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Square-free Gröbner degenerations. (English) Zbl 07233317
Let \(K\) denote a field and let \(S = K[x_1,\ldots,x_n]\) denote the polynomial ring equipped with the standard grading. For a finitely generated graded \(S\)-module \(M\) let \(h^{ij}(M)\) denote the \(K\)-dimension of the degree \(j\) component of the \(i\)-th local cohomology module \(H^i_{\mathfrak{m}}(M)\) supported in the maximal ideal \(\mathfrak{m} = (x_1,\ldots,x_n)\). Let \(I \subset S\) be a homogeneous ideal with \(J\) an initial ideal with respect to a term order. Jürgen Herzog conjectured that the extremal Betti numbers of \(S/I\) and \(S/J\) coincide provided \(J\) is square-free. In their main result the authors prove that \(h^{ij}(S/I) = h^{ij}(S/J)\) provided \(J\) is square-free. Among others this solves Herzog’s Conjecture in the affirmative and has various interesting further consequences. Moreover, they discuss properties of ideals admitting square-free initial ideals and consequences of the main result about \(\operatorname{depth},\operatorname{reg}\) and the cohomological dimension of \(S/I\) and \(S/J\) resp. It is shown that ideals defining ASLs, Cartwright-Sturm ideals and Knutson ideals are families with square-free initial ideals. Finally further developments and questions are discussed. The main technical tool for the authors’ proof is the notion of cohomological full singularity as introduced by A. De Stefani [“Cohomologically full rings”, Preprint, arXiv:1806.00536].

13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
13D45 Local cohomology and commutative rings
CoCoA; Macaulay2
Full Text: DOI
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