Determinants in triangulated categories

Avishay Vaknin

March 19 2000

Abstract:
In this paper we define the determinant function

  det: {Automorphisms in T} ----> K_1(T),

where  T  is a small triangulated category, 
and  K_1  is Neeman's first  K-theory group for triangulated 
categories. This function is compatible with the classical 
determinant map

  det: {Automorphisms in A} ----> K_1(A),

where  A  is a small abelian category.
We prove that this determinant is a multiplicative additive 
function. As an application, we prove that the alternating 
sum determinant of bounded complex automorphisms over an 
abelian category  A  is an additive multiplicative homotopy 
invariant with values in  K_1(A).