Ischebeck's formula, grade and quasi-homological dimensions

TL;DR

This paper explores generalized homological invariants called quasi-homological dimensions, establishing formulas and inequalities that extend classical results like Ischebeck's formula, and introduces the concept of quasi-perfect modules.

Contribution

It generalizes classical homological formulas to quasi-homological dimensions and introduces quasi-perfect modules, expanding the understanding of module theory over local rings.

Findings

Established conditions for Ischebeck's formula involving quasi-homological dimensions.

Introduced the concept of quasi-perfect modules and proved related properties.

Derived grade formulas and inequalities for modules with finite quasi-injective and quasi-projective dimensions.

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.