Forcing the strong Lefschetz and the maximal rank properties

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Three basic properties which standard graded artinian -algebras may or may not enjoy are the Weak and Strong Lefschetz Properties and the Maximal Rank Property (respectively WLP, SLP, and MRP).

In this paper we will assume that the base field has characteristic zero. It is known that SLP implies MRP, which in turn implies WLP, but that both implications are strict. However, it surprisingly turned out (see [T. Harima, J. Migliore, U. Nagel, J. Watanabe, The weak and strong Lefschetz properties for artinian -algebras, J. Algebra 262 (2003) 99–126]) that the set of Hilbert functions admitting any algebras with WLP coincides with the corresponding set for SLP (and therefore with that for MRP). In [J. Migliore, F. Zanello, The Hilbert functions which force the weak Lefschetz property, J. Pure Appl. Algebra 210 (2) (2007) 465–471], J. Migliore and the first author, using Green’s theorem and a result of Wiebe, characterized the Hilbert functions forcing all algebras to enjoy WLP. The purpose of this note is to prove the corresponding characterizations for both SLP and MRP. Unsurprisingly (or surprisingly??), the two characterizations coincide, but they define a class of Hilbert functions strictly smaller than that determined for WLP in the above cited reference.

Our methods include the Herzog–Popescu theorem on quotients of -algebras modulo a general form, a result of Wiebe, and gins and stable ideals. At the end, we will also discuss the importance of assuming that the characteristic be zero, and we will exhibit a class of codimension 2 monomial complete intersections for which SLP (but not MRP) fails in positive characteristic.

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Copyright 2008 Elsevier B.V. All rights reserved. Publisher’s version of record: https://doi.org/10.1016/j.jpaa.2008.11.015

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Journal of Pure and Applied Algebra