Resolutions over strict complete intersections
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910628048797696 |
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| author | Puthenpurakal, Tony J. |
| author_facet | Puthenpurakal, Tony J. |
| contents | Let $(Q, \mathfrak{n})$ be a regular local ring and let $f_1, \ldots, f_c \in \mathfrak{n}^2$ be a $Q$-regular sequence. Set $(A, \mathfrak{m}) = (Q/(\mathbf{f}), \mathfrak{n}/(\mathbf{f}))$. Further assume that the initial forms
$f_1^*, \ldots, f_c^*$ form a $G(Q) = \bigoplus_{n \geq 0}\mathfrak{n}^i/\mathfrak{n}^{i+1}$-regular sequence. Without loss of any generality assume $ord_Q(f_1) \geq ord_Q(f_2) \geq \cdots \geq ord_Q(f_c)$. Let $M$ be a finitely generated $A$-module and let $(\mathbb{F}, \partial)$ be a minimal free resolution of $M$. Then we prove that $ord(\partial_i) \leq ord_Q(f_1) - 1$ for all $i \gg 0$. We also construct an MCM $A$-module $M$ such that $ord(\partial_{2i+1}) = ord_Q(f_1) - 1$ for all $i \geq 0$. We also give a considerably simpler proof regarding the periodicity of ideals of minors of maps in a minimal free resolution of modules over arbitrary complete intersection rings (not necessarily strict). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_11877 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Resolutions over strict complete intersections Puthenpurakal, Tony J. Commutative Algebra Primary 13D02, Secondary 13C14 Let $(Q, \mathfrak{n})$ be a regular local ring and let $f_1, \ldots, f_c \in \mathfrak{n}^2$ be a $Q$-regular sequence. Set $(A, \mathfrak{m}) = (Q/(\mathbf{f}), \mathfrak{n}/(\mathbf{f}))$. Further assume that the initial forms $f_1^*, \ldots, f_c^*$ form a $G(Q) = \bigoplus_{n \geq 0}\mathfrak{n}^i/\mathfrak{n}^{i+1}$-regular sequence. Without loss of any generality assume $ord_Q(f_1) \geq ord_Q(f_2) \geq \cdots \geq ord_Q(f_c)$. Let $M$ be a finitely generated $A$-module and let $(\mathbb{F}, \partial)$ be a minimal free resolution of $M$. Then we prove that $ord(\partial_i) \leq ord_Q(f_1) - 1$ for all $i \gg 0$. We also construct an MCM $A$-module $M$ such that $ord(\partial_{2i+1}) = ord_Q(f_1) - 1$ for all $i \geq 0$. We also give a considerably simpler proof regarding the periodicity of ideals of minors of maps in a minimal free resolution of modules over arbitrary complete intersection rings (not necessarily strict). |
| title | Resolutions over strict complete intersections |
| topic | Commutative Algebra Primary 13D02, Secondary 13C14 |
| url | https://arxiv.org/abs/2409.11877 |