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

Şentürk B., Ünlü Ö.

Journal of Pure and Applied Algebra, vol.223, no.6, pp.2562-2584, 2019 (SCI-Expanded, Scopus) identifier

  • Publication Type: Article / Article
  • Volume: 223 Issue: 6
  • Publication Date: 2019
  • Doi Number: 10.1016/j.jpaa.2018.09.007
  • Journal Name: Journal of Pure and Applied Algebra
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.2562-2584
  • Keywords: Borel orbit, Projective variety, Rank conjecture, Square-zero matrices
  • TED University Affiliated: No

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.