GL-equivariant modules over polynomial rings in infinitely many variables
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- by Steven V Sam and Andrew Snowden PDF
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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
- MR Author ID: 836995
- ORCID: 0000-0003-1940-9570
- 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
- MR Author ID: 788741
- Email: asnowden@umich.edu
- 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
- © Copyright 2015 American Mathematical Society
- 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
- MathSciNet review: 3430359