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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Vespri, Vincenzo / Marano, Salvatore Angelo


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Volume 15, Issue 1

Issues

Volume 13 (2015)

Simple modules over Auslander regular rings

Chonghui Huang / Lijing Zheng
Published Online: 2017-12-29 | DOI: https://doi.org/10.1515/math-2017-0138

Abstract

In this paper, we discuss the properties of simple modules over Auslander regular rings with global dimension at most 3. Using grade theory, we show the right projective dimension of ExtΛ1(S,Λ) is equal to 1 for any simple Λ-module S with gr S = 1. As a result, we give some equivalent characterization of diagonal Auslander regular rings.

Keywords: Auslander regular ring; Diagonal ring; Simple module; Grade

MSC 2010: 16E05; 16E10

1 Introduction

Recall from [1, Theorem 3.7] that an Artin algebra Λ is called to be n-Gorenstein if l.pd Ii(Λ) ≤ i holds for any integer 0 ≤ in−1.Λ is an Auslander-regular ring if Λ is n-Gorenstein for any integer n ≥ 0 and gl. dim Λ < ∞. It is known to all that n-Gorenstein rings admit left-right symmetric properties and this kind of rings play an important role in homological algebras and representation theory of algebras as well as many other fields (see [2, 3, 4, 5, 6, 7, 8, 9] among others). In [9], 0. Iyama introduced a kind of special n-Gorenstein rings–diagonal Auslander regular rings, that is, Auslander regular ring Γ is said to be diagonal if any non-zero direct summand I of Ii(Γ) satisfies the left flat dimension of Ii(Γ) = i. These rings are closely related to the representation theory of partially ordered sets, vector space categories, and orders over complete discrete valuation rings (See [10-11]).

Simple modules are important research objects and many properties of rings are determined by them. So, it is nice to study properties of simple modules over Auslander regular rings. Moreover, motivated by the work of 0. Iyama, we use grade theory to study the properties of diagonal Auslander regular rings. In section 2, we study the properties of simple modules over Auslander regular ring with gl. dimΛ ≤ 3. As an application, we give some equivalent characterization of diagonal Auslander regular rings. Our results will be helpful to understand Auslander regular rings better.

Throughout this paper, Λ is an artin algebra and all the modules we considered are finitely generated. We use mod Λ(resp.mod Λop) to denote the category of finitely generated Λ-modules(resp. Λop-modules). We put ()*: = HomΛ (, Λ) and for any integer n ≥ 1 En:=ExtΛn(,Λ):modΛmodΛop,Fn:=soc ExtΛn(,Λ):mod ΛmodΛop.

We use the following symbols:

gl.dim Λ:the global dimension of Λ
l. pd M:the projective dimension of a left module M
r.pd M:the projective dimension of a right module M
Ii(M):the (i + 1)-th term in a minimal injective resolution of a module M
gr M:the grade of M, defined by inf{i0|ExtΛi(M,Λ)0}
s.gr M:the strong grade of M, defined by inf{gr N for any NM}

2 Properties of simple modules and diagonal Auslander regular ring

The following results are useful in our statements, for the sake of the readability, we list them as lemmas.

Lemma 2.1

([12, Lemma 1]). Let R be any ring. M is a R-module and 0 → MI0(M) → I1(M) → ⋯ a minimal injective resolution for M. Then, for a simple R-module S and n > 0, ExtRn(S,M)HomR(S,In(M)).

Lemma 2.2

([12, Proposition 7]). Let R be a Gorenstein ring with self-injective dimension n. Then for a simple left R-module S, if S is a submodule of In(R), S never appears in soc(Ii(R)) for all in.

Lemma 2.3

([8, Theorem 1.2]). Let Λ be an n-Gorenstein noetherian algebra and 0 ≤ l ≤ n−2. Then 𝓔l gives a duality between Cl(Λ) and Cl(Λop), where Cl(Λ) is the subcategory consisting of all module X = 𝓔l(Y) for some Y ∈ mod Λop with gr Yl, and 𝓔l ∘ 𝓔l is isomorphic to the identity functor.

Lemma 2.4

([8, Theorem 1.3]). Let Λ be an n-Gorenstein noetherian algebra and 0 ≤ ln. Then 𝓕l gives a duality between simple Λ-modules X with gr X = l and that of Λop, and 𝓕l ∘ 𝓕l is isomorphic to the identity functor. Moreover, gr 𝓔l(X)/𝓕l(X) > l holds.

For any A ∈ mod Λ, there exists a projective presentation in mod Λ: P1fP0A0.

Then we get an exact sequence 0AP0fP1Coker f0

in mod Λop. Recall from [13] that Coker f* is called a transpose of A, and denoted by Tr A. We remark that the transpose of A depends on the choice of the projective presentation of A, but it is unique up to projective equivalence (see [13]). By [13, Proposition 2.6], we have the following two useful exact sequences: 0ExtΛop1(Tr A,Λ)AσAAExtΛop2(Tr A,Λ)0(1) 0ExtΛ1(A,Λ)Tr AσTr A(Tr A)ExtΛ2(A,Λ)0(2)

The following result plays an important role in this paper.

Proposition 2.5

Let Λ be an Auslander regular ring with gl.dim Λ ≤ 3 and S be a simple Λ-module with gr S = 1. Then r. pd 𝓔1(S) = gr 𝓔1(S) = 1.

Proof

Let ⋯ → P1P0 → 𝓔1(S) → 0 be a projective resolution of 𝓔1(S) in mod Λop and L = Im(P1P0). Then by [1, Theorem 3.7], gr 𝓔1(S) ≥ 1 and we get an exact sequence 0P0LE1E1(S)0 in mod Λ. Talking (, Λ), we have the following exact sequence in mod Λop: 0LP0E1E1E1(S)E1(L)0.

Put H = Coker (L**P0). We get the following commutative diagram with exact rows: 0LP0E1(S)0,σLσP0h0LP0H0

where h is induced by the natural morphisms σ L and σ P0. It is easy to see the morphism h is an epimorphism. Moreover, there is an exact sequence 0 → 𝓔2 (Tr L) → 𝓔1(S) → H → 0 by snake lemma and the sequence (1).

Denote Sr = 𝓕1(S), we claim that 𝓔2 (Tr L) = 0. By Lemma 2.4, 𝓕1 is a duality between simple Λ-module S with gr S = 1 and that of Λop. So we have gr Sr = 1. If 𝓔2 (Tr L) ≠ 0, Sr will be a submodule of 𝓔2 (Tr L) and hence gr Sr ≥ 2 by [1, Theorem 3.7] and [9, Lemma 2.3(2)], a contradiction will arise. So we have LL** and H ≅ 𝓔1(S).

By Lemma 2.3, we get 𝓔1(S) ≅ 𝓔1𝓔1𝓔1(S). This derives that 𝓔3 (Tr L) ≅ 𝓔1(L*) = 0 by the exactness of the sequence 0 → H → 𝓔1𝓔1𝓔1(S) → 𝓔1(L*) → 0. Then the facts 𝓔i (Tr L) = 0 for any i ≥ 1 and gl. dim Λ ≤ 3 yield that Tr L is projective. So we have L is projective and r.pd 𝓔1(S) ≤ 1. It follows that r.pd 𝓔1(S) = gr𝓔1(S) = 1. □

We have the following useful result.

Corollary 2.6

Let Λ be an Auslander regular ring with gl. dim Λ ≤ 3 and S be a simple Λ-module with gr S = 1. Then 1. pd 𝓔1𝓔1(S) = gr 𝓔1𝓔1(S) = 1.

Proof

By Proposition 2.5, we get r.pd 𝓔1(S) = gr 𝓔1(S) = 1. Let 0 → Q1Q0 → 𝓔1(S) → 0 be any projective resolution of 𝓔1(S). Taking (, Λ), we have a projective resolution of 𝓔1𝓔1(S) in mod Λ: 0 → Q0Q1 𝓔1𝓔1(S) → 0. Because gr 𝓔1(S) = 1, we get that l.pd 𝓔1𝓔1(S) = 1. Moreover, we have gr 𝓔1𝓔1(S) ≥ 1 by [1, Theorem 3.7]. So we are done. □

Let S be any simple module with gr S = 1. Lemma 2.4 shows that the socle of 𝓔1(S) is simple, denoted by Sr, and there is an exact sequence 0 → Sr → 𝓔1(S) → N → 0 with gr N ≥ 2. On the other hand, we always have 0 → SS** → 𝓔2 (Tr S) → 0 for any S with gr S = 0. It is easy to see that gr 𝓔2 (Tr S) ≥ gr S+2. Then, it is natural to ask: does this hold true for any simple module with gr S = 1? The following result shows the relationship between S and 𝓔1𝓔1(S). That is, we have

Theorem 2.7

Let Λ be an Auslander regular ring with gl. dim Λ ≤ 3 and S be a simple Λ-module with gr S = 1. Then there exists K ∈ mod Λ with gr K ≥ 3 such that 0 → S → 𝓔1𝓔1(S) → K → 0 is exact.

Proof

Put Sr = 𝓕1(S) and N = 𝓔1(Sr)/𝓕1(Sr). By Lemma 2.4, we have gr Sr = 1 and gr N ≥ 2. Applying (−, Λ) on the exact sequence 0 → S iS 𝓕1(Sr) → N → 0, since gr N ≥ 2, we get the following exact sequence in mod Λop: 0E1E1(Sr)E1(S)E2(N)0.

Taking (−, Λ) again, we get an exact sequence 0 → 𝓔1𝓔1(S) → 𝓔1𝓔1𝓔1(Sr) → 𝓔2𝓔2(N) → 0 in mod Λ by the fact gr 𝓔2(N) ≥ 2. By Lemma 2.3, 𝓔1(S) ≅ 𝓔1𝓔1𝓔1(S). Suppose f and g are the isomorphisms 𝓔1(S) → 𝓔1𝓔1𝓔1(S) and 𝓔1𝓔1(S) → Ker(𝓔1𝓔1𝓔1(Sr) → 𝓔2𝓔2(N)), respectively. Then we have the following commutative diagram with exact rows: 0SE1(Sr)N0ufv0E1E1(S)E1E1E1(Sr)E2E2(N)0,

where u = g−1fiS and v is the induced morphism. Put K = Ker v, then we have exact sequences in mod Λ: 0SE1E1(S)K0()0ΛNE2E2(N)0()

by snake lemma. By [9, Lemma 2.3(2)] and the exactness of sequence (**), we have gr K ≥ 2. Applying (−, Λ) on the exact sequence (*), we get the following long exact sequence in mod Λop: 0E1(K)E1E1E1(S)E1(S)E2(K)E2E1E1(S).

By Corollary 2.6, we have l.pd 𝓔1𝓔1(S) = 1 and, therefore, 𝓔2𝓔1𝓔1(S) = 0. So the facts 𝓔1(K) = 0 and 𝓔1𝓔1𝓔1(S)≅ 𝓔1(S) yield that 𝓔2(K) = 0. That is, gr K ≥ 3.□

Now we fix our sight on diagonal rings. We have the following

Theorem 2.8

Let Λ be an Auslander regular ring. Then the following statements are equivalent.

  1. Λ is diagonal.

  2. Any simple Λ-module S with gr S = k satisfies 1. pd S = k for any k ≥ 0.

  3. Any simple Λ-module S with 1. pd S = k satisfies S≅ 𝓔, 𝓔, (S) for any k ≥ 0.

Proof

(1)(2) is trivial by the definition of diagonal ring and Lemma 2.1. We need to show (1) ⇔ (3).

(1) ⇒ (3) Suppose Λ is diagonal. Let S be any simple Λ-module with l.pd S = k. Then one gets that 𝓔k(S) ≠ 0. By Lemma 2.1, S can be embedded into Ik(Λ). So, S will never appear in Ii(Λ) for any ik. Applying Lemma 2.1 again, we get gr S = k. That is, S≅ 𝓔k𝓔k(S).

(3) ⇒ (1) Suppose any simple Λ-module S with 1. pd S = k satisfies S ≅ 𝓔k𝓔k(S) for any k ≥ 0. Then by [1, Theorem 3.7], we get gr Sk. Hence S can not be embedded into any term of the minimal injective resolution except for Ik(Λ). That is to say, Λ is diagonal.□

Actually, we have a strong version of the above result.

Proposition 2.9

Let Λ be an Auslander regular ring with global dimension n. Then, Λ is diagonal if and only if any simple Λ-module S with gr S = k satisfies 1. pd S = k for any kn−2.

Proof

We need only to show the necessity. It is known that any simple module S with gr S = n must have a projective dimension n. Let S be any simple module with gr S = n−1. Then the projective dimension of S must not be n by Lemma 2.2. So we have l.pd S = n−1. That is, Λ is diagonal by Proposition 2.5 (2).□

Let Λ be an Auslander regular ring with global dimension at most 3. By Corollary 2.6, we get that any simple Λ-module S with gr S = 1 satisfies l.pd (𝓔1𝓔1(S)) = 1. So we have the following

Corollary 2.10

Let Λ be an Auslander regular ring with global dimension at most 3. Then the following statements are equivalent.

  1. Λ is diagonal.

  2. Any simple Λ-module S with gr S = 0 is projective and any simple Λ-module S with gr S = 1 satisfies S ≅ 𝓔1𝓔1(S).

Acknowledgement

This research was partially supported by NSFC (Grant No. 11701488, 11201220, 11101217, 11126169) and Hunan Provincial Natural Science Foundation of China (Grant Nos. 14JJ3099, 2016JJ6124).

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About the article

Received: 2017-01-02

Accepted: 2017-11-28

Published Online: 2017-12-29


Competing interests: The authors declare that there is no conflict of interest regarding the publication of this paper.


Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 1618–1622, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2017-0138.

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© 2017 Huang and Zheng. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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