Journal of the Ramanujan Mathematical Society
Volume 41, Issue 1, March 2026 pp. 1–8.
Linear resolution of products of monomial ideals related to maximal minors
Authors:
Arindam Banerjee, Dipankar Ghosh and S Selvaraja
Author institution:Department of Mathematics, Indian Institute of Technology Kharagpur, West~Bengal~721~302, India.
Summary:
Let X be an m × n matrix of distinct indeterminates over a field K ,
where m ≤ n. Set the polynomial ring K[X] := K[X{ij} : 1 ≤ i
≤ m, 1 ≤ j ≤ n]. Let 1 ≤ k ˂ l ≤ n be such
that l - k + 1 ≥ 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 × 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}} ⋯ J{k{s}
l{s}} has a linear free resolution, where k{1} ≤ k{2} ≤ ⋯
≤ k{s} and l{1} ≤ l{2} ≤ ⋯ ≤ l{s}. This result
is a variation of a theorem due to Bruns and Conca. Moreover, our proof is
self-contained, elementary and combinatorial.
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