Carlsson's rank conjecture and a conjecture on square-zero upper triangular matrices
Embargo Lift Date: 2021-06-01
Journal of Pure and Applied Algebra
2562 - 2584
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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 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.