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 $\text{dim} S=2$. 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 $\text{dim} S$ 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.
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