arXiv subjects

Markus Spitzweck

Publications and source records attributed to Markus Spitzweck.

31 records · Page 2Linked to original sources

arXiv2010

Derived Fundamental Groups for Tate Motives

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.

math.AG↗

arXiv2010

Motivic twisted K-theory

This paper sets out basic properties of motivic twisted K-theory with respect to degree three motivic cohomology classes of weight one. Motivic twisted K-theory is defined in terms of such motivic cohomology classes by taking pullbacks along the universal principal BG_m-bundle for the classifying space of the multiplicative group scheme. We show a Kuenneth isomorphism for homological motivic twisted K-groups computing the latter as a tensor product of K-groups over the K-theory of BG_m. The proof employs an Adams Hopf algebroid and a tri-graded Tor-spectral sequence for motivic twisted K-theory. By adopting the notion of an E-infinity ring spectrum to the motivic homotopy theoretic setting, we construct spectral sequences relating motivic (co)homology groups to twisted K-groups. It generalizes various spectral sequences computing the algebraic K-groups of schemes over fields. Moreover, we construct a Chern character between motivic twisted K-theory and twisted periodized rational motivic cohomology, and show that it is a rational isomorphism. The paper includes a discussion of some open problems.

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arXiv2009

Motivic Landweber Exactness

We prove a motivic version of Landweber's exact functor theorem from topology. The main result is that the assignment given by a Landweber-type formula using the MGL-homology of a motivic spectrum defines a homology theory on the stable motivic homotopy category and is representable by a Tate-like (or cellular) spectrum. Using the universal coefficient spectral sequence of Dugger-Isaksen we deduce formulas for operations of motivic Landweber spectra of a certain type including homotopy algebraic K-theory. Finally we construct a Chern character as a map between motivic spectra.

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arXiv2009

Periodizable motivic ring spectra

We show that the cellular objects in the module category over a motivic E infinity ring spectrum E can be described as the module category over a graded topological spectrum if E is strongly periodizable in our language. A similar statement is proven for triangulated categories of motives. Since MGL is strongly periodizable we obtain topological incarnations of motivic Landweber spectra. Under some categorical assumptions the unit object of the model category for triangulated motives is as well strongly periodizable giving motivic cochains whose module category models integral triangulated categories of Tate motives.

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arXiv2009

Motivic strict ring models for K-theory

It is shown that the K-theory of every noetherian base scheme of finite Krull dimension is represented by a strict ring object in the setting of motivic stable homotopy theory. The adjective `strict' is used to distinguish between the type of ring structure we construct and one which is valid only up to homotopy. Both the categories of motivic functors and motivic symmetric spectra furnish convenient frameworks for constructing the ring models. Analogous topological results follow by running the same type of arguments as in the motivic setting.

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arXiv2009

Relations between slices and quotients of the algebraic cobordism spectrum

We prove a relative statement about the slices of the algebraic cobordism spectrum. If the map from MGL to a certain quotient of MGL introduced by Hopkins and Morel is the map to the zero-slice then a relative version of Voevodsky's conjecture on the slices of MGL holds true. We outline the picture for K-theory and rational slices.

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arXiv2009

Slices of motivic Landweber spectra

We show that the Conjecture of Voevodsky concerning slices of the algebraic cobordism spectrum MGL implies a general statement about the slices of motivic Landweber spectra. In particular it confirms the possible approach suggested by Voevodsky for the computation of the slices of the homotopy algebraic K-theory spectrum KGL via a Conner-Floyd isomorphism complementing Levine's unconditional proof of these slices over perfect fields. A similar result, and Voevodsky's conjecture over fields of char. 0, are also announced by Hopkins-Morel.

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arXiv2008

Chern classes, K-theory and Landweber exactness over nonregular base schemes

The purpose of this paper is twofold. First, we use the motivic Landweber exact functor theorem to deduce that the Bott inverted infinite projective space is homotopy algebraic $K$-theory. The argument is considerably shorther than any other known proofs and serves well as an illustration of the effectiveness of Landweber exactness. Second, we dispense with the regularity assumption on the base scheme which is often implicitly required in the notion of oriented motivic ring spectra. The latter allows us to verify the motivic Landweber exact functor theorem and the universal property of the algebraic cobordism spectrum for every noetherian base scheme of finite Krull dimension.

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arXiv2007

Duality for topological abelian group stacks and T-duality

v2: An additional assumption was added in Theorem 4.8. In order to show that a connected abelian group is admissible on the site of locally compact spaces we must in addition assume that it is locally topologically divisible. This condition is used in the proof of Lemma 4.62.

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arXiv2007

Sheaf theory for stacks in manifolds and twisted cohomology for S^1-gerbes

This is the first of a series of papers on sheaf theory on smooth and topological stacks and its applications. The main result of the present paper is the characterization of the twisted (by a closed integral three-form) de Rham complex on a manifold. As an object in the derived category it will be related with the push-forward of the constant sheaf from a S^1-gerbe with Dixmier-Douady class represented by the three-form. In order to formulate and prove this result we develop in detail the foundations of sheaf theory for smooth stacks.

math.KT↗

arXiv2006

Inertia and delocalized twisted cohomology

We show that the inertia stack of a topological stack is again a topological stack. We further observe that the inertia stack of an orbispace is again an orbispace. We show how a U(1)-banded gerbe over an orbispace gives rise to a flat line bundle over its inertia stack. Via sheaf theory over topological stacks it gives rise to the twisted delocalized cohomology of the orbispace. With these results and constructions we generalize concepts, which are well-known in the smooth framework, to the topological case. In the smooth case we show, that our sheaf-theoretic definition of twisted delocalized cohomology of orbispaces coincides with former definitions using a twisted de Rham complex.

math.KT↗

arXiv2005

Homological algebra with locally compact abelian groups

In this article we study locally compact abelian (LCA) groups from the viewpoint of derived categories, using that their category is quasi-abelian in the sense of J.-P. Schneiders. We define a well-behaved derived Hom-complex with values in the derived category of Hausdorff topological abelian groups. Furthermore we introduce a smallness condition for LCA groups and show that such groups have a natural tensor product and internal Hom which both admit derived versions.

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arXiv2001

Operads, Algebras and Modules in General Model Categories

In this paper we develop the theory of operads, algebras and modules in cofibrantly generated symmetric monoidal model categories. We give J-semi model strucures, which are a slightly weaker version of model structures, for operads and algebras and model structures for modules. In a second part we develop the thoery of S-modules of [EKMM]., which allows a general homotopy theory for commutative algebras and pseudo unital symmetric monoidal categories of modules over them. Finally we prove a base change and projection formula.

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