Browsing by Subject "Rank conjecture"

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    Carlsson's rank conjecture and a conjecture on square-zero upper triangular matrices
    (Elsevier, 2018) Şentürk, Berrin; Ünlü, Özgün
    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.
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    Minimal models of some differential graded modules
    (Springer Science and Business Media B.V., 2023-01-03) Şentürk, B.; Ünlü, Özgün
    Minimal models of chain complexes associated with free torus actions on spaces have been extensively studied in the literature. In this paper, we discuss these constructions using the language of operads. The main goal of this paper is to define a new Koszul operad that has projections onto several of the operads used in these minimal model constructions.