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Hilbert functions of graded modules over an exterior algebra: an algorithmic approach. (English) Zbl 1430.13002

Authors’ abstract: Let \(K\) be a field, \(E\) the exterior algebra of a finite dimensional \(K\)-vector space, and \(F\) a finitely generated graded free \(E\)-module with homogeneous basis \(g_1,\dots,g_r\) such that \(\deg g_1\leq \deg g_2\leq \dots \leq\deg g_r\). Given the Hilbert function of a graded \(E\)-module of the type \(F/M\), with \(M\) graded submodule of \(F\), the existence of the unique lexicographic submodule of \(F\) with the same Hilbert function as \(M\) is proved by a new algorithmic approach. Such an approach allows us to establish a criterion for determining if a sequence of nonnegative integers defines the Hilbert function of a quotient of a free \(E\)-module only via the combinatorial Kruskal-Katona’s theorem.

MSC:

13A02 Graded rings
15A75 Exterior algebra, Grassmann algebras
18G10 Resolutions; derived functors (category-theoretic aspects)
68W30 Symbolic computation and algebraic computation
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References:

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