𝑆𝑝-equivariant modules over polynomial rings in infinitely many variables

Abstract

We study the category of S p \mathbf {Sp} -equivariant modules over the infinite variable polynomial ring, where S p \mathbf {Sp} denotes the infinite symplectic group. We establish a number of results about this category: for instance, we show that every finitely generated module M M fits into an exact triangle T → M → F → T \to M \to F \to where T T is a finite length complex of torsion modules and F F is a finite length complex of “free” modules; we determine the Grothendieck group; and we (partially) determine the structure of injective modules. We apply these results to show that the twisted commutative algebras Sym ⁡ ( C ∞ ⊕ ⋀ 2 C ∞ ) \operatorname {Sym}(\mathbf {C}^{\infty } \oplus \bigwedge ^2{\mathbf {C}^{\infty }}) and Sym ⁡ ( C ∞ ⊕ Sym 2 ⁡ C ∞ ) \operatorname {Sym}(\mathbf {C}^{\infty } \oplus \operatorname {Sym}^2{\mathbf {C}^{\infty }}) are noetherian, which are the strongest results to date of this kind. We also show that the free 2-step nilpotent twisted Lie algebra and Lie superalgebra are noetherian.