Overview
My research is mostly in commutative homological algebra. The questions I’m interested in are usually about understanding modules and their free resolutions.
Links
Papers
Unstable elements in cohomology and a question of Lescot (with Srikanth Iyengar and Sarasij Maitra) https://arxiv.org/abs/2507.23213
Sri, Sarasij, and I reinterpret an invariant introduced by Lescot through the lens of stable cohomology. We provide some new classes of rings where the invariant is finite by utilizing the relevant multiplicative structures. This project started with a question about Bass numbers of modules and their syzygies. We realized that Lescot had considered similar questions and this led us to his sigma invariant.
Tensor products of d-fold matrix factorizations (with Richie Sheng) https://doi.org/10.1017/nmj.2025.12
Richie and I study the tensor product of d-fold matrix factorizations. Our main goal was to understand the decomposability of the tensor product operation. This work is motivated by Yoshino’s paper “Tensor product of matrix factorizations”. In the last section of the paper we show how to construct maximal Cohen-Macaulay and Ulrich modules over hypersurface domains. Constructing these modules is not new, though the tensor product gives a convenient way to work with them. What is new is being able to tell when these modules are indecomposable (or not) while also keeping track of their ranks, minimal number of generators, and multiplicity.
*Richie’s work on this project was funded through the Undergraduate Research Opportunities Program (UROP) at the University of Utah (summer 2024). He was also funded by the Utah Math Department’s NSF RTG Grant #1840190.
A family of simplicial resolutions which are DG-algebras (with James Cameron, Trung Chau, and Sarasij Maitra) https://arxiv.org/abs/2412.21120
James, Trung, Sarasij, and I introduced and studied a construction which prunes the Taylor resolution in a way that preserves the DG-algebra structure. Our resolution is still rarely minimal but it is always “smaller” than the Taylor resolution and still has an DG-algebra structure which is induced from the multiplication on the Taylor resolution. Some similar results were found independently in https://arxiv.org/abs/2502.00591v1 (shoutout to Syracuse commutative algebra!)
Branched covers and matrix factorizations (with Graham Leuschke) https://doi.org/10.1112/blms.12901
In this paper, Graham and I study the connection between the d-fold branched cover of a hypersurface ring and the category of matrix factorizations with d factors. We show that, from a representation theory perspective, these ideas are closely related. Namely, the category of MCM modules over the branched cover is representation finite if and only if the same is true of the category of matrix factorizations with d factors.
Matrix factorizations with more than two factors https://arxiv.org/abs/2102.06819
This paper is a study of the category of matrix factorizations with more than two factors. I start by describing a naturally occurring exact structure which turns out to be Frobenius. I determine the indecomposable projective-inejctive objects and I give explicit formulas for syzygies, cosyzygies, and cones.
The majority of the rest of the paper is focused on establishing two module theoretic descriptions of the category of matrix factorizations with d factors. This builds upon work of Knörrer and Solberg.
This paper is in the process of a complete rewrite. Sometime soon, there will be improved results and many removed hypotheses. It will remain confusingly named the same as my thesis; the old statements can mostly be found there.