On the vanishing of self extensions over algebras
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Archiv der Mathematik
On the vanishing of self extensions over algebras Ali Mahin Fallah
Abstract. Recently, Araya, Celikbas, Sadeghi, and Takahashi proved a theorem about the vanishing of self extensions of finitely generated modules over commutative Noetherian rings. The aim of this paper is to obtain extensions of their result over algebras. Mathematics Subject Classification. 13D07, 13E05, 13H10, 16G30, 18G25. Keywords. Auslander–Reiten conjecture, Orders, n-canonical orders.
1. Introduction. Throughout the paper, we assume that R is a d-dimensional commutative Noetherian local ring with maximal ideal m and Λ is a Noetherian R-algebra, that is, an R-algebra which is finitely generated as an R-module and we denote by modΛ the category of finitely generated left Λ-modules. One of the most important open problems in representation theory of Artin algebras is the so-called Auslander–Reiten conjecture. It was proposed by Auslander and Reiten in [3] and can be formulated as follows: Conjecture. Let Λ be an Artin algebra and M a finitely generated Λmodule. If ExtiΛ (M, M ⊕ Λ) = 0 for all i > 0, then M is projective. This long-standing conjecture holds for several classes of algebras, including algebras of finite representation type [3] and symmetric Artin algebras with radical cube zero [14]. The Auslander–Reiten conjecture is rooted in Nakayama’s conjecture [18]. It is also closely related to other important homological conjectures in representation theory of Artin algebras, for instance, the finitistic dimension conjecture [13] and the Tachikawa conjecture [21]. There are already some results in the study of the Auslander–Reiten conjecture for commutative Noetherian rings, see for instance [1,7,9,12,15,19] and, more recently, for Noetherian algebras over commutative Gorenstein rings [6]. This work was jointly supported by the Iran National Science Foundation (INSF) and Alzahra University Grant No. 97024145. This research was also in part supported by a Grant from IPM (No. 98130019).
A. Mahin Fallah
Arch. Math.
There is a valuable result about the Auslander–Reiten conjecture, which is due to Araya. For a non-negative integer n and an R-module M , we put X n (R) = {p ∈ Spec(R) : ht(p) n} and say M is locally projective on X n (R) if Mp is a projective Rp -module for each prime ideal p ∈ X n (R). Also we put (−)∨ = HomΛ (−, Λ) and (−)† = HomR (−, ωR ) in which ωR is a canonical module of R. Theorem 1.1 (Araya [1]). Let R be a Gorenstein local ring with d = dim R 2 and let M ∈ modR. Then M is projective provided that the following hold: (i) M is locally projective on X d−1 (R). (ii) M is a Gorenstein projective R-module. (iii) Extd−1 R (M, M ) = 0. Ono and Yoshino [17] relaxed the condition that M is locally projective on X d−1 (R) in Araya’s theorem under the hypothesis that ExtiR (M, M ) vanishes for i = d−2, i = d−1 and M is locally projective on X d−2 (R) . More precisely they proved: Theorem 1.2 (Ono and Yoshino [17]). Let R be a Gorenstein local ring with d = dim R 3 and let M ∈ modR. Then M is projective
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