Congruence modules and Wiles defects of determinantal rings of maximal minors

TL;DR

This paper computes the congruence module and Wiles defect of determinantal rings of maximal minors over a discrete valuation ring, relating them to minors of matrices associated with algebra maps.

Contribution

It provides explicit formulas for the congruence module and Wiles defect of determinantal rings at specific algebra maps, extending previous work by Iyengar, Khare, and Manning.

Findings

Explicit formulas for congruence modules in terms of minors.

Explicit formulas for Wiles defects in terms of minors.

Connections between algebra maps and rank-deficient matrices.

Abstract

Let $O$ be a discrete valuation ring and $A := O[X_{m \times n}]/I_{m}(X)$ the determinantal ring of maximal minors. We consider algebra maps $\lambda \colon A \to O$, which is tantamount to choosing rank-deficient matrices $a \in O^{m \times n}$. Following Iyengar--Khare--Manning, we compute the congruence module and the Wiles defect of $A$ at $\lambda$, expressing them in terms of the $(m - 1)$-sized minors of $a$.