𝑆𝑝-equivariant modules over polynomial rings in infinitely many variables

Sam, Steven; Snowden, Andrew

doi:10.1090/tran/8496

$\mathrm {Sp}$-equivariant modules over polynomial rings in infinitely many variables
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Trans. Amer. Math. Soc. 375 (2022), 1671-1701 Request permission

Abstract:

We study the category of $\mathbf {Sp}$-equivariant modules over the infinite variable polynomial ring, where $\mathbf {Sp}$ denotes the infinite symplectic group. We establish a number of results about this category: for instance, we show that every finitely generated module $M$ 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 $\operatorname {Sym}(\mathbf {C}^{\infty } \oplus \bigwedge ^2{\mathbf {C}^{\infty }})$ and $\operatorname {Sym}(\mathbf {C}^{\infty } \oplus \operatorname {Sym}^2{\mathbf {C}^{\infty }})$ 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.

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