Abstract
Twisted commutative algebras (tca’s) have played an important role in the nascent field of representation stability. Let $A_{d}$ be the tca freely generated by $d$ indeterminates of degree 1. In a previous paper, we determined the structure of the category of $A_{1}$-modules (which is equivalent to the category of $\mathbf{FI}$-modules). In this paper, we establish analogous results for the category of $A_{d}$-modules, for any $d$. Modules over $A_{d}$ are closely related to the structures used by the authors in previous works studying syzygies of Segre and Veronese embeddings, and we hope the results of this paper will eventually lead to improvements on those works. Our results also have implications in asymptotic commutative algebra.
Highlights
We prove a duality theorem for local cohomology and saturation with respect to the Fourier transform
In [Sn], the second author introduced a notion of Hilbert series for twisted commutative algebras and their modules, and proved a rationality result for the tca’s considered in this paper
We can take the local cohomology of the A(Cn)-module M(Cn) with respect to the ideal ar (Cn). We show that these two constructions are canonically isomorphic for n 0 when M is finitely generated
Summary
In recent years, twisted commutative algebras (tca’s) have played an important role in the nascent field of representation stability. Every object of Dbfg(ModA) admits a finite filtration where the graded pieces are shifts of modules of this form. In [Sn], the second author introduced a notion of Hilbert series for twisted commutative algebras and their modules, and proved a rationality result for the tca’s considered in this paper. We can take the local cohomology of the A(Cn)-module M(Cn) with respect to the ideal ar (Cn) We show that these two constructions are canonically isomorphic for n 0 when M is finitely generated. These results are mostly well known; we include this material to recall salient facts and set notation.
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