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Register a new userThe K-Theory Spectrum of Varieties
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Authors
Jonathan A. Campbell
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Using a construction closely related to Waldhausens $S_bullet$-construction, we produce a spectrum $K(mathbf{Var}_{/k})$ whose components model the Grothendieck ring of varieties (over a field $k$) $K_0 (mathbf{Var}_{/k})$. We then produce liftings of various motivic measures to spectrum-level maps, including maps into Waldhausens $K$-theory of spaces $A(ast)$ and to $K(mathbf{Q})$. We end with a conjecture relating $K(mathbf{Var}_{/k})$ and the doubly-iterated $K$-theory of the sphere spectrum.
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In this paper we introduce an equivariant extension of the Chern-Simons form, associated to a path of connections on a bundle over a manifold M, to the free loop space LM, and show it determines an equivalence relation on the set of connections on a bundle. We use this to define a ring, loop differential K-theory of M, in much the same way that differential K-theory can be defined using the Chern-Simons form [SS]. We show loop differential K-theory yields a refinement of differential K-theory, and in particular incorporates holonomy information into its classes. Additionally, loop differential K-theory is shown to be strictly coarser than the Grothendieck group of bundles with connection up to gauge equivalence. Finally, we calculate loop differential K-theory of the circle.
We show that Mandells inverse $K$-theory functor is a categorically-enriched multifunctor. In particular, it preserves algebraic structures parametrized by operads. As applications, we describe how ring categories, bipermutative categories, braided ring categories, and $E_n$-monoidal categories arise as the images of inverse $K$-theory.
We establish an equivalence of homotopy theories between symmetric monoidal bicategories and connective spectra. For this, we develop the theory of $Gamma$-objects in 2-categories. In the course of the proof we establish strictfication results of independent interest for symmetric monoidal bicategories and for diagrams of 2-categories.
Let $R$ be the homogeneous coordinate ring of a smooth projective variety $X$ over a field $k$ of characteristic~0. We calculate the $K$-theory of $R$ in terms of the geometry of the projective embedding of $X$. In particular, if $X$ is a curve then we calculate $K_0(R)$ and $K_1(R)$, and prove that $K_{-1}(R)=oplus H^1(C,cO(n))$. The formula for $K_0(R)$ involves the Zariski cohomology of twisted Kahler differentials on the variety.
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 McCarthys 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 homology (THH) and explain its connection to traces in bicategories with shadow (also known as trace theories).
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