arXiv subjects
Publications and source records attributed to Markus Spitzweck.
We prove an equivalence between the real $K$-theory genuine $C_2$-spectra of Calm\`es et al. for Poincar\'e $\infty$-categories and the one of the authors of this work for Waldhausen $\infty$-categories with genuine duality.
We compute the 2-line of stable homotopy groups of motivic spheres over fields of characteristic not two in terms of motivic cohomology and hermitian K-groups.
We develop a new framework to study real $K$-theory in the context of $\infty$-categories. For this, we introduce Waldhausen $\infty$-categories with genuine duality, which will be the input for such $K$-theory. These are Waldhausen $\infty$-categories in the sense of Barwick equipped with a compatible duality and a refinement of their (lax) hermitian objects generalizing the concept of Poincar\'e $\infty$-categories of Lurie. They may also be thought of as a version of complete Segal spaces enriched in genuine $C_2$-spaces whose underlying $\infty$-category carries a compatible Waldhausen structure, since we show that their respective $\infty$-categories are equivalent. We define the real $K$-theory genuine $C_2$-spaces by means of an enriched version of the $S_\bullet$-construction, defined for Waldhausen $\infty$-categories with genuine duality. Moreover, we prove an Additivity Theorem for this $S_\bullet$-construction which leads to an Additivity Theorem for real $K$-theory. Furthermore, such real $K$-theory satisfy a universal property -- analogous to that proved by Barwick for algebraic $K$-theory of Waldhausen $\infty$-categories --: We prove that every theory can be universally turned into an additive theory and identify our real K-theory with the universal additive theory associated to the functor that associates to a Waldhausen $\infty$-category with genuine duality its maximal subspace. Finally, we promote the real $K$-theory genuine $C_2$-spaces to genuine $C_2$-spectra.
Motivated by calculations of motivic homotopy groups, we give widely attained conditions under which operadic algebras and modules thereof are preserved under (co)localization functors
We compute the 1-line of stable homotopy groups of motivic spheres over fields of characteristic not two in terms of hermitian and Milnor K-groups. This is achieved by solving questions about convergence and differentials in the slice spectral sequence.
Thomason's \'{e}tale descent theorem for Bott periodic algebraic $K$-theory \cite{aktec} is generalized to any $MGL$ module over a regular Noetherian scheme of finite dimension. Over arbitrary Noetherian schemes of finite dimension, this generalizes the analog of Thomason's theorem for Weibel's homotopy $K$-theory. This is achieved by amplifying the effects from the case of motivic cohomology, using the slice spectral sequence in the case of the universal example of algebraic cobordism. We also obtain integral versions of these statements: Bousfield localization at \'etale motivic cohomology is the universal way to impose \'etale descent for these theories. As applications, we describe the \'etale local objects in modules over these spectra and show that they satisfy the full six functor formalism, construct an \'etale descent spectral sequence converging to Bott-inverted motivic Landweber exact theories, and prove cellularity and effectivity of the \'{e}tale versions of these motivic spectra.
The dual motivic Steenrod algebra with mod $\ell$ coefficients was computed by Voevodsky over a base field of characteristic zero, and by Hoyois, Kelly, and {\O}stv{\ae}r over a base field of characteristic $p \neq \ell$. In the case $p = \ell$, we show that the conjectured answer is a retract of the actual answer. We also describe the slices of the algebraic cobordism spectrum $MGL$: we show that the conjectured form of $s_n MGL$ is a retract of the actual answer.
We solve affirmatively the homotopy limit problem for $K$-theory over fields of finite virtual cohomological dimension. Our solution employs the motivic slice filtration and the first motivic Hopf map.
We show that any preadditive infinity category with duality gives rise to a direct sum hermitian K-theory spectrum. This assignment is lax symmetric monoidal, thereby producing E-infinity ring spectra from preadditive symmetric monoidal infinity categories with duality. To have examples of preadditive symmetric monoidal infinity categories with duality we show that any preadditive symmetric monoidal infinity category, in which every object admits a dual, carries a canonical duality. Moreover we classify and twist the dualities in various ways and apply our definitions for example to finitely generated projective modules over E-infinity ring spectra.
We construct a Grothendieck-Witt space for any stable infinity category with duality. If we apply our construction to perfect complexes over a commutative ring in which 2 is invertible we recover the classical Grothendieck-Witt space. Our Grothendieck-Witt space is a grouplike E-infinity space which is part of a genuine C_2-spectrum, the connective real K-theory spectrum.
Hermitian K-theory and Witt-theory are cellular in the sense of stable motivic homotopy theory over any base scheme without points of characteristic two.
We show that algebraic K-theory KGL, the motivic Adams summand ML and their connective covers acquire unique E-infinity structures refining naive multiplicative structures in the motivic stable homotopy category. The proofs combine Gamma-homology computations and work due to Robinson giving rise to motivic obstruction theory. As an application we employ a motivic to simplicial delooping argument to show a uniqueness result for E-infinity structures on the K-theory Nisnevich presheaf of spectra.
We compute the geometric part of algebraic cobordism over Dedekind domains of mixed characteristic after inverting the positive residue characteristics and prove cases of a Conjecture of Voevodsky relating this geometric part to the Lazard ring for regular local bases. The method is by analyzing the slice tower of algebraic cobordism, relying on the Hopkins-Morel isomorphism from the quotient of the algebraic cobordism spectrum by the generators of the Lazard ring to the motivic Eilenberg-MacLane spectrum, again after inverting the positive residue characteristics.
We construct a motivic Eilenberg-MacLane spectrum with a highly structured multiplication over smooth schemes over Dedekind domains which represents Levine's motivic cohomology. The latter is defined via Bloch's cycle complexes. Our method is by gluing p-completed and rational parts along an arithmetic square. Hereby the finite coefficient spectra are obtained by truncated étale sheaves (relying on the now proven Bloch-Kato conjecture) and a variant of Geisser's version of syntomic cohomology, and the rational spectra are the ones which represent Beilinson motivic cohomology. As an application the arithmetic motivic cohomology groups can be realized as Ext-groups in a triangulated category of Tate sheaves with integral coefficients. These can be modelled as representations of derived fundamental groups. Our spectrum is compatible with base change giving rise to a formalism of six functors for triangulated categories of motivic sheaves over general base schemes including the localization triangle. Further applications include a generalization of the Hopkins-Morel isomorphism and a structure result for the dual motivic Steenrod algebra in the case where the coefficient characteristic is invertible on the base scheme.
Colored operads were introduced in the 1970's for the purpose of studying homotopy invariant algebraic structures on topological spaces. In this paper we introduce colored operads in motivic stable homotopy theory. Our main motivation is to uncover hitherto unknown highly structured properties of the slice filtration. The latter decomposes every motivic spectrum into its slices, which are motives, and one may ask to what extend the slice filtration preserves highly structured objects such as algebras and modules. We use colored operads to give a precise solution to this problem. Our approach makes use of axiomatic setups which specialize to classical and motivic stable homotopy theory. Accessible t-structures are central to the development of the general theory. Concise introductions to colored operads and Bousfield (co)localizations are given in separate appendices.
We give an interpretation of J-spaces in terms of symmetric spectra in symmetric sequences. As application we show how one can define graded endomorphism objects in a general situation. As example we discuss the motivic bigraded endomorphisms of a motivic E-infinity ring spectrum. Finally we give an infinity categorical interpretation of our result.
The initial motivation of this work was to give a topological interpretation of two-periodic twisted de-Rham cohomology which is generalizable to arbitrary coefficients. To this end we develop a sheaf theory in the context of locally compact topological stacks with emphasis on the construction of the sheaf theory operations in unbounded derived categories, elements of Verdier duality and integration. The main result is the construction of a functorial periodization functor associated to a U(1)-gerbe. As applications we verify the $T$-duality isomorphism in periodic twisted cohomology and in periodic twisted orbispace cohomology.
We construct derived fundamental group schemes for Tate motives over connected smooth schemes over fields. We show that there exists a pro affine derived group scheme over the rationals such that its category of perfect representations models the triangulated category of rational mixed Tate motives. Under a hypothesis which is weaker than an integral version of the Beilinson-Soule vanishing conjecture we show that there is an affine derived group scheme over the integers such that its perfect representations model Tate motives with integral coefficients. The hypothesis is for example fulfilled for number fields. This generalizes previous non-derived constructions of fundamental group schemes for Tate motives with rational coefficients.