Connecting homomorphisms in localization sequences:II
The goals of this paper are two-fold: First, given an exact category P, and two commuting automorphisms f, g, of an object P of P, we construct an element f * g ε K (P)- Second, suppose that: A is an abelian category; S is a Serre subcategory; α and β are commuting monic endomorphisms of an object A of A having the property that coker α and coker β lie in S. Then α and β give rise to automorphisms aα, β, of the corresponding object Ā of the quotient category A/S, hence give rise to an element ᾱ* β̄ ε K (A/S). If we denote by ∥ the connecting homomorphism K (A/S) → K (S) in the localization sequence for A and S, then our main result is an explicit formula for ∥(ᾱ *β̄). This formula complements the main result of , which dealt with the connecting homomorphism K (A/S) → K (S). Furthermore, just as in that paper, our formula turns out to be the analog of the corresponding formula for the localization sequence for projective modules, which was worked out some years ago by Grayson in ; as before, a simple modification of our proof applies in that context as well.
Algebraic K-theory, Localization sequences
Sherman, Clayton. "Connecting homomorphisms in localization sequences: II." K-Theory 32, no. 4 (2004): 365-389.