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In $k$-linear triangulated categories, there is an evaluation map

$$\oplus_i \text{Hom}(E,A[i])\otimes_k E[-i]\to A .$$

I've learned that in the derived categorie of coherent sheafs on a scheme $X$, the tensor product $V\otimes F[-i]$ is just the direct sum of $\text{dim }V$ copies of $F[-i]$.

However, I do not understand the above construction in general triangulated $k$-linear categories. Could anyone help me to understand how the tensorproduct and the evaluation map are defined? Thank you.

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Looking at Bondal "Representation of associative algebras and coherent sheaves " suggests that the $\text{coh X}$ case is the general definition, not a consequence. – Carsten Jan 18 '12 at 11:45

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