This paper generalizes the normally ordered tensor product from Tate vector spaces to Tate
objects over arbitrary exact categories. We show how to lift bi-right exact monoidal …

We generalize the theory of Contou-Carr\ere symbols to higher dimensions. To an $(n+ 1) $-
tuple $ f_0,\dots, f_n\in A ((t_1))\cdots ((t_n))^{\times} $, where $ A $ denotes a commutative …

Tate's central extension originates from 1968 and has since found many applications to
curves. In the 80s Beilinson found an n-dimensional generalization: cubically decomposed …

Tate gave a famous construction of the residue symbol on curves by using some non-
commutative operator algebra in the context of algebraic geometry. We explain Beilinson's …

We generalize Contou-Carrère symbols to higher dimensions. To an $(n+ 1) $-tuple $
f_0,\dots, f_n\in A ((t_1))\cdots ((t_n))^{\times} $, where $ A $ denotes an algebra over a field …

Let K denote an n-dimensional local field. The aim of this expository paper is to survey the
basic arithmetic theory of the n-dimensional local field K together with its Milnor K-theory and …

Tate gave a famous construction of the residue symbol on curves by using some non-
commutative operator algebra in the context of algebraic geometry. We explain Beilinson's …

We work with completed adelic structures on an arithmetic surface and justify that the
construction under consideration is compatible with Arakelov geometry. The ring of …

For an arithmetic surface X→ B= Spec OK, the Deligne pairing⟨,⟩: Pic (X)× Pic (X)→ Pic (B)
gives the “schematic contribution” to the Arakelov intersection number. We present an idelic …

This thesis is naturally split into two parts. In the first part, we develop the theory of multi-
linear algebra for Tate objects over exact categories endowed with an exact tensor product …