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Equivalences of triangulated categories and Fourier-Mukai transforms. (English) Zbl 0937.18012
Using the notion of indecomposability and spanning class of triangulated categories the author proves that a fully faithful exact functor $$F:{\mathcal A}\to{\mathcal B}$$ of triangulated categories, where $${\mathcal B}$$ is indecomposable, is an equivalence if and only if $$F$$ has a left adjoint $$G$$ and a right adjoint $$H$$, s.t. for any object $$B$$ of $${\mathcal B}$$ with $$H(B)\cong 0$$ it follows $$G(B)\cong 0$$. This result will be applied to Fourier-Mukai transforms of smooth projective varieties $$X$$ and $$Y$$. The main result is given as follows: Let $${\mathcal O}_y$$ be the structure sheaf of $$y\in Y$$ with reduced scheme structure and $$F(-):={\mathbb R}\pi_{X,*}({\mathcal P}\otimes\pi^*(-)):D(X)\to D(Y)$$ be the so-called integral functor where $${\mathcal P}$$ is an object in the triangulated category $$D(X\times Y)$$ of sheaves on $$X\times Y$$. Then $$F$$ is an equivalence if and only if $$F({\mathcal O}_y)\otimes\omega_X\cong F({\mathcal O}_y)$$ for every point $$y\in Y$$.

##### MSC:
 18E30 Derived categories, triangulated categories (MSC2010) 14J28 $$K3$$ surfaces and Enriques surfaces
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