Sp-equivariant modules over polynomial rings in infinitely many variables

Abstract

We study the category of Sp-equivariant modules over the infinite variable polynomial ring, where Sp denotes the infinite symplectic group. We establish a number of results about this category: for instance, we show that every finitely generated moduleM fits into an exact triangle T → M → F → where T is a finite length complex of torsion modules and 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) and Sym(C⊕Sym C) 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.