Componentwise linear ideals from sums

Ideals with linear resolutions arise naturally across commutative algebra and combinatorics. For example, the Cohen-Macaulay property was a crucial ingredient in the solution of the Upper Bound Conjecture, and it is well-known that a simplicial complex is Cohen-Macaulay if and only if the Stanley-Reisner ideal of its Alexander dual has a linear resolution. Componentwise linear ideals, introduced by Herzog and Hibi, generalize the notion of ideals with linear resolutions: an ideal I in a polynomial ring S is componentwise linear if the ideal generated by each component of I has a linear resolution.

In this talk, after introducing componentwise linear ideals and reviewing several of their characterizations, we will explain some necessary and sufficient conditions for the sum of two componentwise linear ideals to be componentwise linear. A complete answer in dimension two helps us to establish the equivalence between componentwise linear monomial ideals and ideals having linear quotients in k[x,y]. When the dimension is arbitrary, we describe a method to build a componentwise linear ideal from a given collection of componentwise linear monomial ideals satisfying some mild compatibility conditions, using only sum and squarefree translations. This is a joint work with Prof. Hailong Dao.

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https://sreehari183.github.io/