Reduction of some homological conjectures to excellent unique factorization domains

TL;DR

This paper demonstrates that several major homological conjectures in commutative algebra can be reduced to the case of excellent unique factorization domains, simplifying their study.

Contribution

It introduces a quasi-Gorenstein analogue of Ulrich's deformation and proves reductions of key conjectures to excellent UFDs, including new cases for the Monomial Conjecture.

Findings

Reduces homological conjectures to excellent UFDs

Proves new reductions of the Monomial Conjecture

Shows certain almost complete intersections satisfy the Monomial Conjecture

Abstract

In this article, applying the quasi-Gorenstein analogous of the Ulrich's deformation of certain Gorenstein rings we show that some homological conjectures, including the Monomial Conjecture, Big Cohen-Macaulay Algebra Conjecture as well as the Small Cohen-Macaulay Conjecture reduce to the excellent unique factorization domains. Some other reductions of the Monomial Conjecture are also proved. We, moreover, show that certain almost complete intersections adhere the Monomial Conjecture.