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The two-dimensional Contou-Carrère symbol and reciprocity laws. (English) Zbl 1346.19003
In a previous article [Algebra Number Theory 5, No. 3, 289–337 (2011; Zbl 1237.19007)] the authors of this paper gave a new conceptual proof of Parshin-type reciprocity laws for algebraic surfaces. They used the machinery of categorical central extensions and generalized commutators to construct the appropriate two-dimensional tame symbols needed for the formulation of these reciprocity laws. The purpose of the current paper is to extend the results of that paper from a ground field $$k$$ to a general commutative ground ring $$R$$.
The notion of central extension is extended to the situation of more general sheaves on a topos and the $$2$$-Tate vector spaces of the previous article are replaced with Drinfeld’s notion of a Tate $$R$$-module. In this way, analogously to the previous article, from the existence of a central extension of $$\mathcal{P}ic^{\mathbb{Z}}$$ by $$L^2\mathbb{G}_m$$ is deduced a tri-multiplicative map of functors (the two-dimensional Contou-Carrére symbol) $C_3: L^2\mathbb{G}_m\times L^2\mathbb{G}_m\times L^2\mathbb{G}_m\to \mathbb{G}_m.$ Here $$\mathcal{P}ic^{\mathbb{Z}}$$ is the Picard groupoid of graded lines and $$L^2\mathbb{G}_m$$ is the group-valued functor $$L^2\mathbb{G}_m(R)=R((u))((t))^*$$ for a ring $$R$$. With this construction, the authors show that $$C_3$$ is invariant under change of local parameters $$(u,t)\to (u',t')$$.
In section $$6$$ of the paper the authors prove the following two-dimensional reciprocity laws (Theorem 6.1): Let $$X$$ be a smooth algebraic surface over a perfect field $$k$$ and let $$R$$ be a local finite $$k$$-algebra. Then
For any closed point $$x\in X$$ and for any $$f,g,h\in (K_x\otimes_kR)^*$$ we have $\prod_{C\ni x}(f,g,h)_{x,C}=1,$ the product being taken over all irreducible curves $$C$$ containing $$x$$
Let $$C$$ be a projective irreducible curve on $$X$$. Then for any $$f,g,h\in (K_C\otimes_kR)^*$$ we have $\prod_{x\in C}\mathrm{Nm}_{k(x)/k}(f,g,h)_{x,C}=1,$ where the product is taken over all closed points $$x\in C$$.
Here the elements $$(f,g,h)_{x,C}\in (k(x)\otimes R)^*$$ are defined using the map $$C_3$$.
In the final section of the paper, the authors use the Contou-Carrére symbol over the ring $$R=\mathbb{F}_q[s]/s^{n+1}$$ to obtain Parshin’s local symbol which is used to construct the generalized Artin-Schreier-Witt duality for the two-dimensional local field $$\mathbb{F}_q((u))((t))$$.
A considerable portion of the paper, however, is devoted to obtaining explicit descriptions of the Contou-Carrére symbol analogous the much simpler one-dimensional case. In particular, in section 3 the authors show that when $$\mathbb{Q}\subset R$$ and for appropriate $$f,g,h\in R((u))((t))^*$$ we have $C_3(f,g,h)=\exp\mathrm{Res}\left(\log f \frac{dg}{g}\wedge \frac{dh}{h}\right)$ In section 7, the authors discuss a $$K$$-theoretic definition of the Contou-Carrére symbols from which the first of the above reciprocity laws also follows naturally.

##### MSC:
 19D45 Higher symbols, Milnor $$K$$-theory
##### Keywords:
Contou-Carrére symbol; reciprocity law
Full Text:
##### References:
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