Research

We show that Segal's K-theory of symmetric monoidal categories can be factored through Waldhausen categories. In particular, given a symmetric monoidal category C, we produce a Waldhausen category whose K-theory is weakly equivalent to the Segal K-theory of C. As a consequence, we show that every connective spectrum may be obtained via Waldhausen K-theory.

We compute the spectrum of prime ideals in the Burnside Tambara functor over an arbitrary finite group.

In the past decades, one of the most fruitful approaches to the study of algebraic K-theory has been trace methods, which construct and study trace maps from algebraic K-theory to topological Hochschild homology and related invariants. In recent years, theories of equivariant algebraic K-theory have emerged, but thus far few tools are available for the study and computation of these theories. In this paper, we lay the foundations for a trace methods approach to equivariant algebraic K-theory.  For G a finite group, we construct a Dennis trace map from equivariant algebraic K-theory to a G-equivariant version of topological Hochschild homology; for G the trivial group this recovers the ordinary Dennis trace map. We show that upon taking fixed points, this recovers the trace map of Adamyk--Gerhardt--Hess--Klang--Kong, and gives a trace map from the fixed points of coarse equivariant A-theory to the free loop space. We also establish important properties of equivariant topological Hochschild homology, such as Morita invariance, and explain why it can be considered as a multiplicative norm.

abstract: Tambara functors are the analogue of commutative rings in equivariant algebra. Nakaoka defined ideals in Tambara functors, leading to the definition of the Nakaoka spectrum of prime ideals in a Tambara functor. In this work, we continue the study of the Nakoaka spectra of Tambara functors. We describe, in terms of the Zariski spectra of ordinary commutative rings, the Nakaoka spectra of many Tambara functors. In particular: we identify the Nakaoka spectrum of the fixed point Tambara functor of any G-ring with the GIT quotient of its classical Zariski spectrum; we describe the Nakaoka spectrum of the complex representation ring Tambara functor over a cyclic group of prime order p; we describe the affine line (the Nakaoka spectra of free Tambara functors on one generator) over a cyclic group of prime order p in terms of the Zariski spectra of Z[x], Z[x, y], and the ring of cyclic polynomials Z[x_0, . . . , x_{p-1}]^{C_p} . 

To obtain these results, we introduce a “ghost construction” which produces an integral extension of any C_p-Tambara functor, the Nakaoka spectrum of which is describable. To relate the Nakaoka spectrum of a Tambara functor to that of its ghost, we prove several new results in equivariant commutative algebra, including a weak form of the Hilbert basis theorem, going up, lying over, and levelwise radicality of prime ideals in Tambara functors. These results also allow us to compute the Krull dimensions of many Tambara functors.

abstract: We compute the RO(G)-graded equivariant algebraic K-groups of a finite field with an action by its Galois group G. Specifically, we show these K-groups split as the sum of an explicitly computable term and the well-studied RO(G)-graded coefficient groups of the equivariant Eilenberg--MacLane spectrum HZ. Our comparison between the equivariant K-theory spectrum and HZ further shows they share the same Tate spectra and geometric fixed point spectra. In the case where G has prime order, we provide an explicit presentation of the equivariant K-groups. 

abstract: We provide a unifying approach to different constructions of the algebraic K-theory of equivariant symmetric monoidal categories. A consequence of our work is that the K-theory functor of Bohmann--Osorno models all connective genuine G-spectra.

abstract: We introduce a version of algebraic K-theory for coefficient systems of rings which is valued in genuine G-spectra for a finite group G. We use this construction to build a genuine G-spectrum associated to a G-space X, which provides a home for equivariant versions of classical invariants like the Wall finiteness obstruction. We provide a comparison between our K-theory spectrum and the equivariant A-theory of Malkiewich--Merling via a genuine equivariant linearization map.

abstract: We introduce two definitions of G-equivariant partitions of a finite G-set, both of which yield G-equivariant partition complexes. By considering suitable notions of equivariant trees, we show that G-equivariant partitions and G-trees are G-homotopy equivalent, generalizing existing results for the non-equivariant setting.  Along the way, we develop equivariant versions of Quillen's Theorems A and B, which are of independent interest.

abstract: Tambara functors are an equivariant generalization of rings that appear as the homotopy groups of genuine equivariant commutative ring spectra. In recent work, Blumberg and Hill have studied the corresponding algebraic structures, called bi-incomplete Tambara functors, that arise from ring spectra indexed on incomplete G-universes. In this paper, we answer a conjecture of Blumberg and Hill by proving a generalization of the Hoyer–Mazur theorem in the bi-incomplete setting. Bi-incomplete Tambara functors are characterized by indexing categories which parameterize incomplete systems of norms and transfers. In the course of our work, we develop several new tools for studying these indexing categories. In particular, we provide an easily checked, combinatorial characterization of when two indexing categories are compatible in the sense of Blumberg and Hill.

abstract: We show the homological Serre spectral sequence with coefficients in a field is a spectral sequence of coalgebras.  We also identify the comultiplication on the E^2 page of the spectral sequence as being induced by the usual comultiplication in homology.  At the end, we provide some example computations highlighting the use the co-Leibniz rule.