Equivalence between derived categories preserve distinguished triangles

Question

I have a problem: Is it true that every equivalence between derived categories preserve their distinguished triangles? Thanks very much!

Answer 1

Not necessarily. For instance, the shift functor is an automorphism of any triangulated category, but it usually does not preserve distinguished triangles (because the shift of a distinguished triangle need not be distinguished, only the shift with the signs of all the maps reversed is distinguished). For a general discussion of equivalences of triangulated categories as categories equipped with a shift functor that do not preserve the triangulation, see the answers to this question on MO. In particular, the accepted answer there details how the shift fails to preserve distinguished triangles in the derived category of $\mathbb{Z}$.

If you don't care whether your equivalence preserves the shift functor, there are even easier examples. For instance, the derived category of a semisimple ring $A$ is equivalent to the category of $\mathbb{Z}$-graded $A$-modules, and any bijection $\mathbb{Z}\to\mathbb{Z}$ induces an automorphism of the latter category via the grading. Almost all of these fail to commute with the shift, so they trivially cannot preserve distinguished triangles.