By Daniel Huybrechts
This seminal textual content on Fourier-Mukai Transforms in Algebraic Geometry via a number one researcher and expositor relies on a direction given on the Institut de Mathematiques de Jussieu in 2004 and 2005. aimed toward postgraduate scholars with a easy wisdom of algebraic geometry, the major element of this ebook is the derived class of coherent sheaves on a delicate projective kind. together with notions from different components, e.g. singular cohomology, Hodge thought, abelian forms, K3 surfaces; complete proofs are given and workouts reduction the reader all through.
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52 A triangulated category D is decomposed into triangulated subcategories D1 , D2 ⊂ D if the following three conditions are satisﬁed: i) Both categories D1 and D2 contain objects non-isomorphic to 0. ii) For all A ∈ D there exists a distinguished triangle B1 G A G B2 G B1  with Bi ∈ Di , i = 1, 2. iii) Hom(B1 , B2 ) = Hom(B2 , B1 ) = 0 for all B1 ∈ D1 and B2 ∈ D2 . A triangulated category that cannot be decomposed is called indecomposable. 10). 53 Show that condition ii) in the presence of iii) just says that A is the direct sum of B1 and B2 .
G. higher direct images. Here, we shall stay in the general G Ab for situation and only consider the covariant functor Hom(A, ) : A an arbitrary object A ∈ A and its contravariant relative Hom( , A). Clearly, Hom(A, ) is left exact and if A contains enough injectives, one deﬁnes Exti (A, ) := H i ◦ RHom(A, ). 56 Suppose A, B ∈ A are objects of an abelian category containing enough injectives. Then there are natural isomorphisms ExtiA (A, B) HomD(A) (A, B[i]), where A and B are considered as complexes concentrated in degree zero.
This leads to the deﬁnition of morphisms in the derived category as diagrams of the form qis ||| | || }| | A• C• C CC CC CC C3 B•, G A• is a quasi-isomorphism. where C • In order to make this a sensible deﬁnition of morphisms, one has to explain when two such roofs are considered equal and how to deﬁne the composition in the derived category. The natural context for both problems is the homotopy category of complexes. This will be an intermediate step in passing from Kom(A) to D(A): G D(A) q Kom(A) C K(A) By abuse of notation, we shall again write Q : K(A) functor.