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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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