(modR).
n
(5) T (modRop) ⊆ ⊥R .
n R
(6) ⊥nR = ⊥R .
R R
Proof. (2) ⇒ (1) and (2) ⇒ (3) are trivial, and (1) ⇔ (6) follows from [Hu3, Lemma
2.1]. Note that M and TrTrM are projectively equivalent for any M ∈ modR or modRop.
Then it is not difficult to verify (2) ⇔ (5) and (4) ⇔ (6). So it suffices to prove (1) ⇒ (2)
and (3) ⇒ (2).
(1) ⇒ (2) Assume that M ∈ ⊥n R. Then M is torsionless by (1). So, by Proposition
R
2.1, we have an exact sequence 0 → M → P0 → M1 → 0 in modR with P0 projective and
M1 ∈ ⊥1RR, which yields that M1 ∈ ⊥n+1RR. Then M1 is torsionless by (1), and again by
Proposition 2.1, we have an exact sequence 0 → M1 → P1 → M2 → 0 in modR with P1
projective and M2 ∈ ⊥1RR, which yields that M1 ∈ ⊥n+2RR. Repeating this procedure, we
get an exact sequence:
0 → M → P0 → P1 → ··· → Pi → ···
in modR with all Pi projective and Im(Pi → Pi+1) ∈ ⊥n+i+1RR ⊆ ⊥i+1RR, which implies
that M is ∞-torsionfree by Proposition 2.1.
(3) ⇒ (2) Assume that M ∈ ⊥n R. Then T − dim M ≤ n by assumption. By
R R
Proposition 3.4, there exists an exact sequence:
0 → H → T → M → 0 (1)
10