Relative Tate Objects and Boundary Maps in the K-Theory of Coherent Sheaves

TL;DR

This paper explores relative Tate objects in exact categories, introduces a relative index map, and applies it to describe boundary morphisms in the K-theory of coherent sheaves on schemes.

Contribution

It develops the theory of relative Tate objects and boundary maps, providing new criteria for categories to be abelian and a novel description of K-theory boundary morphisms.

Findings

Criteria for categories to be abelian

Introduction of a relative index map

Description of boundary morphisms in K-theory

Abstract

We investigate the properties of relative analogues of admissible Ind, Pro, and elementary Tate objects for pairs of exact categories, and give criteria for those categories to be abelian. A relative index map is introduced, and as an application we deduce a description for boundary morphisms in the K-theory of coherent sheaves on Noetherian schemes.