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A categorical proof of the Parshin reciprocity laws on algebraic surfaces. (English) Zbl 1237.19007
Let $$X$$ be an algebraic surface over a perfect field $$k$$. For any pair $$(x,C)$$, where $$C$$ is an algebraic curve in $$X$$ and $$x\in C$$ is a closed point, there is a well-defined ring $$K_{x,C}$$ such that $$K_{x,C}$$ is either a two-dimensional local field (if $$x$$ is a smooth point) or a finite direct sum of two-dimensional local fields. For any two-dimensional local field, one can define so-called two-dimensional tame symbols of three variables. The whole idea goes back to Parshin, who also formulated (and indicated the proofs of) certain reciprocity laws for such two-dimensional tame symbols [A. N. Parshin, “Local class field theory”, Proc. Steklov Inst. Math. 165, 157–185 (1985; Zbl 0579.12012)].
In the paper under review, the authors obtain a complete intrinsic proof of Parshin’s reciprocity laws for two-dimensional tame symbols on an algebraic surface. This is achieved by giving a generalization of J. Tate’s proof of the Weil reciprocity law on a projective algebraic curve [“Residues of differentials on curves”, Ann. Sci. Éc. Norm. Supér. (4) 1, No. 1, 149–159 (1968; Zbl 0159.22702)] in a modern, very powerful categorical framework.
More precisely, the paper is organized as follows. After a detailed introduction to the problem of reciprocity laws on algebraic surfaces in Section 1, the subsequent section is devoted to the description of some categorical constructions applied later on, including Picard groupoids, the 2-category of torsors over a Picard groupoid, and the Picard 2-groupoid of central extensions of a group $$G$$ by a Picard groupoid $${\mathcal P}$$. Also, the authors define and study properties of the commutator category of such a central extension. This section is general enough to be of independent interest in category theory. Section 3 recalls the theory of graded-determinantal theories of $$n$$-Tate vector spaces as developed by the first author [“Adeles on $$n$$-dimensional schemes and categories $$C_n$$”, Int. J. Math. 18, No. 3, 269–279 (2007; Zbl 1126.14004)]. Section 4 discusses applications of these categorical constructions to one-dimensional and two-dimensional local fields. In particular, the Parshin two-dimensional tame symbols are obtained as certain commutators of three elements in some central extension of the multiplicative group of the local field by some explicit Picard groupoid.
Finally, Section 5 presents the various reciprocity laws. In the first part of this section, a new proof of the classical Weil reciprocity law on a curve is given by using the previous categorical constructions and adèle complexes on a curve. In the second part, and in a similar vein, an intrinsic proof of Parshin’s reciprocity laws on an algebraic surfaces is derived via so-called semilocal ade\`le complexes on an algebraic surface. As the authors point out, their approach can be generalized to Artinian ground rings and to reciprocity laws for two-dimensional Contou-Carrère symbols, which will be discussed in a future paper. Certainly, the authors’ approach developed in the current paper is highly interesting and inspiring, thereby providing many stimulations for further research in this direction.

##### MSC:
 19F15 Symbols and arithmetic ($$K$$-theoretic aspects) 18D05 Double categories, $$2$$-categories, bicategories and generalizations (MSC2010) 11R37 Class field theory 14G20 Local ground fields in algebraic geometry 14J20 Arithmetic ground fields for surfaces or higher-dimensional varieties 18B40 Groupoids, semigroupoids, semigroups, groups (viewed as categories)
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