Related papers: The rigid syntomic ring spectrum

We prove the Riemann-Roch theorem for homotopy invariant $K$-theory and projective local complete intersection morphisms between finite dimensional noetherian schemes, without smoothness assumptions. We also prove a new Riemann-Roch theorem…

We extend Homotopy Type Theory with a novel modality that is simultaneously a monad and a comonad. Because this modality induces a non-trivial endomap on every type, it requires a more intricate judgemental structure than previous modal…

In this paper, we construct and study a Serre-type spectral sequence for motivic cohomology associated to a map of bisimplicial schemes with motivically cellular fiber. Then, we show how to apply it in order to approach the computation of…

We prove that the $\infty$-category of motivic spectra satisfies Milnor excision: if $A\to B$ is a morphism of commutative rings sending an ideal $I\subset A$ isomorphically onto an ideal of $B$, then a motivic spectrum over $A$ is…

Working over an algebraically closed field $k$ of characteristic $0$, we show that the motivic stable homotopy groups of the sphere spectrum can be determined entirely from the motivic homotopy groups of the $p$-completed sphere spectra and…

In this paper we discuss a general notion of Weil cohomology theories, both in algebraic geometry and in rigid analytic geometry. We allow our Weil cohomology theories to have coefficients in arbitrary commutative ring spectra. Using the…

We consider slice filtrations in logarithmic motivic homotopy theory. Our main results establish conjectured compatibilities with the Beilinson, BMS, and HKR filtrations on (topological, log) Hochschild homology and related invariants. In…

We construct a motivic spectral sequence for the relative homotopy invariant K-theory of a closed immersion of schemes $D \subset X$. The $E_2$-terms of this spectral sequence are the cdh-hypercohomology of a complex of equi-dimensional…

Let $X$ be a Noetherian separated scheme of finite Krull dimension. We show that the layers of the slice filtration in the motivic stable homotopy category $\stablehomotopy$ are strict modules over Voevodsky's algebraic cobordism spectrum.…

Let kq denote the very effective cover of the motivic Hermitian K-theory spectrum. We analyze the ring of cooperations $\pi^\mathbb{R}_{**}(\text{kq} \otimes \text{kq})$ in the stable motivic homotopy category $\text{SH}(\mathbb{R})$,…

We prove the following result of V. Voevodsky. If $S$ is a finite dimensional noetherian scheme such that $S=\cup_\alpha\Spec(R_\alpha)$ for {\em countable} rings $R_\alpha$, then the stable motivic homotopy category over $S$ satisfies…

Over any field of characteristic not 2, we establish a 2-term resolution of the $\eta$-periodic, 2-local motivic sphere spectrum by shifts of the connective 2-local Witt K-theory spectrum. This is curiously similar to the resolution of the…

The aim of this article is to give a concise algebraic treatment of the modular symbols formalism, generalised from modular curves to Hecke triangle surfaces. A sketch is included of how the modular symbols formalism gives rise to the…

This paper sets up the foundations for derived algebraic geometry, Goerss--Hopkins obstruction theory, and the construction of commutative ring spectra in the abstract setting of operadic algebras in symmetric spectra in an (essentially)…

The proof of the coincidence of the Gysin morphism in motivic cohomology and the usual pushout on Chow groups has been improved (see Lemma 3.3 and Proposition 3.11)

By a theorem of Mandell-May-Schwede-Shipley the stable homotopy theory of classical $S^1$-spectra is recovered from orthogonal spectra. In this paper general linear, special linear, symplectic, orthogonal and special orthogonal motivic…

We compute the generalized slices (as defined by Spitzweck-{\O}stv{\ae}r) of the motivic spectrum KO (representing hermitian K-theory) in terms of motivic cohomology and (a version of) generalized motivic cohomology, obtaining good…

To an Adams-type homology theory we associate a notion of a synthetic spectrum, this is a product-preserving sheaf on the site of finite spectra with projective $E$-homology. We prove that the $\infty$-category $Syn_{E}$ of synthetic…

In this note, we provide an axiomatic framework that characterizes the stable $\infty$-categories that are module categories over a motivic spectrum. This is done by invoking Lurie's $\infty$-categorical version of the Barr--Beck theorem.…

We determine the A(1)-homotopy of the topological cyclic homology of the connective real K-theory spectrum ko. The answer has an associated graded that is a free F_2[v_2^4]-module of rank 52, on explicit generators in stems -1 \le * \le 30.…