Systems of ideals parametrized by combinatorial structures

2023/04/07 by Laudone, Robert P., Snowden, Andrew · 1 citation
#Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2304.03686

Abstract

A symmetric chain of ideals is a rule that assigns to each finite set S an ideal I_S in the polynomial ring ℂ[x_i]_{i ∈ S} such that if ϕ\colon S → T is an embedding of finite sets then the induced homomorphism ϕ_* maps I_S into I_T. Cohen proved a fundamental noetherian result for such chains, which has seen intense interest in recent years due to a wide array of new applications. In this paper, we consider similar chains of ideals, but where finite sets are replaced by more complicated combinatorial objects, such as trees. We give a general criterion for a Cohen-like theorem, and give several specific examples where our criterion holds. We also prove similar results for certain limiting situations, where a permutation group acts on an infinite variable polynomial ring. This connects to topics in model theory, such as Fraïssé limits and oligomorphic groups.

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