Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.