GL-EQUIVARIANT MODULES OVER POLYNOMIAL RINGS IN INFINITELY MANY VARIABLES. II

By Steven V. Sam, Andrew Snowden. Forum of Mathematics, Sigma, 2012

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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.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1493eb8f5363eca0f68acd68738a7c6d4909d01c

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Citations in the corpus

FI- and OI-modules with varying coefficients (2017, background): “over a field K that is invariant under the action of the symmetric group Sym(N) is generated by finitely many Sym(N)-orbits (see [1, 5, 7, 14] and also [6, 10, 22] for related results).”
Topological Noetherianity of polynomial functors (2017, background): “…a resolution of the Artinian conjecture [PS14] and to a resolution of a conjecture by Rauh-Sullivant on iterated toric fibre products [DO16], and finiteness results for secant varieties of Segre and Segre-Veronese embeddings; see [LM04, Rai12, Sam15, Sam16] and the notion of inheritance in [LO13].”