Jonas McCandless has written a paper [M] on ‘curves’ in $\K$-theory and the connection to $\TR$, topological restriction theory. This is a funny name: $\TR$ is named according to the convention by which one prepends `topological’ to denote a theory computed over the sphere spectrum $\bS$ as opposed to the ring of integers $\bZ$ (e.g., $\THH$ instead of $\HH$). However, there is no non-$\T$ version of $\TR$, which creates a kind of pickle in terms of writing it out or speaking it aloud.

Anyways, topological restriction theory was for decades the computational Yerba Buena between $\THH$ and $\TC$. If $A$ is a connective $\bE_1$-ring, one wants to compute $\TC(A)$ as the best possible accessible approximation to algebraic $\K$-theory of $A$ and it was done in two steps for a long time: from $\THH(A),$ compute $\TR(A)$ which admits a residual Frobenius operator $\F$. Then, $\TC(A)\simeq\TR(A)^{\F=1}$ is the spectrum of $\F$-fixed points. Specifically, the Dundas–Goodwillie–McCarthy theorem asserts that if $\widetilde{A}\rightarrow A$ is a map of connective $\bE_1$-rings such that the kernel of $\pi_0\widetilde{A}\rightarrow\pi_0 A$ is nilpotent, then the fiber of $\K(\widetilde{A})\rightarrow\K(A)$ is naturally equivalent to the fiber of $\TC(\widetilde{A})\rightarrow\TC(A)$. Put another way, the commutative diagram

$\begin{CD} \K(\widetilde{A})@>>> \K(A)\\ @VVV @VVV\\ \TC(\widetilde{A}) @>>> \TC(A). \end{CD}$

is cartesian. A significant class of known computations of algebraic $\K$-groups uses this fact to reduce to simpler rings from more complicated ones. This is the case for example with the calculation of $\K(\bF_p[x]/(x^2))$ from $\K(\bF_p)$ (due to Quillen), $\TC(\bF_p)$ (due I suppose to Bökstedt), and $\TC(\bF_p[x]/(x^2))$.

Added 20 July 2021: Lars Hesselholt corrected me on the history of the computation of $\TC(\bF_p)$. He wrote, “The history is that Bökstedt and Madsen had already done the much harder calculation of $\TC(\bZ_p)$. Ib presented this calculation at a K-theory conference in Strasbourg in 1992 and, a week later, at the first European Congress in Paris. They published the calculation of $\TC(\bZ_p)$ in the proceedings from the Strasbourg conference, but Ib also needed something to put in the proceedings from the Paris congress, so he realized that the same arguments, but in a much easier form, also gave $\TC(\bF_p)$. Therefore the first published account is on mathscinet.”

Nikolaus and Scholze bypassed all the tolls by creating the Transbay Tube [NS] of $\TC$, which gives a conceptually and computationally easier way to compute $\TC(A)$ from $\THH(A)$ using (topological) negative cyclic and periodic cyclic homology. However, the importance of $\TR$ has persisted. For example, Thomas and I found in [AN] that the homotopy groups of $\TR$ compute the cyclotomic homotopy groups and that $p$-typical $\TR$ is the cofree $p$-typical topological Cartier module on $\TR$. Similarly, Krause and Nikolaus showed in [KN] that if $X$ is a $p$-typical cyclotomic spectrum, then $\TR(X,p)$ is the cofree $p$-typical cyclotomic spectrum with Frobenius lifts on $X$.

The $p$-typical story is clearly a part of an integral story and Thomas and I defined the $\infty$-category of integral topological Cartier modules, but we did not study the theory.

McCandless introduces this $\infty$-category and gives some basic properties on the way to proving several theorems generalizing work of Hesselholt and of Blumberg–Mandell.

# Cyclotomic spectra and all that

There are three closely related $\infty$-categories of relevance to cyclotomic spectra. The contemporary definition of a cyclotomic spectrum, due to Nikolaus–Scholze, is a spectrum $X$ with $S^1$-action together with Frobenius maps

$\varphi_p\colon X\rightarrow X^{\t C_p}$

for each prime $p$, where $X^{\t C_p}$ is given the residual $S^1/C_p$-action. The maps are suitable maps $X\rightarrow Y$ which commute with the Frobenius maps in the appropriate sense. Let $\CycSp$ be the $\infty$-category of cyclotomic spectra.

A cyclotomic spectrum with Frobenius lifts is a spectrum $X$ with $S^1$-action and Frobenius maps

$\psi_n\colon X\rightarrow X^{\h C_n}$

for every integer $n\geq 1$. These are required to satisfy some coherence conditions, which we will suppress for the moment. By restricting to $\psi_p$ and composing with the natural maps $X^{\h C_p}\rightarrow X^{\t C_p}$, one sees that every cyclotomic spectrum with Frobenius lifts gives rise to a cyclotomic spectrum. Let $\CycSp^\Fr$ denote the $\infty$-category of cyclotomic spectra with Frobenius lifts. In fact, this is the $\infty$-category of functors $\B\bW^\op\rightarrow\Sp$, where $\bW$ is the Witt monoid, the $\bE_1$-monoid given as a semidirect product $\bT\rtimes\bN^\times$, where $k\in \bN^\times$ acts on $\bT$ by $x\mapsto x^k.$ The importance of the Witt monoid was pointed out by Ayala, Mazel-Gee, and Rozenblyum in [AMR].

Finally, one has topological Cartier modules. These can be defined as follows. Let $\Sp_\bT$ denote the $\infty$-category of ‘genuine’ $\bT=S^1$-spectra, what Barwick and Glasman call cyclonic spectra in [BG]. These have genuine fixed points spectra for each finite subgroup $C_n\subseteq\bT$. Given an integer $n\geq 1$, there are two endofunctors of $\Sp_\bT$, one given by the fixed points functor $(-)^{C_n}$ and the other given by the geometric fixed points functor $(-)^{\Phi C_n}$. These assemble into two $\bN^\times$-actions on $\Sp_\bT$. The $\infty$-category $\TCart$ of topological Cartier modules is

$(\Sp_{\bT})^{\h\bN^\times},$

using the fixed points action of $\bN^\times$, while the $\infty$-category $\CycSp^{\mathrm{gen}}$ of genuine cyclotomic spectra is

$(\Sp_{\bT})^{\h\bN^\times},$

using the geometric fixed points action. Roughly speaking, these $\infty$-categories consist of genuine $\bT$-spectra $X$ equipped with compatible families of equivalences of genuine $\bT$-spectra $X\we X^{C_n}$ or $X\we X^{\Phi C_n}$ for $n\geq 1$.

There are some real theoretical gems in this paper, for example McCandless’ observation that $\B\bW$ is an orbital $\infty$-category in the sense of Barwick and that $\TCart$ is equivalent to the $\infty$-category of spectral Mackey functors on $\B\bW^{\op}$. Or, the following Nikolaus–Scholze-style equalizer formula for $\TR$ of a cyclotomic spectrum $X$:

$\TR(X)\we\mathrm{fib}\left(\prod_{k\geq 1}X^{\h C_k}\rightrightarrows\prod_{\text{p prime}}\prod_{k\geq 1}(X^{\t C_p})^{\h C_k}\right),$

for appropriate parallel maps. But, these are assistants for the following main theorems.

# The representability theorem

Blumberg and Mandell proved in [BM] that the functor

$\TR\colon\CycSp^{\mathrm{gen}}\rightarrow\Sp$

is representable by the cyclotomic spectrum $\widetilde{\THH}(\bS[\bN])$, the fiber of the natural augmentation map $\THH(\bS[\bN])\rightarrow\bS$. Here, $\bS[\bN]$ is the spherical monoid algebra of $\bN$.

However, $\TR(A)$ has additional structure: it is a cyclotomic spectrum itself, with Frobenius lifts. The first main theorem of McCandless is the representability theorem, which says that $\widetilde{\THH}(\bS[\bN])$ is an internal cyclotomic spectrum with Frobenius lifts in cyclotomic spectra, and that computing the mapping spectrum out yields $\TR$ with the functoriality above.

# The curves theorem

Work of Hesselholt [H] and Betley–Schlichtkrull [BS] connects $\TR(A)$, where $A$ is an associative ring, to

$\lim_k\Omega\K(A[t]/t^k,(t)),$

after profinite completion, a limit involving relative $\K$-groups. The curves theorem of McCandless is that this is true (1) for any connective $\bE_1$-ring and (2) that it holds integrally, i.e., without profinite completion. The proof is additionally rather different than that of previous authors and uses a serious analysis of the fiber computation of $\TR(A)$ mentioned above. Specifically, the main idea is to use the natural graded structure on

$\TR(A)\we\mathbf{Map}_{\CycSp}(\widetilde{\THH}(\bS[t]),\THH(A))$

to compare it to

$\lim_k\Omega(\THH(A)\otimes\widetilde{\THH}(\bS[t]/t^k))\we\lim_k\Omega\widetilde{\THH}(A[t]/t^k).$

Together with the Dundas–Goodwillie–McCarthy theorem, this is enough for the theorem.

[AN] Antieau, Nikolaus, Topological Cartier modules and cyclotomic spectra, JAMS 34(1) (2021), 1-78, arXiv:1809.01714.

[AMR] Ayala, Mazel-Gee, Rozenblyum, A naive approach to genuine G-spectra and cyclotomic spectra, arXiv:1710.06416.

[BG] Barwick, Glasman, Cyclonic spectra, cyclotomic spectra, and a conjecture of Kaledin, arXiv:1602.02163.

[BS] Betley, Schlichtkrull, The cyclotomic trace and curves on $K$-theory, Topology 44(4) (2005), 845-874.

[BM] Blumberg, Mandell, The homotopy theory of cyclotomic spectra, G&T 19(6) (2016), 3105-3147, arXiv:1303.1694.

[H] Hesselholt, On the $p$-typical curves in Quillen’s $\K$-theory, Acta 177(1) (1996), 1-53, pdf.

[KN] Krause, Nikolaus, Lectures on topological Hochschild homology and cyclotomic spectra, pdf.

[M] McCandless, On curves in K-theory and TR, arXiv:2102.08281.

[NS] Nikolaus, Scholze, On topological cyclic homology, Acta 221(2) (2018), 203-409, arXiv:1707.01799.