Non-commutative Castelnuovo-Mumford regularity and AS-regular algebras

JOURNAL OF ALGEBRA

卷:322

Article

Dong, Z. -C.¹  Wu, Q. -S.¹ 

[1] Fudan Univ, Sch Math Sci, Shanghai 200433, Peoples R China

关键词: Non-commutative graded algebras;    Non-commutative projective geometry;    Castelnuovo-Mumford regularity;    AS-Gorenstein algebras;    AS-regular algebras;   

DOI  :  10.1016/j.jalgebra.2009.03.013

来源: Elsevier

【 摘 要 】

Let A be a connected graded k-algebra with a balanced dualizing complex. We prove that A is a Koszul AS-regular algebra if and only if that the Castelnuovo-Mumford regularity and the Extregularity coincide for all finitely generated A-modules. This can be viewed as a non-commutative version of [T. Romer, On the regularity over positively graded algebras, J. Algebra 319 (2008) 1-15, Theorem 1.3]. By using Castelnuovo-Mumford regularity, we prove that any Koszul standard AS-Gorenstein algebra is AS-regular. As a preparation to prove the main result, we also prove that the following statements are equivalent: (1) A is AS-Gorenstein; (2) A has finite left injective dimension: (3) the dualizing complex has finite left projective dimension. This generalizes [I. Mori, Homological properties of balanced Cohen-Macaulay algebras, Trans. Amer. Math. Soc. 355 (2002) 1025-1042, Corollary 5.9]. (C) 2009 Elsevier Inc. All rights reserved.

【 授权许可】

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