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GL-equivariant modules over polynomial rings in infinitely many variables


Authors: Steven V Sam and Andrew Snowden
Journal: Trans. Amer. Math. Soc. 368 (2016), 1097-1158
MSC (2010): Primary 13A50, 13C05, 13D02, 05E05, 05E10, 16G20
DOI: https://doi.org/10.1090/tran/6355
Published electronically: June 17, 2015
MathSciNet review: 3430359
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Abstract: Consider the polynomial ring in countably infinitely many variables over a field of characteristic zero, together with its natural action of the infinite general linear group $ G$. We study the algebraic and homological properties of finitely generated modules over this ring that are equipped with a compatible $ G$-action. We define and prove finiteness properties for analogues of Hilbert series, systems of parameters, depth, local cohomology, Koszul duality, and regularity. We also show that this category is built out of a simpler, more combinatorial, quiver category which we describe explicitly.

Our work is motivated by recent papers in the literature which study finiteness properties of infinite polynomial rings equipped with group actions. (For example, the paper by Church, Ellenberg and Farb on the category of FI-modules, which is equivalent to our category.) Along the way, we see several connections with the character polynomials from the representation theory of the symmetric groups. Several examples are given to illustrate that the invariants we introduce are explicit and computable.


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Additional Information

Steven V Sam
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Address at time of publication: Department of Mathematics, University of California, Berkeley, California 94720
Email: svs@math.berkeley.edu

Andrew Snowden
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Address at time of publication: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: asnowden@umich.edu

DOI: https://doi.org/10.1090/tran/6355
Received by editor(s): June 25, 2012
Received by editor(s) in revised form: October 16, 2013, and December 16, 2013
Published electronically: June 17, 2015
Additional Notes: The first author was supported by an NDSEG fellowship while this work was being done
Article copyright: © Copyright 2015 American Mathematical Society