<?xml version="1.0" encoding="utf-8"?> <feed xmlns="http://www.w3.org/2005/Atom"> <title type="text">Recent zbMATH articles in MSC 13H05</title> <id>https://zbmath.org/atom/cc/13H05</id> <updated>2025-04-04T17:10:03.436181Z</updated> <link href="https://zbmath.org/" /> <link href="https://zbmath.org/atom/cc/13H05" rel="self" /> <generator>Werkzeug</generator> <entry xml:base="https://zbmath.org/atom/cc/13H05"> <title type="text">Notes on Lefschetz properties and linear elements of maximal height with applications</title> <id>https://zbmath.org/1553.13017</id> <updated>2025-04-04T17:10:03.436181Z</updated> <link href="https://zbmath.org/1553.13017" /> <author> <name>&quot;Smith, Larry&quot;</name> <uri>https://zbmath.org/authors/?q=ai:smith.larry.1|smith.larry</uri> </author> <content type="text">Summary: For a Poincar茅 duality quotient \(A\) of \(S=\mathbb{F}[x_1, \dots, x_n]\) we show that the strong Lefschetz property for \(A\) implies there is a linear form of maximal height and use Macaulay's Double Duality Theorem in the guise of inverse systems to provide a criterion for the existance of such a form. We adopt the viewpoint that the presence or lack of a linear form of maximal height is a test for the strong Lefschetz property in a Poincar茅 duality algebra. In characteristic zero such a form always exists. The test using Macaulay's inverse systems leads to a number of non examples in nonzero characteristic. By using Wu's formula, Steenrod operations, and Mitchell's Theorem we show that rings of coinvariants of reflection groups with polynomial invariants need not contain a linear element of maximal height if the rank of the coinvariants is two or more and the ground field \(\mathbb{F}\) has characteristic \(p \ne 0\). We apply this to show that Dickson and symmetric coinvariant algebras do not have the strong Lefschetz property over their defining fields, respectively the prime field, if their rank is at least two. Finally we investigate the connection of Lefschetz properties with the graded Frobenius functor for \(S\)-modules and show there are no forms of maximal height after applying this functor to \(A\) based on our criteria using Macaulay's theory so the Frobenius functor breaks the strong Lefschetz property. For the entire collection see [Zbl 1544.13003].</content> </entry> </feed>