Certain monomial ideals whose numbers of generators of powers descend

2020/05/20 by Abdolmaleki, Reza, Kumashiro, Shinya
#Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2005.09991

Abstract

This paper studies the numbers of minimal generators of powers of monomial ideals in polynomial rings. For a monomial ideal I in two variables, Eliahou, Herzog, and Saem gave a sharp lower bound μ(I²)≥ 9 for the number of minimal generators of I² with μ(I)≥ 6. Recently, Gasanova constructed monomial ideals such that μ(I)>μ(Iⁿ) for any positive integer n. In reference to them, we construct a certain class of monomial ideals such that μ(I)>μ(I²)>⋯ >μ(Iⁿ)=(n+1)² for any positive integer n, which provides one of the most unexpected behaviors of the function μ(I^k). The monomial ideals also give a peculiar example such that the Cohen-Macaulay type (or the index of irreducibility) of R/Iⁿ descends.

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