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Register a new userInterpreting the Bokstedt smash product as the norm
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This note compares two models of the equivariant homotopy type of the smash powers of a spectrum, namely the Bokstedt smash product and the Hill-Hopkins-Ravenel norm.
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In this note we compute the topological Hochschild homology of quotients of DVRs. Along the way we give a short argument for Bokstedt periodicity and generalizations over various other bases. Our strategy also gives a very efficient way to redo the computations of THH (resp. logarithmic THH) of complete DVRs originally due to Lindenstrauss-Madsen (resp. Hesselholt-Madsen).
We define a map of simplicial presheaves, the Chern character, that assigns to every sequence of composable non connection preserving isomorphisms of vector bundles with holomorphic connections an appropriate sequence of holomorphic forms. We apply this Chern character map to the Cech nerve of a good cover of a complex manifold and assemble the data by passing to the totalization to obtain a map of simplicial sets. In simplicial degree 0, this map gives a formula for the Chern character of a bundle in terms of the clutching functions. In simplicial degree 1, this map gives a formula for the Chern character of bundle maps. In each simplicial degree beyond 1, these invariants, defined in terms of the transition functions, govern the compatibilities between the invariants assigned in previous simplicial degrees. In addition to this, we also apply this Chern character to complex Lie groupoids to obtain invariants of bundles on them in terms of the simplicial data. For group actions, these invariants land in suitable complexes calculating various Hodge equivariant cohomologies. In contrast, the de Rham Chern character formula involves additional terms and will appear in a sequel paper. In a sense, these constructions build on a point of view of characteristic classes in terms of transition functions advocated by Raoul Bott, which has been addressed over the years in various forms and degrees, concerning the existence of formulae for the Hodge and de Rham characteristic classes of bundles solely in terms of their clutching functions.
Building on work of Livernet and Richter, we prove that E_n-homology and E_n-cohomology of a commutative algebra with coefficients in a symmetric bimodule can be interpreted as functor homology and cohomology. Furthermore we show that the associated Yoneda algebra is trivial.
We show that K_{2i}(Z[x,y]/(xy),(x,y)) is free abelian of rank 1 and that K_{2i+1}(Z[x,y]/(xy),(x,y)) is finite of order (i!)^2. We also compute K_{2i+1}(Z[x,y]/(xy),(x,y)) in low degrees.
We give a new proof of a result of Lazarev, that the dual of the circle $S^1_+$ in the category of spectra is equivalent to a strictly square-zero extension as an associative ring spectrum. As an application, we calculate the topological cyclic homology of $DS^1$ and rule out a Koszul-dual reformulation of the Novikov conjecture.
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