Top dickson class and annihilators of cohomology over invariant rings
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| Format: | Preprint |
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2025
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| _version_ | 1866912753542758400 |
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| author | Puthenpurakal, Tony J. |
| author_facet | Puthenpurakal, Tony J. |
| contents | Let $\mathbb{F}_q$ denote the finite field with $q = p^r$ elements. Let $V$ be a finite dimensional vector space of dimension $d$ over $\mathbb{F}_q$ and let $G \subseteq GL(V)$ be a group. Let $R = \mathbb{F}_q[V] = \text{Sym}(V^*)$ and let $S = R^G$.
Let $\mathbf{d}_{d,0} $ be the top Dickson class, i.e., $\mathbf{d}_{d,0} = \prod_{0\neq v \in V^*}v$. Surprisingly (a power of) $\mathbf{d}_{d,0}$ annihilates many cohomological modules.
(a) Let $H^i(G, R)$ be the $i^{th}$-group cohomology of $R$ considered as a $S$-module. Set $J_i = \text{ann}_S \ H^i(G, R)$. We show that $\mathbf{d}_{d,0} \in \sqrt{J_i}$ for all $i \geq 1$.
(b) We also show that $\mathbf{d}_{d,0} \in \sqrt{ \text{ann}_S \ H^j_{S_+}(S)}$ for all $ 0 \leq j \leq d - 1$ (here $H^j_{S_+}(S)$ is the $j^{th}$ local cohomology of $S$ with respect to $S_+$).
As an application we get that there exists a fixed power of $\mathbf{d}_{d,0} $ which works as a cohomological annihilator. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_07157 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Top dickson class and annihilators of cohomology over invariant rings Puthenpurakal, Tony J. Commutative Algebra Primary 13A50, 13D03, Secondary 13D07, 13D45 Let $\mathbb{F}_q$ denote the finite field with $q = p^r$ elements. Let $V$ be a finite dimensional vector space of dimension $d$ over $\mathbb{F}_q$ and let $G \subseteq GL(V)$ be a group. Let $R = \mathbb{F}_q[V] = \text{Sym}(V^*)$ and let $S = R^G$. Let $\mathbf{d}_{d,0} $ be the top Dickson class, i.e., $\mathbf{d}_{d,0} = \prod_{0\neq v \in V^*}v$. Surprisingly (a power of) $\mathbf{d}_{d,0}$ annihilates many cohomological modules. (a) Let $H^i(G, R)$ be the $i^{th}$-group cohomology of $R$ considered as a $S$-module. Set $J_i = \text{ann}_S \ H^i(G, R)$. We show that $\mathbf{d}_{d,0} \in \sqrt{J_i}$ for all $i \geq 1$. (b) We also show that $\mathbf{d}_{d,0} \in \sqrt{ \text{ann}_S \ H^j_{S_+}(S)}$ for all $ 0 \leq j \leq d - 1$ (here $H^j_{S_+}(S)$ is the $j^{th}$ local cohomology of $S$ with respect to $S_+$). As an application we get that there exists a fixed power of $\mathbf{d}_{d,0} $ which works as a cohomological annihilator. |
| title | Top dickson class and annihilators of cohomology over invariant rings |
| topic | Commutative Algebra Primary 13A50, 13D03, Secondary 13D07, 13D45 |
| url | https://arxiv.org/abs/2512.07157 |