Non-commutative Castelnuovo-Mumford Regularity and AS-regular Algebras

Details Non-commutative Castelnuovo-Mumford Regularity and AS-regular Algebras Non-commutative Castelnuovo-Mumford Regularity and AS-regular Algebras

2008-08-04

arxiv.org/abs/0808.0407v1

Mathematics

Quantum Algebra

17 pages

Scientific paper

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 Ext-regularity coincide for all finitely generated $A$-modules. This can be viewed as a non-commutative version of \cite[Theorem 1.3]{ro}. 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 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 \cite[Corollary 5.9]{mori}.

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