Deformations of Algebraic Schemes by Edoardo Sernesi PDF

By Edoardo Sernesi

ISBN-10: 3540306080

ISBN-13: 9783540306085

ISBN-10: 3540306153

ISBN-13: 9783540306153

This account of deformation concept in classical algebraic geometry over an algebraically closed box provides for the 1st time a few effects formerly scattered within the literature, with proofs which are fairly little identified, but suitable to algebraic geometers. Many examples are supplied. many of the algebraic effects wanted are proved. the fashion of exposition is stored at a degree amenable to graduate scholars with a regular heritage in algebraic geometry.

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FORMAL DEFORMATION THEORY o(R/Λ) = (0) (resp. if o(R/Λ) = (0)); R is said to be unobstructed (resp. obstructed) if o(R) = (0) (resp. if o(R) = (0)). Given a homomorphism f : R → S in A∗Λ we denote by o(f /Λ) : o(S/Λ) → o(R/Λ) the linear map induced by pullback: o(f /Λ)([η]) = [f ∗ η] ∈ ExΛ (R, k) for all [η] ∈ ExΛ (S, k). Since this definition is functorial we have a contravariant functor: o(−/Λ) : A∗Λ → (vector spaces/k) When Λ = k we write o(f ) instead of o(f /k). If µ is such that o(µ) is injective one simetimes says that R is less obstructed than Λ.

Ii) If oξ (e) = 0 then there is a natural transitive action of H 1 (X, TX ) on ˜ the set of isomorphism classes of liftings of ξ to A. (iii) The correspondence e → oξ (e) defines a k-linear map oξ : Exk (A, k) → H 2 (X, TX ) Proof. Let U = {Ui }i∈I be an affine open cover of X. We have isomorphisms θi : Ui × Spec(A) → X|Ui and consequently θij := θi−1 θj is an automorphism of the trivial deformation Uij × Spec(A). 2. LOCALLY TRIVIAL DEFORMATIONS 37 on Uijk × Spec(A). To give a lifting ξ˜ of ξ to A˜ it is necessary and sufficient to give a collection of automorphisms {θ˜ij } of the trivial deformations Uij × ˜ such that Spec(A) (a) θ˜ij θ˜jk = θ˜ik (b) θ˜ij restricts to θij on Uij × Spec(A) In fact from such data we will be able to define X˜ by patching the local ˜ along the open subsets Uij × Spec(A) ˜ in the usual way.

When S = Spec(A) with A in ob(A∗ ) and s ∈ S is the closed point we have a local family of deformations (shortly a local deformation) of X over A. The deformation η will be also denoted (S, η) or (A, η) when S = Spec(A). The local deformation (A, η) is infinitesimal (resp. first order) if A ∈ ob(A) (resp. A = k[ ]). e. such that the following diagram is commutative: X X φ −→ S Y 22 CHAPTER 1. INFINITESIMAL DEFORMATIONS By a pointed scheme we will mean a pair (S, s) where S is a scheme and s ∈ S.

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Deformations of Algebraic Schemes by Edoardo Sernesi


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