We consider categories C n which are very close to the iterated functor lim ←→ , which was introduced by A.A.Beilinson in [2]. We prove that an adelic space on n -dimensional Noetherian scheme is an object of C n . IntroductionIn this note we want to introduce by induction some class of infinite-dimensional vector spaces and morphisms between them. These spaces depend on integer n , we call such spaces as C n -spaces.then we have a filtration . . . O(m) ⊂ O(m − 1) ⊂ O(m − 2) . . . and every factor space O(m − k)/O(m) is a finite dimensional vector space over k .We can consider the morphisms between two such spaces as continuous linear maps. We remark that these morphisms can be described only in terms of filtration O n , without considering topology on k((t)) . From this point of view, the space of adeles on an algebraic curve has a structure of C 1 -space, which is filtered by partially ordered set of coherent sheaves on the curve.We construct an iterated version of C 1 -spaces, which we call a C n -space. But in our construction we do not consider the structure of completion. So, we consider the filtered vector spaces. For example, the discrete valuation field is also a C 1 -space, the space of rational adeles from [14] is also a C 1 -space.The constructions of similar categories were introduced also in [2], [7], [6]. Our construction of C n is very close to the iterated functor lim ←→ , which was introduced by A.A.Beilinson in appendix to [2]. The main difference is that we consider noncompleted version of lim ←→ , i.e., filtered spaces, but with morphisms which come from lim ←→ . From this point of view the categories C n are rather closed to the dir-inv modules which were considered by A. Yekutieli in [15] for n = 1 .

Several algebro-geometric properties of commutative rings of partial differential operators as well as several geometric constructions are investigated. In particular, we show how to associate a geometric data by a commutative ring of partial differential operators, and we investigate the properties of these geometric data. This construction is in some sense similar to the construction of a formal module of Baker-Akhieser functions. On the other hand, there is a recent generalization of Sato's theory which belongs to the third author of this paper. We compare both approaches to the commutative rings of partial differential operators in two variables. As a by-product we get several necessary conditions on geometric data describing commutative rings of partial differential operators.

We develop harmonic analysis in some categories of filtered abelian groups and vector spaces. These categories contain as objects local fields and adelic spaces arising from arithmetical surfaces. Some structure theorems are proven for quotients of the adelic groups of algebraic and arithmetical surfaces.where the group G 0 is the connected component of the group G which contains the identity element e , the group K is a maximal compact subgroup of the group G 0 , the group G tor /G 0 is the maximal torsion subgroup of the discrete group G/G 0 . Then we have the following isomorphisms

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

Abstract. We define a two-dimensional Contou-Carrère symbol, which is a deformation of the two-dimensional tame symbol and is a natural generalization of the (usual) one-dimensional Contou-Carrère symbol. We give several constructions of this symbol and investigate its properties. Using higher categorical methods, we prove reciprocity laws on algebraic surfaces for this symbol. We also relate the two-dimensional Contou-Carrère symbol to the two-dimensional class field theory.