The derived depth formula for modules of finite quasi-projective dimension

TL;DR

This paper establishes new formulas relating depth, width, and other invariants for modules with finite quasi-projective or quasi-injective dimensions over Noetherian local rings, extending classical results.

Contribution

It introduces the Derived Depth Formula and related results, extending classical formulas to modules with finite quasi-projective or quasi-injective dimensions.

Findings

Derived Depth Formula generalizes Auslander's depth formula

Extended versions of Ischebeck's Formula are proved

New special cases under stronger assumptions are identified

Abstract

Let $R$ be a commutative Noetherian local ring. We prove a variety of new formulae for modules of finite quasi-projective or finite quasi-injective dimension. These include the Derived Depth Formula, itself an extension of Auslander famous depth formula, a variation of the Derived Depth Formula for width, an extended version of Ischebeck's Formula, and a Dependency formula in the vein of Jorgensen. Several special cases of our main results are new even under stronger assumptions on the vanishing of various complete intersection dimensions.