Preserving Lefschetz properties after extension of variables

2025/03/21 by Kling, Filip Jonsson · 1 citation
#13C40 #13D40 #13E10 #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2503.16990

Abstract

Consider a standard graded artinian k-algebra B and an extension of B by a new variable, A=B⊗_k k[x]/(x^d) for some d≥ 1. We will show how maximal rank properties for powers of a general linear form on A can be determined by maximal rank properties for different powers of general linear forms on B. This is then used to study Lefschetz properties of algebras that can be obtained via such extensions. In particular, it allows for a new proof that monomial complete intersections have the strong Lefschetz property over a field of characteristic zero. Moreover, it gives a recursive formula for the determinants that show up in that case. Finally, for algebras over a field of characteristic zero, we give a classification for what properties B must have for all extensions B⊗_k k[x]/(x^d) to have the weak or the strong Lefschetz property.

Cited by

• The Weak Lefschetz Property for Tensor Products of Artinian Monomial Algebras and Its Applications to Lollipop Graphs

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