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A triangulated category is an additive category equipped with the additional structure of an autoequivalence (called the translation functor) and a class of of triangles satisfying certain axioms.
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Cone of morphism induced by Serre duality
For a smooth projective variety $X$, Serre duality gives an exact autoequivalence on the derived category :
$$
S_X : D^\flat(X) \to D^\flat(X), \hspace{3em} S_X(-) = - \otimes \omega_X[\dim X]
$$
sati …