A Generalized Contou-Carrère Symbol and its Reciprocity Laws in Higher Dimensions

13 Oct 2014  ·  Oliver Braunling, Michael Groechenig, Jesse Wolfson ·

We generalize the theory of Contou-Carr\`ere symbols to higher dimensions. To an $(n+1)$-tuple $f_0,\dots,f_n \in A((t_1))\cdots((t_n))^{\times}$, where $A$ denotes a commutative algebra over a field $k$, we associate an element $(f_0,\dots,f_n) \in A^{\times}$, compatible with the higher tame symbol for $k = A$, and earlier constructions for $n = 1$, by Contou-Carr\`ere, and $n = 2$ by Osipov--Zhu... Our definition is based on the notion of \emph{higher commutators} for central extensions of groups by spectra, thereby extending the approach of Arbarello--de Concini--Kac and Anderson--Pablos Romo. Following Beilinson--Bloch--Esnault for the case $n=1$, we allow $A$ to be arbitrary, and do not restrict to artinian $A$. Previous work of the authors on Tate objects in exact categories, and the index map in algebraic $K$-theory is essential in anchoring our approach to its predecessors. We also revisit categorical formal completions, in the context of stable $\infty$-categories. Using these tools, we describe the higher Contou-Carr\`ere symbol as a composition of boundary maps in algebraic $K$-theory, and conclude the article by proving a version of Parshin--Kato reciprocity for higher Contou-Carr\`ere symbols. read more

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Algebraic Geometry K-Theory and Homology Number Theory Representation Theory 19D45 (Primary), 19E99 (Secondary)