Sreehari Suresh Babu: Componentwise linear ideals from sums

Abstract.

A componentwise linear ideal in a polynomial ring \(S\) is an ideal \(I\) such that the ideal generated by each component of \(I\) has a linear resolution. Given two componentwise linear ideals \(I\) and \(J\), 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, we show that any componentwise linear monomial ideal in \(k[x,y]\) has linear quotients using generators in non-decreasing degrees. When \(\dim S\) is arbitrary, we describe how one can build a componentwise linear ideal from a given collection of componentwise linear monomial ideals, satisfying some mild compatibility conditions, using only sum and product with square-free monomials. This is a joint work with Prof. Hailong Dao.