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- W2592100031 abstract "Let $S = k[x_1, dotsc, x_n]$ be a polynomial ring over a field $k$ and let $M$ be a graded $S$-module with minimal free resolution $mathbb{F}_bullet$. Its linear part $lin(mathbb{F}_bullet)$ is obtained by deleting all non-linear entries from the differential of $mathbb{F}_bullet$. Our first result is an elementary description of $lin(mathbb{F}_bullet)$ in the case that $M$ is the Stanley-Reisner ring of a simplicial complex $Delta$. Indeed, the differential of $lin(mathbb{F}_bullet)$ is simply a compilation of restriction maps in the simplicial cohomology of induced subcomplexes of $Delta$. Our second result concerns the linearity defect of $M$, which is the largest $i$ such that $H_i(lin(mathbb{F}_bullet)) neq 0$. We show that the linearity defect of $M$ is also the largest $i$, such that there exists a $g in mathbb{F}_{i}$ with $gnotinmathfrak{m}mathbb{F}_{i}$ and $mathrm{d} ginmathfrak{m}^2mathbb{F}_{i-1}$. Combining these two results, we obtain a description of the linearity defect of (the Stanley-Reisner ideal of) a simplicial complex, as well as a characterization of its componentwise linearity. Along the way, we also show that if a monomial ideal has at least one generator of degree $2$, then the linear strand of its resolution can be written using only $pm 1$ coefficients." @default.
- W2592100031 created "2017-03-16" @default.
- W2592100031 creator A5032425086 @default.
- W2592100031 date "2017-02-24" @default.
- W2592100031 modified "2023-09-27" @default.
- W2592100031 title "Linearity in minimal resolutions of monomial ideals" @default.
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