GL-equivariant modules over polynomial rings in infinitely many variables. II

Abstract

Twisted commutative algebras (tca's) have played an important role in the nascent field of representation stability. Let A_d be the complex 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 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.

51 pages; v2: many revisions

References in corpus (3)

• Regularity and cohomology of determinantal thickenings
• Syzygies of bounded rank symmetric tensors are generated in bounded degree
• Hilbert series for twisted commutative algebras

Cited by in corpus (8)

• $\mathrm{VI}$ modules in non-describing characteristic, Part I
• Sp-equivariant modules over polynomial rings in infinitely many variables
• Generalizations of Stillman's conjecture via twisted commutative algebras
• Topological Noetherianity of polynomial functors II: base rings with Noetherian spectrum
• Regularity bounds for twisted commutative algebras
• GL-algebras in positive characteristic I: the exterior algebra
• Stillman's question for twisted commutative algebras
• A note on projective dimension over twisted commutative algebras