Linear resolution of products of monomial ideals related to maximal minors
Arindam Banerjee, Dipankar Ghosh, S. Selvaraja
TL;DR
This work studies when products of monomial ideals $J_{kl}$, generated by diagonal monomials of maximal minors of submatrices $Y_{kl}$ of a generic $m\times n$ matrix, have linear free resolutions. It proves that the product $J_{k_1l_1}\cdots J_{k_sl_s}$ has a linear free resolution under the natural ordering $k_1\le\cdots\le k_s$ and $l_1\le\cdots\le l_s$ with $l_i-k_i+1\ge m$, by first establishing that each $J_{kl}$ has linear quotients through a careful colon-ideal analysis, and then performing an induction on the number of factors using short exact sequences to bound regularity. A key technical identity for the induction expresses $(J+\langle f_1,\dots,f_u\rangle : f_{u+1})$ in terms of the remaining product $J_{k_2l_2}\cdots J_{k_sl_s}$ plus a monomial-generated part, enabling the conclusion that $\operatorname{reg}(J)=sm$ and hence a linear free resolution. The results are self-contained, combinatorial, and extend Bruns-Conca type findings to a new family of determinantal-related monomial ideals, with implications for initial ideals and Gröbner basis analyses in this context.
Abstract
Let $ X $ be an $ m \times n $ matrix of distinct indeterminates over a field $ K $, where $ m \le n $. Set the polynomial ring $ K[X] := K[X_{ij} : 1 \le i \le m, 1 \le j \le n] $. Let $ 1 \le k < l \le n $ be such that $ l - k + 1 \ge m $. Consider the submatrix $ Y_{kl} $ of consecutive columns of $ X $ from $ k $th column to $ l $th column. Let $ J_{kl} $ be the ideal generated by `diagonal monomials' of all $ m \times m $ submatrices of $ Y_{kl} $, where the diagonal monomial of a square matrix means product of its main diagonal entries. We show that $ J_{k_1 l_1} J_{k_2 l_2} \cdots J_{k_s l_s} $ has a linear free resolution, where $ k_1 \le k_2 \le \cdots \le k_s $ and $ l_1 \le l_2 \le \cdots \le l_s $. This result is a variation of a theorem due to Bruns and Conca. Moreover, our proof is self-contained, elementary and combinatorial.