Ischebeck's formula, grade and quasi-homological dimensions

TL;DR

This work extends classical Ischebeck-type phenomena to the framework of quasi-homological dimensions, introducing and exploiting the notions of quasi-projective and quasi-injective dimensions as refinements of projective and injective dimensions. It defines quasi-perfect modules (finite $\operatorname{qpd}$ with $\operatorname{grade} = \operatorname{qpd}$) and develops a cohesive theory linking $\operatorname{grade}$, $\operatorname{P}_R(-,-)$, and Ext-vanishing through quasi-projective/injective approximations. The main contributions include three Ischebeck-type results under different finiteness hypotheses, a formula for the grade of modules with finite $\operatorname{qid}$, and several Cohen–Macaulayness criteria for rings and modules arising from quasi-perfectness. Collectively, the results broaden the AB-dimension landscape, offering tools to detect Cohen–Macaulayness and to compare quasi-homological dimensions with classical invariants in commutative algebra.

Abstract

The quasi-projective dimension and quasi-injective dimension are recently introduced homological invariants that generalize the classical notions of projective dimension and injective dimension, respectively. For a local ring $R$ and finitely generated $R$-modules $M$ and $N$, we provide conditions involving quasi-homological dimensions where the equality $\sup \lbrace i\geq 0: \operatorname{Ext}_R^i(M,N)\not=0 \rbrace =\operatorname{depth} R-\operatorname{depth} M$, which we call Ischebeck's formula, holds. One of the results in this direction generalizes a well-known result of Ischebeck concerning modules of finite injective dimension, considering the quasi-injective dimension. On the other hand, we establish an inequality relating the quasi-projective dimension of a finitely generated module to its grade and introduce the concept of a quasi-perfect module as a natural generalization of a perfect module. We prove several results for this new concept similar to the classical results. Additionally, we provide a formula for the grade of finitely generated modules with finite quasi-injective dimension over a local ring, as well as grade inequalities for modules of finite quasi-projective dimension. In our study, Cohen-Macaulayness criteria are also obtained.

Theorems & Definitions (67)

• Theorem 1.1: See Theorems \ref{['teo:qproj']}, \ref{['g-dim']} and \ref{['isch']}
• Theorem 1.2: See Theorem \ref{['theo4.5']}
• Theorem 1.3: See Theorem \ref{['teo:forinj']}
• Corollary 1.4: See Corollary \ref{['corol:criteria']}
• proof
• Definition 2.8
• Theorem 2.10
• Theorem 2.11
• Theorem 2.12
• Definition 2.13