Thomas H. Geisser
We show that the cyclotomic trace map for smooth varieties over number rings can be interpreted as a regulator map and hence are related to special values of $ζ$-functions.
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We prove that for a quasi-regular semiperfectoid $\mathbb{Z}_p^{\rm cycl}$-algebra $R$ (in the sense of Bhatt-Morrow-Scholze), the cyclotomic trace map from the $p$-completed $K$-theory spectrum $K(R;\mathbb{Z}_p)$ of $R$ to the topological cyclic homology $\mathrm{TC}(R;\mathbb{Z}_p)$ of $R$ identifies on $π_2$ with a $q$-deformation of the logarithm.
The purpose of this paper is to present a simple and explicit construction of the Bokstedt-Hsiang-Madsen cyclotomic trace relating algebraic K-theory and topological cyclic homology. Our construction also incorporates Goodwillie's idea of a global cyclotomic trace.
In this article, we suggest a categorification procedure in order to capture an analogy between Crystalline Grothendieck-Lefschetz trace formula and the cyclotomic trace map $K\rightarrow TC$ from the algebraic $K$-theory to the topological cyclic homology $TC$. First, we categorify the category of schemes to the $(2, \infty)$-category of noncommuatative schemes a la Kontsevich. This gives a categorification of the set of rational points of a scheme. Then, we categorify the Crystalline Grothendieck-Lefschetz trace formula and find an analogue to the Crystalline cohomology in the setting of non…
We provide a new construction of the topological cyclic homology $TC(C)$ of any spectrally-enriched $\infty$-category $C$, which affords a precise algebro-geometric interpretation of the cyclotomic trace map $K(X) \to TC(X)$ from algebraic K-theory to topological cyclic homology for any scheme $X$. This construction rests on a new identification of the cyclotomic structure on $THH(C)$, which we find to be a consequence of (i) the geometry of 1-manifolds, and (ii) linearization (in the sense of Goodwillie calculus). Our construction of the cyclotomic trace likewise arises from the linearization…
Let $p\in \mathbb{Z}$ be an odd prime. We prove a spectral version of Tate-Poitou duality for the algebraic $K$-theory spectra of number rings with $p$ inverted. This identifies the homotopy type of the fiber of the cyclotomic trace $K(\mathcal{O}_{F})^{\scriptscriptstyle\wedge}_{p} \to TC(\mathcal{O}_{F})^{\wedge}_{p}$ after taking a suitably connective cover. As an application, we identify the homotopy type at odd primes of the homotopy fiber of the cyclotomic trace for the sphere spectrum in terms of the algebraic $K$-theory of $\mathbb{Z}$.
Let $p\in \mathbb Z$ be an odd prime. We show that the fiber sequence for the cyclotomic trace of the sphere spectrum $\mathbb S$ admits an "eigensplitting" that generalizes known splittings on $K$-theory and $TC$. We identify the summands in the fiber as the covers of $\mathbb Z_{p}$-Anderson duals of summands in the $K(1)$-localized algebraic $K$-theory of $\mathbb Z$. Analogous results hold for the ring $\mathbb Z$ where we prove that the $K(1)$-localized fiber sequence is self-dual for $\mathbb Z_{p}$-Anderson duality, with the duality permuting the summands by $i\mapsto p-i$ (indexed mod …
Making use of the theory of noncommutative motives, we characterize the topological Dennis trace map as the unique multiplicative natural transformation from algebraic K-theory to topological Hochschild homology (THH) and the cyclotomic trace map as the unique multiplicative lift through topological cyclic homology (TC). Moreover, we prove that the space of all multiplicative structures on algebraic K-theory is contractible. We also show that the algebraic K-theory functor from small stable infinity categories to spectra is lax symmetric monoidal, which in particular implies that E_n ring spec…
We prove that the Farrell-Jones assembly map for connective algebraic K-theory is rationally injective, under mild homological finiteness conditions on the group and assuming that a weak version of the Leopoldt-Schneider conjecture holds for cyclotomic fields. This generalizes a result of Bökstedt, Hsiang, and Madsen, and leads to a concrete description of a large direct summand of $K_n(\mathbb{Z}[G])\otimes_{\mathbb{Z}}\mathbb{Q}$ in terms of group homology. In many cases the number theoretic conjectures are true, so we obtain rational injectivity results about assembly maps, in particular fo…
We do three things in this paper: (1) study the analog of localization sequences (in the sense of algebraic $K$-theory of stable $\infty$-categories) for additive $\infty$-categories, (2) define the notion of nilpotent extensions for suitable $\infty$-categories and furnish interesting examples such as categorical square-zero extensions, and (3) use (1) and (2) to extend the Dundas-Goodwillie-McCarthy theorem for stable $\infty$-categories which are not monogenically generated (such as the stable $\infty$-category of Voevodsky's motives or the stable $\infty$-category of perfect complexes on s…
We extend Geisser and Hesselholt's result on ``bi-relative K-theory'' from discrete rings to connective ring spectra. That is, if $\mathcal A$ is a homotopy cartesian $n$-cube of ring spectra (satisfying connectivity hypotheses), then the $(n+1)$-cube induced by the cyclotomic trace $$K(\mathcal A)\to TC(\mathcal A)$$ is homotopy cartesian after profinite completion. In other words, the fiber of the profinitely completed cyclotomic trace satisfies excision.
We show that the characteristic polynomial and the Lefschetz zeta function are manifestations of the trace map from the $K$-theory of endomorphisms to topological restriction homology (TR). Along the way we generalize Lindenstrauss and McCarthy's map from $K$-theory of endomorphisms to topological restriction homology, defining it for any Waldhausen category with a compatible enrichment in orthogonal spectra. In particular, this extends their construction from rings to ring spectra. We also give a revisionist treatment of the original Dennis trace map from $K$-theory to topological Hochschild …
Let A be an arbitrary ring. We introduce a Dennis trace map mod n, from K_1(A;Z/n) to the Hochschild homology group with coefficients HH_1(A;Z/n). If A is the ring of integers in a number field, explicit elements of K_1(A,Z/n) are constructed and the values of their Dennis trace mod n are computed. If F is a quadratic field, we obtain this way non trivial elements of the ideal class group of A. If F is a cyclotomic field, this trace is closely related to Kummer logarithmic derivatives; this trace leads to an unexpected relationship between the first case of Fermat last theorem, K-theory and th…
Let $K$ be a complete discrete valuation field with finite residue field of characteristic $p$, and let $D$ be a central division algebra over $K$ of finite index $d$. Thirty years ago, Suslin and Yufryakov showed that for all prime numbers $\ell$ different from $p$ and integers $j \geq 1$ , there exists a "reduced norm" isomorphism of $\ell$-adic $K$-groups $\operatorname{Nrd}_{D/K} \colon K_j(D,\mathbb{Z}_{\ell}) \to K_j(K,\mathbb{Z}_{\ell})$ such that $d \cdot \operatorname{Nrd}_{D/K}$ is equal to the norm homomorphism $N_{D/K}$. The purpose of this paper is to prove the analogous result fo…
If A is a homotopy cartesian square of ring spectra satisfying connectivity hypotheses, then the cube induced by Goodwillie's integral cyclotomic trace from K(A) to TC(A) is homotopy cartesian. In other words, the homotopy fiber of the cyclotomic trace satisfies excision. The method of proof gives as a spin-off new proofs of some old results, as well as some new results, about periodic cyclic homology, and - more relevantly for our current application - the T-Tate spectrum of topological Hochschild homology, where T is the circle group
We use the methods of topological Hochschild homology to shed new light on the groups satisfying the Bass trace conjecture. We show that the factorization of the Hattori-Stallings rank map through the Bokstedt-Hsiang-Madsen cyclotomic trace map leads to Linnell's restriction on such groups. As a new consequence of this restriction, we show that the conjecture holds for any group G with the property that every subgroup that is isomorphic to the additive group of rational numbers has nontrivial and central image in some quotient of G.
Given a henselian pair $(R, I)$ of commutative rings, we show that the relative $K$-theory and relative topological cyclic homology with finite coefficients are identified via the cyclotomic trace $K \to \mathrm{TC}$. This yields a generalization of the classical Gabber-Gillet-Thomason-Suslin rigidity theorem (for mod $n$ coefficients, with $n$ invertible in $R$) and McCarthy's theorem on relative $K$-theory (when $I$ is nilpotent). We deduce that the cyclotomic trace is an equivalence in large degrees between $p$-adic $K$-theory and topological cyclic homology for a large class of $p$-adic ri…
Placing a Dirac-Schrödinger operator along the orbit of a flow on a compact manifold \(M\) defines an \(\R\)-equivariant spectral triple over the algebra of smooth functions on \(M\). We study some of the properties of these triples, especially their zeta functions, which have the form \(\trace (fH^{-s})\) with \(f\) the restriction to \(\R\) of a function on \(M\) and \(H = -\frac{\partial^2}{\partial x^2} + x^2\) the harmonic oscillator. The meromorphic continuation property and pole structure of these zeta functions is related to ergodic time averages in dynamics. The construction reproduce…
$c$-structures and trace methods beyond connective rings
本文提出由连通环谱控制的$c$-范畴概念，证明其幂零扩张诱导截断不变量同构，并将Dundas--Goodwillie--McCarthy定理推广到完美复形范畴为$c$-范畴的栈上。
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