Numerical homological regularities over positively graded algebras

Abstract

We study numerical regularities for complexes over noncommutative noetherian locally finite $N$-graded algebras A such as CM-regularity, Tor-regularity (Ext-regularity) and Ex-regularity, which are the supremum or infimum degrees of some associated canonical complexes. We introduce their companions—lowercase named regularities, which are defined by taking the infimum or supremum degrees of the respective canonical associated complexes. We show that for any right bounded complex X with finitely generated cohomologies, the supremum degree of $R{\underset{\_}{\mathrm{Hom}}}_A(X,A_0)$ coincides with the opposite of the infimum degree of X if $A_0$ is semisimple. If A has a balanced dualizing complex and $A_0$ is semisimple, we prove that the CM-regularity of X coincides with the supremum degree of $R{\underset{\_}{\mathrm{Hom}}}_A(A_0,X)$ for any left bounded complex X with finitely generated cohomologies.

Several inequalities concerning the numerical regularities and the supremum or infimum degrees of derived Hom or derived tensor complexes are given for noncommutative noetherian locally finite $N$-graded algebras. Some of these are generalizations of Jørgensen's results on the inequalities between the CM-regularity and Tor-regularity, some are new even in the connected graded case. Conditions are given under which the inequalities become equalities by establishing two technical lemmas.

Following Kirkman, Won and Zhang, we also use the numerical AS-regularity (resp. little AS-regularity) to study Artin-Schelter regular property (finite-dimensional property) for noetherian $N$-graded algebras. We prove that the numerical AS-regularity of A is zero if and only if that A is an $N$-graded AS-regular algebra under some mild conditions, which generalizes a result of Dong-Wu and a result of Kirkman-Won-Zhang. If A has a balanced dualizing complex and $A_0$ is semisimple, we prove that the little AS-regularity of A is zero if and only if A is finite-dimensional.