On the normally ordered tensor product and duality for Tate objects

TL;DR

This paper extends the normally ordered tensor product to Tate objects over arbitrary exact categories, establishing duality, external Homs, and applications in duality, adeles, and intersection theory.

Contribution

It introduces a generalized framework for tensor products and duality in Tate objects, enabling new applications in algebraic and arithmetic geometry.

Findings

Pontryagin duality extends to n-Tate objects

Adeles of a flag can be expressed as ordered tensor products

Intersection numbers can be interpreted through these tensor products

Abstract

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 structures, duality functors, and construct external Homs. We list some applications: (1) Pontryagin duality uniquely extends to n-Tate objects in locally compact abelian groups; (2) Adeles of a flag can be written as ordered tensor products; (3) Intersection numbers can be interpreted via these tensor products.