VImodules in nondescribing characteristic, part I.
 [Abstract] [arXiv]
Fix a finite field F. Let VI be the category of finite dimensional Fvector spaces with injections, and let k be a noetherian ring. We study the category of functors from VI to kmodules in the case when the characteristic of F is invertible in k. Our results include a structure theorem, finiteness of regularity, and a description of the hilbert series.
Linear and quadratic ranges in representation stability with Thomas Church, Jeremy Miller and Jens Reinhold.
 [Abstract] [arXiv]
We prove two general results concerning spectral sequences of FImodules. These results can be used to significantly improve stable ranges in a large portion of the stability theorems for FImodules currently in the literature. We work this out in detail for the cohomology of configuration spaces where we prove a linear stable range and the homology of congruence subgroups of general linear groups where we prove a quadratic stable range. Previously, the best stable ranges known in these examples were exponential. Up to an additive constant, our work on congruence subgroups verifies a conjecture of Djament.
Periodicity in the cohomology of symmetric groups via divided powers with Andrew Snowden.
 Under revision at Proc. Lond. Math. Soc.
 [Abstract] [arXiv]
A famous theorem of Nakaoka asserts that the cohomology of the symmetric group stabilizes. The first author generalized this theorem to nontrivial coefficient systems, in the form of FImodules over a field, though one now obtains periodicity of the cohomology instead of stability. In this paper, we further refine these results. Our main theorem states that if M is a finitely generated FImodule over a noetherian ring k then ⊕_{n≥0} H^{t}(S_{n}, M_{n}) admits the structure of a Dmodule, where D is the divided power algebra over k in a single variable, and moreover, this Dmodule is "nearly" finitely presented. This immediately recovers the periodicity result when k is a field, but also shows, for example, how the torsion varies with n when k=Z. Using the theory of connections on Dmodules, we establish sharp bounds on the period in the case where k is a field. We apply our theory to obtain results on the modular cohomology of Specht modules and the integral cohomology of unordered configuration spaces of manifolds.
Regularity of FImodules and local cohomology with Steven V Sam and Andrew Snowden.
 [Abstract] [arXiv]
We resolve a conjecture of Li and Ramos that relates the regularity of an FImodule to its local cohomology groups. This is an analogue of the familiar relationship between regularity and local cohomology in commutative algebra.
Noetherianity of some degree two twisted skewcommutative algebras with Steven V Sam and Andrew Snowden.
 [Abstract] [arXiv]
A major open problem in the theory of twisted commutative algebras (tca's) is proving noetherianity of finitely generated tca's. For bounded tca's this is easy, in the unbounded case, noetherianity is only known for Sym(Sym^{2}(C^{∞})) and Sym(∧^{2}(C^{∞})). In this paper, we establish noetherianity for the skewcommutative versions of these two algebras, namely ∧(Sym^{2}(C^{∞})) and ∧(∧^{2}(C^{∞})). The result depends on work of Serganova on the representation theory of the infinite periplectic Lie superalgebra, and has found application in the work of MillerWilson on "secondary representation stability" in the cohomology of configuration spaces.
The module theory of divided power algebras with Andrew Snowden.
 [Abstract] [arXiv]
We study modules for the divided power algebra D in a single variable over a commutative noetherian ring k. Our first result states that D is a coherent ring. In fact, we show that there is a theory of Gröbner bases for finitely generated ideals, and so computations with finitely presented Dmodules are in principle algorithmic. We go on to determine much about the structure of finitely presented Dmodules, such as: existence of certain nice resolutions, computation of the Grothendieck group, results about injective dimension, and how they interact with torsion modules. Our results apply not just to the classical divided power algebra, but to its qvariant as well, and even to a much broader class of algebras we introduce called "generalized divided power algebras." On the other hand, we show that the divided power algebra in two variables over Z_{p} is not coherent.
Gröbnercoherent rings and modules with Andrew Snowden.
 Journal of commutative algebra, to appear.
 [Abstract] [arXiv]
Let R be a graded ring. We introduce a class of graded Rmodules called Gröbnercoherent modules. Roughly, these are graded Rmodules that are coherent as ungraded modules because they admit an adequate theory of Gröbner bases. The class of Gröbnercoherent modules is formally similar to the class of coherent modules: for instance, it is an abelian category closed under extension. However, Gröbnercoherent modules come with tools for effective computation that are not present for coherent modules.
FImodules and the cohomology of modular representations of the symmetric groups.
 Thesis work done under Jordan S. Ellenberg.

[Abstract] [arXiv]
An FImodule V over a commutative ring k encodes a sequence (V_{n})_{n≥0} of representations of the symmetric groups (S_{n})_{n≥0} over k. In this paper, we show that for a finitely generated FImodule V over a field of characteristic p, the cohomology groups H^{t}(S_{n} , V_{n}) are eventually periodic in n. We describe a recursive way to calculate the period and the periodicity range, and show that the period is always a power of p. As an application, we show that if M is compact, connected, oriented manifold of dimension ≥ 2, and conf_{n}(M) is the configuration space of unordered ntuples of distinct points in M then the dimensions of the modp cohomology groups H^{t}(conf_{n}(M), k) are eventually periodic in n with period a power of p.
Noetherianity of some degree two twisted commutative algebras with Steven V Sam and Andrew Snowden.
 Selecta Mathematica (N.S.) 222 (2016), 913–937.
 [Abstract] [arXiv]
In recent years, researchers have discovered various large algebraic structures that have surprising finiteness properties, such as FImodules and Deltamodules. In this paper, we add another example to the growing list: we show that certain degree two twisted commutative algebras are noetherian. This example appears to have some fundamental differences from previous examples, and is therefore especially interesting. Reflective
of this, our proof introduces new methods for establishing noetherianity that are likely to be applicable in other situations. The algebras considered in this paper are closely related to the stable representation theory of classical groups, which is one source of motivation for their study.
FImodules over Noetherian rings with Thomas Church, Jordan S. Ellenberg and Benson Farb.
 Geometry and Topology 185 (2014), 2951–2984.
 [Abstract] [arXiv]
FImodules were introduced by Church, Ellenberg and Farb in [CEF] to encode sequences of representations of symmetric groups. Over a field of characteristic 0, finite generation of an FImodule implies representation stability for the corresponding sequence of S_{n}representations. In this paper we prove the Noetherian property for FImodules over
arbitrary Noetherian rings: any subFImodule of a finitelygenerated FImodule is finitely
generated. This lets us extend many of the results of [CEF] to representations in positive characteristic, and even to integral coefficients. We focus on three major applications of the main theorem: on the integral and mod p cohomology of configuration spaces; on diagonal coinvariant algebras in positive characteristic; and on an integral version of Putman's central stability for homology of congruence subgroups.