Componentwise Linear Ideals From Sums
Hailong Dao, Sreehari Suresh-Babu
TL;DR
The paper addresses the problem of when the sum of componentwise linear ideals remains componentwise linear, and provides a general framework leveraging full sets of squarefree monomials to build componentwise linear sums under mild compatibility. It develops reg-related inequalities via short exact sequences and a constructive criterion, then specializes to the two-variable case to obtain a complete characterization for $I+J$ in $S=k[x,y]$, including a corollary that componentwise linear monomial ideals in two variables admit linear quotients with generators in nondecreasing degrees. The work yields a broad, practical method to assemble componentwise linear ideals from simpler pieces and sharp conditions showing when such sums fail. It thus enhances understanding of how componentwise linearity behaves under sums and provides precise, low-dimensional classifications with concrete examples.
Abstract
Let $I,J$ be componentwise linear ideals in a polynomial ring $S$. We study necessary and sufficient conditions for $I+J$ to be componentwise linear. We provide a complete characterization when $\dim S=2$. As a consequence, any componentwise linear monomial ideal in $k[x,y]$ has linear quotients using generators in non-decreasing degrees. In any dimension, we show that under mild compatibility conditions, one can build a componentwise linear ideal from a given collection of componentwise linear monomial ideals using only sum and product with square-free monomials. We provide numerous examples to demonstrate the optimality of our results.