Equivariant resolutions over Veronese rings
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arXiv
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| Hauptverfasser: | , , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866917594324271104 |
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| author | Almousa, Ayah Perlman, Michael Pevzner, Alexandra Reiner, Victor VandeBogert, Keller |
| author_facet | Almousa, Ayah Perlman, Michael Pevzner, Alexandra Reiner, Victor VandeBogert, Keller |
| contents | Working in a polynomial ring $S=\mathbf{k}[x_1,\ldots,x_n]$ where $\mathbf{k}$ is an arbitrary commutative ring with $1$, we consider the $d^{th}$ Veronese subalgebras $R=S^{(d)}$, as well as natural $R$-submodules $M=S^{(\geq r, d)}$ inside $S$. We develop and use characteristic-free theory of Schur functors associated to ribbon skew diagrams as a tool to construct simple $GL_n(\mathbf{k})$-equivariant minimal free $R$-resolutions for the quotient ring $\mathbf{k}=R/R_+$ and for these modules $M$. These also lead to elegant descriptions of $\mathrm{Tor}^R_i(M,M')$ for all $i$ and $\mathrm{Hom}_R(M,M')$ for any pair of these modules $M,M'$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_16342 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Equivariant resolutions over Veronese rings Almousa, Ayah Perlman, Michael Pevzner, Alexandra Reiner, Victor VandeBogert, Keller Commutative Algebra Combinatorics 13D02, 05E40 Working in a polynomial ring $S=\mathbf{k}[x_1,\ldots,x_n]$ where $\mathbf{k}$ is an arbitrary commutative ring with $1$, we consider the $d^{th}$ Veronese subalgebras $R=S^{(d)}$, as well as natural $R$-submodules $M=S^{(\geq r, d)}$ inside $S$. We develop and use characteristic-free theory of Schur functors associated to ribbon skew diagrams as a tool to construct simple $GL_n(\mathbf{k})$-equivariant minimal free $R$-resolutions for the quotient ring $\mathbf{k}=R/R_+$ and for these modules $M$. These also lead to elegant descriptions of $\mathrm{Tor}^R_i(M,M')$ for all $i$ and $\mathrm{Hom}_R(M,M')$ for any pair of these modules $M,M'$. |
| title | Equivariant resolutions over Veronese rings |
| topic | Commutative Algebra Combinatorics 13D02, 05E40 |
| url | https://arxiv.org/abs/2210.16342 |