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      Carlsson's rank conjecture and a conjecture on square-zero upper triangular matrices

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      Embargo Lift Date: 2021-06-01
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      Author(s)
      Şentürk, Berrin
      Ünlü, Özgün
      Date
      2018
      Source Title
      Journal of Pure and Applied Algebra
      Print ISSN
      0022-4049
      Publisher
      Elsevier
      Volume
      223
      Issue
      6
      Pages
      2562 - 2584
      Language
      English
      Type
      Article
      Item Usage Stats
      171
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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.
      Keywords
      Rank conjecture
      Square-zero matrices
      Projective variety
      Borel orbit
      Permalink
      http://hdl.handle.net/11693/53066
      Published Version (Please cite this version)
      https://dx.doi.org/10.1016/j.jpaa.2018.09.007
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