Carlsson's rank conjecture and a conjecture on square-zero upper triangular matrices
| Autor: | ��ent��rk, Berrin, ��nl��, ��zg��n |
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| Rok vydání: | 2017 |
| Předmět: | |
| DOI: | 10.48550/arxiv.1706.03217 |
| Popis: | Let $k$ be an algebraically closed field and $A$ the polynomial algebra in $r$ variables with coefficients in $k$. In case the characteristic of $k$ is $2$, Carlsson conjectured that for any $DG$-$A$-module $M$ of dimension $N$ as a free $A$-module, if the homology of $M$ is nontrivial and finite dimensional as a $k$-vector space, then $2^r\leq N$. Here we state a stronger conjecture about varieties of square-zero upper-triangular $N\times N$ matrices with entries in $A$. Using stratifications of these varieties via Borel orbits, we show that the stronger conjecture holds when $N < 8$ or $r < 3$ without any restriction on the characteristic of $k$. As a consequence, we attain a new proof for many of the known cases of Carlsson's conjecture and give new results when $N > 4$ and $r = 2$. 21 pages, final version to appear in JPAA |
| Databáze: | OpenAIRE |
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