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Research Report MRR00-013
Determinants in triangulated categories
Avishay Vaknin
Abstract:
In this paper we define the determinant function
\[
\det: \{\text{Automorphisms in } \TT\} \lra{} K_1(\TT),
\]
where $\TT$ 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
\[
\det: \{\text{Automorphisms in } \AA\} \lra{} K_1(\AA),
\]
where $\AA$ is a small abelian category.
We prove that this determinant is a multiplicative additive function.
As an application, we provide an explicit formula
for the deteminant of an automorphism of chain complexes.
In this case the automorphism may be thought of as being
in $D^b(\AA)$,
and we know that $K_1(D^b(\AA))=K_1(\AA)$.
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