Carlsson's rank conjecture and a conjecture on square-zero upper triangular matrices
Date
2018Source Title
Journal of Pure and Applied Algebra
Print ISSN
0022-4049
Publisher
Elsevier
Volume
223
Issue
6
Pages
2562 - 2584
Language
English
Type
ArticleItem Usage Stats
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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 [9] 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 2r ≤ N. Here we
state a stronger conjecture about varieties of square-zero upper triangular N × 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 obtain a new proof for
many of the known cases of Carlsson’s conjecture and give new results when N > 4
and r = 2.