Systems of ideals parameterized by combinatorial structures

Abstract A symmetric chain of ideals is a rule that assigns to each finite set an ideal in the polynomial ring such that if is an embedding of finite sets then the induced homomorphism maps into . 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.