Polarization and Gorenstein liaison
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866908720476192768 |
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| author | Faridi, Sara Klein, Patricia Rajchgot, Jenna Seceleanu, Alexandra |
| author_facet | Faridi, Sara Klein, Patricia Rajchgot, Jenna Seceleanu, Alexandra |
| contents | A major open question in the theory of Gorenstein liaison is whether or not every arithmetically Cohen--Macaulay subscheme of $\mathbb{P}^n$ can be G-linked to a complete intersection. Migliore and Nagel showed that, if such a scheme is generically Gorenstein (e.g., reduced), then, after re-embedding so that it is viewed as a subscheme of $\mathbb{P}^{n+1}$, indeed it can be G-linked to a complete intersection. Motivated by this result, we consider techniques for constructing G-links on a scheme from G-links on a closely related reduced scheme.
Polarization is a tool for producing a squarefree monomial ideal from an arbitrary monomial ideal. Basic double G-links on squarefree monomial ideals can be induced from vertex decompositions of their Stanley--Reisner complexes. Given a monomial ideal $I$ and a vertex decomposition of the Stanley--Reisner complex of its polarization $P(I)$, we give conditions that allow for the lifting of an associated basic double G-link of $P(I)$ to a basic double G-link of $I$ itself. We use the relationship we develop in the process to show that the Stanley--Reisner complexes of polarizations of stable Cohen--Macaulay monomial ideals are vertex decomposable.
We then introduce and study polarization of a Gröbner basis of an arbitrary homogeneous ideal and give a relationship between geometric vertex decomposition of a polarization and elementary G-biliaison that is analogous to our result on vertex decomposition and basic double G-linkage. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_19985 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Polarization and Gorenstein liaison Faridi, Sara Klein, Patricia Rajchgot, Jenna Seceleanu, Alexandra Commutative Algebra Algebraic Geometry Combinatorics A major open question in the theory of Gorenstein liaison is whether or not every arithmetically Cohen--Macaulay subscheme of $\mathbb{P}^n$ can be G-linked to a complete intersection. Migliore and Nagel showed that, if such a scheme is generically Gorenstein (e.g., reduced), then, after re-embedding so that it is viewed as a subscheme of $\mathbb{P}^{n+1}$, indeed it can be G-linked to a complete intersection. Motivated by this result, we consider techniques for constructing G-links on a scheme from G-links on a closely related reduced scheme. Polarization is a tool for producing a squarefree monomial ideal from an arbitrary monomial ideal. Basic double G-links on squarefree monomial ideals can be induced from vertex decompositions of their Stanley--Reisner complexes. Given a monomial ideal $I$ and a vertex decomposition of the Stanley--Reisner complex of its polarization $P(I)$, we give conditions that allow for the lifting of an associated basic double G-link of $P(I)$ to a basic double G-link of $I$ itself. We use the relationship we develop in the process to show that the Stanley--Reisner complexes of polarizations of stable Cohen--Macaulay monomial ideals are vertex decomposable. We then introduce and study polarization of a Gröbner basis of an arbitrary homogeneous ideal and give a relationship between geometric vertex decomposition of a polarization and elementary G-biliaison that is analogous to our result on vertex decomposition and basic double G-linkage. |
| title | Polarization and Gorenstein liaison |
| topic | Commutative Algebra Algebraic Geometry Combinatorics |
| url | https://arxiv.org/abs/2406.19985 |