Asymmetry of Ext-groups

JOURNAL OF ALGEBRA

卷:322

Article

Mori, Izuru

关键词: Ext-groups;    AS-Gorenstein algebras;    Frobenius Koszul algebras;    Noncommutative projective geometry;   

DOI  :  10.1016/j.jalgebra.2009.02.027

来源: Elsevier

【 摘 要 】

In this paper. we will use techniques of noncommutative projective geometry to construct examples of algebras R over a field k not satisfying the following two types of symmetric behaviors of Ext-groups: (EE) For any pair of finitely generated R-modules (M, N). dim(k) Ext(R)(i) (M, N) < infinity for all i epsilon N if and only if dim(k) Ext(R)(i) (N, M) < infinity for all i epsilon N. (ee) For any pair of finitely generated R-modules (M, N), Ext(R)(i) (M, N) = 0 for all i >> 0 if and only if Ext(R)(i) (N. M) = 0 for all i >> 0. In particular, and contrary to the commutative case, we give a simple example of a noncommutative noetherian Gorenstein (Frobenius) local algebra satisfying (uac) (uniform Auslander condition) but not satisfying (ee). (C) 2009 Elsevier Inc. All rights reserved.

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