Guardado en:
| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2020
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2008.12944 |
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- 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 $X$.