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| Format: | Preprint |
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2020
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| Online Access: | https://arxiv.org/abs/2008.12944 |
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| _version_ | 1866909958838157312 |
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| author | Şentürk, Berrin |
| author_facet | Şentürk, Berrin |
| contents | Let $A$ be the polynomial algebra in $r$ variables with coefficients in an algebraically closed field $k$. When the characteristic of $k$ is $2$, Carlsson conjectured that any $\mathrm{dg}$-$A$-module that is free of rank $N$ as an $A$-module and whose homology is nontrivial and finite dimensional as a $k$-vector space satisfies $N\geq 2^r$. In this paper, we examine a stronger conjecture concerning varieties of square-zero upper triangular $N\times N$ matrices. Stratifying these varieties via Borel orbits, we show that the stronger conjecture holds when $N = 8$ without any restriction on the characteristic of $k$. This result also verifies that if $X$ is a product of $3$ spheres of any dimensions, then the elementary abelian $2$-group of rank $4$ cannot act freely on $X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2008_12944 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | The rank 8 case of a conjecture on square-zero upper triangular matrices Şentürk, Berrin Commutative Algebra Algebraic Topology 13Dxx, 55M35 Let $A$ be the polynomial algebra in $r$ variables with coefficients in an algebraically closed field $k$. When the characteristic of $k$ is $2$, Carlsson conjectured that any $\mathrm{dg}$-$A$-module that is free of rank $N$ as an $A$-module and whose homology is nontrivial and finite dimensional as a $k$-vector space satisfies $N\geq 2^r$. In this paper, we examine a stronger conjecture concerning varieties of square-zero upper triangular $N\times N$ matrices. Stratifying these varieties via Borel orbits, we show that the stronger conjecture holds when $N = 8$ without any restriction on the characteristic of $k$. This result also verifies that if $X$ is a product of $3$ spheres of any dimensions, then the elementary abelian $2$-group of rank $4$ cannot act freely on $X$. |
| title | The rank 8 case of a conjecture on square-zero upper triangular matrices |
| topic | Commutative Algebra Algebraic Topology 13Dxx, 55M35 |
| url | https://arxiv.org/abs/2008.12944 |