On a Relation between Euler characteristics of \MakeLowercase{de} Rham cohomology and Koszul cohomology of graded local cohomology modules
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916754718982144 |
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| author | Puthenpurakal, Tony J. Reddy, Rakesh B. T. |
| author_facet | Puthenpurakal, Tony J. Reddy, Rakesh B. T. |
| contents | Let $K$ be a field of characteristic zero. Let $R = K[X_0, X_1,\ldots,X_n]$ be standard graded. Let $A_{n+1}(K)$ be the $(n + 1)^{th}$ Weyl algebra over $K$. Let $I$ be a homogeneous ideal of $R$ and let $M = H^i_I(R)$ for some $i \geq 0$.
By a result of Lyubeznik, $M$ is a graded holonomic $A_{n +1}(K)$-module for each $i \geq 0$. Let $χ^c(\mathbf{\partial}, M)$ ($χ^c(\mathbf{X}, M)$) be the Euler characteristics of de Rham cohomology (resp. Koszul cohomology) of $M$. We prove $χ^c(\mathbf{\partial}, M) = (-1)^{n+1}χ^c(\mathbf{X}, M)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_17597 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a Relation between Euler characteristics of \MakeLowercase{de} Rham cohomology and Koszul cohomology of graded local cohomology modules Puthenpurakal, Tony J. Reddy, Rakesh B. T. Commutative Algebra Primary 13D45, Secondary 13N10 Let $K$ be a field of characteristic zero. Let $R = K[X_0, X_1,\ldots,X_n]$ be standard graded. Let $A_{n+1}(K)$ be the $(n + 1)^{th}$ Weyl algebra over $K$. Let $I$ be a homogeneous ideal of $R$ and let $M = H^i_I(R)$ for some $i \geq 0$. By a result of Lyubeznik, $M$ is a graded holonomic $A_{n +1}(K)$-module for each $i \geq 0$. Let $χ^c(\mathbf{\partial}, M)$ ($χ^c(\mathbf{X}, M)$) be the Euler characteristics of de Rham cohomology (resp. Koszul cohomology) of $M$. We prove $χ^c(\mathbf{\partial}, M) = (-1)^{n+1}χ^c(\mathbf{X}, M)$. |
| title | On a Relation between Euler characteristics of \MakeLowercase{de} Rham cohomology and Koszul cohomology of graded local cohomology modules |
| topic | Commutative Algebra Primary 13D45, Secondary 13N10 |
| url | https://arxiv.org/abs/2505.17597 |