  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 module M fits into an exact triangle $T \to M \to F \to$
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 ${\rm Sym}({\bf
C}^{\infty} \oplus \bigwedge^2{\bf C}^{\infty})$ and ${\rm Sym}({\bf
C}^{\infty} \oplus {\rm Sym}^2{\bf C}^{\infty})$ 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.
