Carlsson's rank conjecture and a conjecture on square-zero upper triangular matrices
Abstract
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 $r = 2$.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2017
- DOI:
- 10.48550/arXiv.1706.03217
- arXiv:
- arXiv:1706.03217
- Bibcode:
- 2017arXiv170603217S
- Keywords:
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- Mathematics - Commutative Algebra;
- 55M35;
- 13D22;
- 13D02
- E-Print:
- 21 pages, final version to appear in JPAA