Noetherianity of Rees algebras for arbitrary S-ideals in the infinite polynomial ring

Determine whether the Rees algebra R_a(R)=⊕_{n≥0} a^n t^n of the infinite-variable polynomial ring R=k[x_i]_{i≥1} (with the natural action of the infinite symmetric group S) is noetherian for every S-stable ideal a⊂R. Establish this for all S-ideals, generalizing the proven case a=h_s and extending the known affirmative result for monomial ideals, in order to support Artin–Rees type properties and the Inj condition for broader localizing subcategories in the module category over R/a.

Background

In Section 7.2, the authors prove that the specific Rees algebra R_{h_s}(R) is noetherian by identifying it with a polynomial ring in two infinite sets of variables, invoking Cohen’s S-noetherianity. This yields an Artin–Rees lemma for h_s and verifies property (Inj) for the corresponding localizing subcategory.

However, for a general S-stable ideal a⊂R, the noetherianity of the Rees algebra R_a(R) remains unresolved. The case when a is a monomial ideal is known to be affirmative via results of Draisma–Eggermont–Krone–Leykin (DEKL), but the methods do not currently extend to arbitrary S-ideals. A resolution would have significant implications for finiteness and derived-functor properties across broader Serre quotient setups in this equivariant commutative algebra framework.