TY - JOUR

T1 - Topological Noetherianity of polynomial functors II: base rings with Noetherian spectrum

AU - Bik, Arthur

AU - Danelon, Alessandro

AU - Draisma, Jan

PY - 2023/4

Y1 - 2023/4

N2 - In a previous paper, the third author proved that finite-degree polynomial functors over infinite fields are topologically Noetherian. In this paper, we prove that the same holds for polynomial functors from free R-modules to finitely generated R-modules, for any commutative ring R whose spectrum is Noetherian. As Erman–Sam–Snowden pointed out, when applying this with R=Z to direct sums of symmetric powers, one of their proofs of a conjecture by Stillman becomes characteristic-independent. Our paper advertises and further develops the beautiful but not so well-known machinery of polynomial laws. In particular, to any finitely generated R-module M we associate a topological space, which we show is Noetherian when Spec(R) is; this is the degree-zero case of our result on polynomial functors.

AB - In a previous paper, the third author proved that finite-degree polynomial functors over infinite fields are topologically Noetherian. In this paper, we prove that the same holds for polynomial functors from free R-modules to finitely generated R-modules, for any commutative ring R whose spectrum is Noetherian. As Erman–Sam–Snowden pointed out, when applying this with R=Z to direct sums of symmetric powers, one of their proofs of a conjecture by Stillman becomes characteristic-independent. Our paper advertises and further develops the beautiful but not so well-known machinery of polynomial laws. In particular, to any finitely generated R-module M we associate a topological space, which we show is Noetherian when Spec(R) is; this is the degree-zero case of our result on polynomial functors.

UR - http://www.scopus.com/inward/record.url?scp=85126793606&partnerID=8YFLogxK

U2 - 10.1007/s00208-022-02386-9

DO - 10.1007/s00208-022-02386-9

M3 - Article

C2 - 37006405

SN - 0025-5831

VL - 385

SP - 1879

EP - 1921

JO - Mathematische Annalen

JF - Mathematische Annalen

IS - 3-4

ER -