Stanley-Reisner ideals with linear powers
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909736714108928 |
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| author | Ficarra, Antonino Moradi, Somayeh |
| author_facet | Ficarra, Antonino Moradi, Somayeh |
| contents | Let $S = K[x_1, \dots, x_n]$ be the standard graded polynomial ring over a field $K$. In this paper, we address and completely solve two fundamental open questions in Commutative Algebra:
(i) For which degrees $d$, does there exist a uniform combinatorial characterization of all squarefree monomial ideals in $S$ having $d$-linear resolutions?
(ii) For which degrees $d$, does having a linear resolution coincide with having linear powers for all squarefree monomial ideals of $S$ generated in degree $d$?
Let $\mathcal{I}_{n,d}(K)$ denote the class of squarefree monomial ideals of $S$ having a $d$-linear resolution. Our main result establishes the equivalence of the following conditions:
(a) Any squarefree monomial ideal $I$ in $S$ generated in degree $d$ has a linear resolution, if and only if, $I$ has linear powers.
(b) $\mathcal{I}_{n,d}(K)$ is independent of the base field $K$.
(c) $d\in\{0,1,2,n{-}2,n{-}1,n\}$.
In each of these degrees, we show that a squarefree monomial ideal has a linear resolution if and only if all of its powers admit linear quotients, and we combinatorially classify such ideals. In contrast, for each degree $3\le d\le n{-}3$, we construct fully-supported squarefree monomial ideals $I$ and $J$ in $S$ generated in degree $d$ such that the linear resolution property of $I$ depends on the choice of the base field, $J$ has a linear resolution and $J^2$ does not have a linear resolution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_10354 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stanley-Reisner ideals with linear powers Ficarra, Antonino Moradi, Somayeh Commutative Algebra Combinatorics Let $S = K[x_1, \dots, x_n]$ be the standard graded polynomial ring over a field $K$. In this paper, we address and completely solve two fundamental open questions in Commutative Algebra: (i) For which degrees $d$, does there exist a uniform combinatorial characterization of all squarefree monomial ideals in $S$ having $d$-linear resolutions? (ii) For which degrees $d$, does having a linear resolution coincide with having linear powers for all squarefree monomial ideals of $S$ generated in degree $d$? Let $\mathcal{I}_{n,d}(K)$ denote the class of squarefree monomial ideals of $S$ having a $d$-linear resolution. Our main result establishes the equivalence of the following conditions: (a) Any squarefree monomial ideal $I$ in $S$ generated in degree $d$ has a linear resolution, if and only if, $I$ has linear powers. (b) $\mathcal{I}_{n,d}(K)$ is independent of the base field $K$. (c) $d\in\{0,1,2,n{-}2,n{-}1,n\}$. In each of these degrees, we show that a squarefree monomial ideal has a linear resolution if and only if all of its powers admit linear quotients, and we combinatorially classify such ideals. In contrast, for each degree $3\le d\le n{-}3$, we construct fully-supported squarefree monomial ideals $I$ and $J$ in $S$ generated in degree $d$ such that the linear resolution property of $I$ depends on the choice of the base field, $J$ has a linear resolution and $J^2$ does not have a linear resolution. |
| title | Stanley-Reisner ideals with linear powers |
| topic | Commutative Algebra Combinatorics |
| url | https://arxiv.org/abs/2508.10354 |