Quasilifting of hulls and depth of tensor product of modules

TL;DR

This paper studies the depth of tensor products of modules over local rings, introducing a lifting construction to establish new bounds and recover known results, with implications for modules over complete intersection rings.

Contribution

It introduces a quasilifting construction for hulls, providing new depth bounds and answering an open question in the context of tensor products over local rings.

Findings

Recovered a known result for complete intersection rings.

Provided a negative answer to an open question.

Established a lower bound for depth of tensor products.

Abstract

We investigate the depth of the tensor product of finitely generated modules over local rings. One of the main ingredients of our approach is a lifting construction introduced by Huneke, Jorgensen, and Wiegand. We recover a result of Celikbas, Sadeghi, and Takahashi for local complete intersection rings. Additionally, we provide a negative answer to a question they asked and establish a corresponding lower bound. We derive a result on the depth of the tensor product of certain modules over local complete $\mathcal{TE}$ rings. Some general conditions on the existence of hulls and approximations are also studied.