A categorical proof of the Parshin reciprocity laws on algebraic surfaces

Abstract

We define and study the 2-category of torsors over a Picard groupoid, a central extension of a group by a Picard groupoid, and commutator maps in this central extension. Using this in the context of two-dimensional local fields and two-dimensional adèle theory we obtain the two-dimensional tame symbol and a new proof of Parshin reciprocity laws on an algebraic surface.Lemma 4.8. The homomorphism C s : P s → Pic Z is isomorphic to the minus (or the inverse) of the following homomorphism

An introduction to affine Grassmannians and the geometric Satake equivalaence

“…Then the above defined det(Λ 1 | Λ 2 ) becomes a graded line bundle on Spec R. Remembering the grading is crucial for the factorisation structure on determinant lines (see Remark 3.1.19 and Remark 3.2.8), which in turn is important for applications to symbols and reciprocity laws (cf. [9,58,59]).…”

“…There is, however, a crucial difference between the isomorphism This reflects the fact that taking the determinant line is a symmetric monoidal functor from the category Proj R of finite projective R-modules (with the tensor product given by direct sums) to the category Pic Z R of graded invertible R-modules. We refer to [58] for further elaborations on this point.…”

Noncommutative reciprocity laws on algebraic surfaces: the case of tame ramification

“…For any object L of the Picard groupoid H 2 (BG, P) , for any g 1 , g 2 ∈ G such that [g 1 , g 2 ] = 1 there is an object C L 2 (g 1 , g 2 ) of the Picard groupoid P (see [7,.13]). The morphism C L 2 is a bimultiplicative and antisymmetric (with respect to g 1 and g 2 ) morphism from the set of pairs of commuting elements of G to the Picard groupoid P .…”

“…We note that the main technical tool for this paper comes from paper [7]. In particular, we use categorical central extensions of groups by the Picard groupoid of Z -graded one-dimensional k -vector spaces.…”

“…In § 3 we recall some calculations of the group H 2 (GL n (K), k * ) (when n ≥ 2 ) and relation of central extensions of this group with the algebraic K -theory of the field K . In § 4 we recall some categorical notions and constructions from [7]. In § 5 we construct the central extension GL n,a (A X ) and study its main properties (see propositions 1 and 2).…”