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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
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