Matching powers of monomial ideals and edge ideals of weighted oriented graphs
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866910385159798784 |
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| author | Erey, Nursel Ficarra, Antonino |
| author_facet | Erey, Nursel Ficarra, Antonino |
| contents | We introduce the concept of matching powers of monomial ideals. Let $I$ be a monomial ideal of $S=K[x_1,\dots,x_n]$, with $K$ a field. The $k$th matching power of $I$ is the monomial ideal $I^{[k]}$ generated by the products $u_1\cdots u_k$ where $u_1,\dots,u_k$ is a monomial regular sequence contained in $I$. This concept naturally generalizes that of squarefree powers of squarefree monomial ideals. We study depth and regularity functions of matching powers of monomial ideals and edge ideals of weighted oriented graphs. We show that the last nonvanishing power of a quadratic monomial ideal is always polymatroidal and thus has a linear resolution. When $I$ is a non-quadratic edge ideal of a weighted oriented forest, we characterize when $I^{[k]}$ has a linear resolution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_13771 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Matching powers of monomial ideals and edge ideals of weighted oriented graphs Erey, Nursel Ficarra, Antonino Commutative Algebra Combinatorics We introduce the concept of matching powers of monomial ideals. Let $I$ be a monomial ideal of $S=K[x_1,\dots,x_n]$, with $K$ a field. The $k$th matching power of $I$ is the monomial ideal $I^{[k]}$ generated by the products $u_1\cdots u_k$ where $u_1,\dots,u_k$ is a monomial regular sequence contained in $I$. This concept naturally generalizes that of squarefree powers of squarefree monomial ideals. We study depth and regularity functions of matching powers of monomial ideals and edge ideals of weighted oriented graphs. We show that the last nonvanishing power of a quadratic monomial ideal is always polymatroidal and thus has a linear resolution. When $I$ is a non-quadratic edge ideal of a weighted oriented forest, we characterize when $I^{[k]}$ has a linear resolution. |
| title | Matching powers of monomial ideals and edge ideals of weighted oriented graphs |
| topic | Commutative Algebra Combinatorics |
| url | https://arxiv.org/abs/2309.13771 |