Carlsson's rank conjecture and a conjecture on square-zero upper triangular matrices |
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Authors: | Berrin ?entürk Özgün Ünlü |
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Institution: | Department of Mathematics, Bilkent University, Ankara, 06800, Turkey |
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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 . Here we state a stronger conjecture about varieties of square-zero upper triangular matrices with entries in A. Using stratifications of these varieties via Borel orbits, we show that the stronger conjecture holds when or 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 and . |
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Keywords: | 55M35 13D22 13D02 Rank conjecture Square-zero matrices Projective variety Borel orbit |
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