Cohen-Macaulay, Gorenstein, Complete intersection and regular defect for the tensor product of algebras

TL;DR

This paper investigates how key algebraic properties like Cohen-Macaulayness, Gorenstein, and regularity are affected when taking tensor products of algebras, by analyzing their homological invariants.

Contribution

It provides formulas and methods to measure the defect of these properties in tensor products based on the invariants of the component algebras.

Findings

Derived relations for Krull dimension, depth, and injective dimension of tensor products.

Established bounds for the type and embedding dimension in tensor products.

Provided criteria to determine when tensor products preserve or fail these properties.

Abstract

This paper main goal is to measure the defect of Cohen-Macaulayness, Gorensteiness, complete intersection and regularity for the tensor product of algebras over a ring. For this sake, we determine the homological invariants which are inherent to these notions, such as the Krull dimension, depth, injective dimension, type and embedding dimension of the tensor product constructions in terms of those of their components.