Deformations of Algebraic Schemes by Edoardo Sernesi

By Edoardo Sernesi

This account of deformation concept in classical algebraic geometry over an algebraically closed box offers for the 1st time a few effects formerly scattered within the literature, with proofs which are quite little identified, but suitable to algebraic geometers. Many examples are supplied. lots 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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The computation we just did immediately implies that the local Kodaira-Spencer map κF,0 : T0 Am−1 → H 1 (Fm , TFm ) is an isomorphism. 5 Higher order deformations - Obstructions Let X be a nonsingular algebraic variety. Consider a small extension e : 0 → (t) → A˜ → A → 0 in A and let ξ: X → X ↓ ↓ Spec(k) → Spec(A) 36 CHAPTER 1. INFINITESIMAL DEFORMATIONS be an infinitesimal deformation of X. A lifting of ξ to A˜ consists in a deformation X → X˜ ˜ ξ: ↓ ↓ ˜ Spec(k) → Spec(A) and an isomorphism of deformations X X φ −→ X˜ ×Spec(A) ˜ Spec(A) Spec(A) If we want to study arbitrary infinitesimal deformations, and not only first order ones, it is important to know whether, given ξ and e, a lifting of ξ to A˜ exists, and how many are there.

S. 1. OBSTRUCTIONS 47 induced by the inclusion J ⊂ (J, (X)n+1 ). e. that o(pn /Λ) is surjective. qed The following theorem gives a characterization of formally smooth homomorphisms in A∗ . 5 Let µ : Λ → R be a homomorphism in A∗ . 4) is commutative. (ii) µ is formally smooth. (iii) dµ : tR → tΛ is surjective and o(µ) is injective. (iv) o(R/Λ) = (0) Proof. (i) ⇒ (ii) is trivial. (ii) ⇒ (iii) Let v ∈ tΛ be given as a k-algebra homomorphism Λ → k[ ]. The formal smoothness of µ implies the existence of a homomorphism w : R → k[ ] which makes the following diagram commutative: k ← R ↑ ↑ k[ ] ← Λ and this means that dµ(w) = v.

To every A-extension ϕ η : 0 → I → R −→ R → 0 we associate a complex c• (η) of R-modules (also denoted c• (ϕ)) defined as follows: c0 (η) = ΩR /A ⊗R R c1 (η) = I cn (η) = (0) n = 0, 1 d1 : c1 (η) → c0 (η) is the map x → d(x)⊗1. In other words c• (η) consists of the first map in the conormal sequence of ϕ. If r : (R , ϕ) → (R , ψ) is a homomorphism of A-extensions then r induces a homomorphism of complexes c• (r) : c• (ϕ) → c• (ψ) in an obvious way. The following is easy to establish: • Let r1 , r2 : (R , ϕ) → (R , ψ) be two homomorphisms of A-extensions of R.

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