Carlsson's rank conjecture and a conjecture on squarezero 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 squarezero uppertriangular $N\times N$ 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 eprints
 Pub Date:
 June 2017
 DOI:
 10.48550/arXiv.1706.03217
 arXiv:
 arXiv:1706.03217
 Bibcode:
 2017arXiv170603217S
 Keywords:

 Mathematics  Commutative Algebra;
 55M35;
 13D22;
 13D02
 EPrint:
 21 pages, final version to appear in JPAA