Download e-book for kindle: Fourier-Mukai Transforms in Algebraic Geometry by Daniel Huybrechts
By Daniel Huybrechts
This seminal textual content on Fourier-Mukai Transforms in Algebraic Geometry by means of a number one researcher and expositor relies on a path given on the Institut de Mathematiques de Jussieu in 2004 and 2005. geared toward postgraduate scholars with a uncomplicated wisdom of algebraic geometry, the foremost element of this booklet is the derived classification of coherent sheaves on a gentle projective style. together with notions from different parts, e.g. singular cohomology, Hodge idea, abelian kinds, K3 surfaces; complete proofs are given and workouts reduction the reader all through.
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10). 53 Show that condition ii) in the presence of iii) just says that A is the direct sum of B1 and B2 . In particular, the deﬁnition is symmetric in D1 and D2 despite the chosen order in ii). 54 Let F : D triangulated categories. Suppose that D contains objects not isomorphic to 0 and that D is indecomposable. Then F is an equivalence of categories if and only if F has a left adjoint G F and a right adjoint F H such that for any object B ∈ D one has: H(B) 0 implies G(B) 0. See . Proof In order to prove the proposition, one introduces two full triangulated subcategories D1 , D2 ⊂ D .
RF Proof i) Let A• ∈ D+ (A) and I • := ι−1 (A• ). The natural transformation G I • in D+ (A), which itself G ι ◦ ι−1 yields a functorial morphism A• id qis • o • • G C I . 38), which, moreover, is unique morphism A independent of the choice of C • . Altogether, one obtains a functorial morphism G K(F )(I • ) = RF (A• ). K(F )(A• ) G D+ (A) is clearly ii) The category K+ (IA ) is triangulated and ι : K+ (IA ) exact. Thus, also the inverse functor ι is exact (cf. 41). Hence, RF is the composition of three exact functors and, therefore, itself exact.
1 Derived category of an abelian category In this section we shall recall the fundamental aspects of derived categories. We begin by stating the existence of the derived category as a theorem, and explain the technical features, necessary for any calculation, later on. 2. In the sequel, we will mostly be interested in the derived category of the abelian category of (coherent) sheaves or of modules over a ring. We recommend the textbooks [39, 61, 70] for more details and other examples. 1 Often, an object in a given abelian category A is studied in terms of its resolutions.