Relative Tate objects and boundary maps in the $K$-theory of coherent sheaves

摘要

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.