We compute the homotopy groups of Chan--Gerhardt--Klang equivariant topological Hochschild homology in the following cases: homotopical cobordism MU_G for finite abelian G, Real cobordism MR, and the constant Green functor with value Z/p when G = C_p.

In six pages* we show that the higher K-groups of the constant C_{p^n}-Tambara fields with value a characteristic p finite field are entirely torsion, and compute all of these groups after inverting p. The punchline is that, after inverting $p$, we get K(\underline{Q}) = K(Q)^{\times n+1} when $G = C_{p^n}$ for any $\F_p$-algebra $Q$.

*plus citations

Algebraic K-theory is a machine that eats categories of modules and spits out ring spectra. Green functors are equviariant algebra analogues of rings and come equipped with a category of modules. Given a Green functor k, one can take its G-theory (the K-theory of its abelian category of finitely generated modules) or its K-theory (the K-theory of its exact category of finitely generated projective modules). When the group is C_{p^n}, we construct a finite filtration for G-theory in terms of G-theories of certain twisted group rings. Under medium-strength conditions, this filtration actually splits as a bunch of summand inclusions. For example, upon taking the Burnside Green functor, this splitting generalizes a result of Greenlees upon specializing to C_p and taking \pi_0.

Classically, if a ring is regular, the resolution theorem implies G-theory and K-theory are equivalent. We are able to interpret various results by other authors as regularity statements about Green functors; consequently our G-theory spectral sequence becomes a K-theory spectral sequence. Using this, we give an explicit calculation of all higher algebraic K-theory groups of the constant C_2-Green functor with value F_2. Additionally, we describe the p-completion of the K-groups of the constant C_{p^n}-Green functor with value Z in terms of the p-completion of the K-groups of Z and of Z[p^{-1},\zeta_p], where \zeta_p is a primitive pth root of unity (which are known explicitly in odd degrees).

In commutative algebra, étale maps are analogues of covering spaces in topology, and are a very important tool (Grothendieck used them to prove some Weil conjectures, Deligne--Lusztig and Lusztig use them to compute the character tables of a lot of the finite simple groups). Hill defined a Tambara functor analogue of being étale and showed that localizations are étale, and Lindenstrauss--Richter--Zou showed how to construct étale Tambara functor maps from Galois field extensions (among other things).

In this paper I establish a lot of basic properties of étale Tambara functor maps: closure under composition, products, coinduction, (flat) base-change (although the flatness assumption seems unnecessary). Using these, show that a construction of Lindenstrauss--Richter--Zou for cyclic groups goes through for arbitrary groups, and classify finite étale extensions of any constant G-Tambara functor with value an algebraically closed field. Using this, I am able to classify finite affine étale group schemes over such constant Tambara functors (and the classification looks exactly like the classical one: the category of finite affine étale group schemes over a field is the category of finite groups with a continuous action of the Galois group of the separable closure).

Tambara functors come equipped with an analogue of the Zariski spectrum: the Nakaoka spectrum, the set of prime ideals with the expected topology. Previous work shows that the change-of-group functor coinduction is special, and I show that an H-Tambara functor and its coinduction have homeomorphic Nakaoka spectra. The adjunction units k --> Coind_H^G Res_H^G k then give rise to maps Spec(Res_H^G k) --> Spec(k) whose images stratify the Nakaoka spectrum of k, I call this the subgroup stratification. In (almost) every case where Spec(k) is explicitly known, I explicitly describe the subgroup stratification; it is typically by closed, nonopen subspaces.

In other work I show that k --> Coind_H^G Res_H^G k is étale. Thus the subgroup stratification gives examples of closed, nonopen étale maps of Tambara functors. This dramatically contrasts the classical fact that étale maps of schemes are always open. I'm not entirely sure what implications this has for Tambara functor algebraic geometry, but it sure is weird.

A coefficient system of abelian groups is a contravariant functor from finite transitive G-sets to abelian groups. A Mackey functor is a coefficient system, plus covariant structure maps called transfers, satisfying compatibility conditions. Sitting in-between these two are incomplete Mackey functors; coefficient systems with some transfers, but not necessarily as many as one needs to make a Mackey functor. The collection of pairs of finite transitive G-sets along which one asks for transfers in an incomplete Mackey functor is called a transfer system. The field of homotopical combinatorics is primarily concerned with enumerating the number of transfer systems for a given group. When G = C_{p^n}, an amazing paper of Balchin--Barnes--Roitzheim shows that you get the Catalan numbers!

Given a set of pairs of finite transitive G-sets, there's a smallest transfer system containing them. One can then ask for a smallest set generating a given transfer system; this is called a basis. First, we show that the size of a basis is an invariant of the transfer system, and then we study how big the size of a basis can be. Namely, when G = C_{p_1 p_2 ... p_n}, we show that there's a transfer system for which the size of its basis is unexpectedly large (formally, we give a lower bound for the minimum size of a basis among all transfer systems which we believe is sharp), and we also study what happens at some other groups.

Coinduction is a change-of-groups functor, right adjoint to restriction, and showed up independently in an essential way in two previous papers. The central point of this paper is that since coinduction is like a G/H-twisted product (taking an H-Tambara functor and producing a G-Tambara functor), it should enjoy all the nice properties that products of rings do. Using this perspective, some useful results in previous papers are organized and generalized with a view towards future work.

For example, we understand what a module over a product of rings is. A Noetherian ring is a finite product of rings which don't contain nontrivial idempotents. These kinds of basic statements have interesting Tambara analogues when we also view coinduction as behaving like a product. In later work, I give even more evidence: we understand ideals in a product of rings, and we understand Tambara ideals in coinductions. Products of étale ring maps are étale, and the canonical map k --> Coind_H^G Res_H^G k is always étale.

Burklund--Schlank--Yuan introduce the concept of "Nullstellensatzian" object of a category as something satisfying the conclusion of Hilbert's Nullstellensatz: every finitely presented algebra admits a section. This actually characterizes algebraically closed fields as objects in the category of rings, so we can take the perspective that the Nullstellensatzian Tambara functors are the right algebraically closed fields (looking back, there's even more evidence for this: I proved Nullstellensatzian Tambara functors have no nontrivial finite étale extensions).

Our headlining result is a straightforward classification of Nullstellensatzian Tambara functors: they are precisely coinductions of algebraically closed fields. However, we have a number of useful lemmas: every restriction in a Tambara functor is integral, checkable conditions for a right adjoint to preserve Nullstellensatzian objects, and compositions of compact morphisms are compact (an abstraction of the statement that being a finitely presented algebra is preserved under composition). 

What's the difference between "Real" complex cobordism MU_R and MU_{C_2}? Both are C_2-spectra with underlying spectrum MU, but support completely different orientation theory. It turns out that there's a C_2 \times C_2 spectrum MR_{C_2} which recovers each of these! Actually, following Schwede's construction, one should think of each MR_G as fitting together into a Real global spectrum.

Global homotopy theory is, roughly speaking, G-equivariant homotopy theory for all G simultaneously. Using global methods, Hausmann showed that equivariant complex cobordism MU_G carries the universal equivariant formal group law, by showing that global MU carries the universal global group law. Real global homotopy theory, roughly speaking, does this with an extra C_2 action (usually by complex conjugation) everywhere. In this paper, we establish Real global analogues of most of Hausmann's results, except we do not show that Schwede's Real global MU carries the universal Real global group law (although we get pretty close to it).

Field-like Tambara functors were introduced by Nakaoka. When G = C_2, the constant Green functor at F_2 is not a field-like Green functor, but it is a field-like Tambara functor. As this is an extremely common choice of Bredon cohomology coefficients, it seems like equivariant topologists should be studying field-like Tambara functors as an equivariant analogue of field coefficients. This paper explicitly classifies all field-like Tambara functor coefficients for Bredon cohomology when G = C_p and gives a recursive formula to determine all field-like Tambara functor coefficients when G = C_{p^n}.

Almost all field-like Tambara functors are fixed-point. This paper discovers a new kind of field-like Tambara functor which is not fixed point; it would be interesting to see if any weird computational stuff happens when taking these as Bredon cohomology coefficients.

The Landweber exact functor theorem is one way to construct a lot of important cohomology theories in chromatic homotopy theory, like Lubin--Tate theories, KU, BP, elliptic cohomology, topological modular forms, and more. It has the form "if a ring map MU^* --> k satisfies certain explicit, checkable algebraic conditions, then MU^*(-) \otimes_{MU^*} k is a cohomology theory." We say a cohomology theory is Landweber exact if it is isomorphic to one arising this way.

Equivariantly it is unclear even what the conclusion of this theorem should be: Lewis tells us equivariant homotopy groups should be considered as Mackey functors, which enjoy their own version of the tensor product called the box product. The box product is very complicated and one would like to avoid dealing with it. The central point of this paper is that it seems like we can, in the sense that given a ring map MU_G^* --> k, asking for the ordinary tensor product MU_G^*(-) \otimes_{MU_G^*} k to be a cohomology theory on G-spaces leads to interesting mathematics. In particular, equivariant versions of KU and BP (and, trivially, MU) all have this form. Also one can prove equivariant analogues of classical properties of Landweber exact cohomology theories using this new definition. E.g. there are no nontrivial phantom maps between KU_G, MU_G, and BP_G.