Ideals with componentwise linear powers
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866913685381840896 |
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| author | Hibi, Takayuki Moradi, Somayeh |
| author_facet | Hibi, Takayuki Moradi, Somayeh |
| contents | Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over a field $K$, and let $A$ be a finitely generated standard graded $S$-algebra. We show that if the defining ideal of $A$ has a quadratic initial ideal, then all the graded components of $A$ are componentwise linear. Applying this result to the Rees ring $\mathcal{R}(I)$ of a graded ideal $I$ gives a criterion on $I$ to have componentwise linear powers. Moreover, for any given graph $G$, a construction on $G$ is presented which produces graphs whose cover ideals $I_G$ have componentwise linear powers. This in particular implies that for any Cohen-Macaulay Cameron-Walker graph $G$ all powers of $I_G$ have linear resolutions. Moreover, forming a cone on special graphs like unmixed chordal graphs, path graphs and Cohen-Macaulay bipartite graphs produces cover ideals with componentwise linear powers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_01677 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Ideals with componentwise linear powers Hibi, Takayuki Moradi, Somayeh Commutative Algebra Combinatorics 13A02, 13P10, 05E40 Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over a field $K$, and let $A$ be a finitely generated standard graded $S$-algebra. We show that if the defining ideal of $A$ has a quadratic initial ideal, then all the graded components of $A$ are componentwise linear. Applying this result to the Rees ring $\mathcal{R}(I)$ of a graded ideal $I$ gives a criterion on $I$ to have componentwise linear powers. Moreover, for any given graph $G$, a construction on $G$ is presented which produces graphs whose cover ideals $I_G$ have componentwise linear powers. This in particular implies that for any Cohen-Macaulay Cameron-Walker graph $G$ all powers of $I_G$ have linear resolutions. Moreover, forming a cone on special graphs like unmixed chordal graphs, path graphs and Cohen-Macaulay bipartite graphs produces cover ideals with componentwise linear powers. |
| title | Ideals with componentwise linear powers |
| topic | Commutative Algebra Combinatorics 13A02, 13P10, 05E40 |
| url | https://arxiv.org/abs/2309.01677 |