Algebraic Geometry: An Introduction (Universitext) by Daniel Perrin

By Daniel Perrin

Aimed essentially at graduate scholars and starting researchers, this publication offers an creation to algebraic geometry that's rather compatible for people with no prior touch with the topic and assumes merely the normal historical past of undergraduate algebra. it's built from a masters direction given on the Université Paris-Sud, Orsay, and focusses on projective algebraic geometry over an algebraically closed base field.

The e-book begins with easily-formulated issues of non-trivial recommendations – for instance, Bézout’s theorem and the matter of rational curves – and makes use of those difficulties to introduce the basic instruments of recent algebraic geometry: measurement; singularities; sheaves; kinds; and cohomology. The therapy makes use of as little commutative algebra as attainable by way of quoting with no facts (or proving simply in particular instances) theorems whose facts isn't really worthwhile in perform, the concern being to advance an realizing of the phenomena instead of a mastery of the strategy. a number of routines is supplied for every subject mentioned, and a range of difficulties and examination papers are accrued in an appendix to supply fabric for additional research.

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We will see in Chapter III that if we embed affine space k n in projective space, then we recover by this method the Zariski topology on k n . c) Let V ⊂ Pn be a projective algebraic set; we associate to V its cone C(V ), which is the inverse image of V under the projection p : k n+1 − {0} → Pn (k), plus the origin of k n+1 . If I is a homogeneous ideal (cf. 2 below) different from R and V = Vp (I), then C(V ) = V (I) ⊂ k n+1 (in the affine category). If I = R, then C(V ) = V (R+ ) = {0}. This type of argument sometimes enables us to reduce a projective problem to a similar affine problem (cf.

13. Assume that k is algebraically closed. The functor Γ is then an equivalence of categories between the category of affine algebraic sets with regular maps and the category of reduced k-algebras of finite type with homomorphisms of k-algebras. (This means that the functor is fully faithful (cf. ) Proof. 7. To prove surjectivity, consider A a reduced k-algebra of finite type. Since A is of finite type, we can write A k[X1 , . . , Xn ]/I (cf. 5), and since A is reduced, the ideal I is radical. We set V = V (I).

Xn ] and set V = Vp (I). 1) Vp (I) = ∅ ⇐⇒ ∃ N such that (X0 , . . , Xn )N ⊂ I ⇐⇒ (X0 , . . , Xn ) = R+ ⊂ rac(I). 2) If Vp (I) = ∅, then Ip (Vp (I)) = rac(I). Proof. If I = R, then V = Vp (I) = ∅ and 1) is trivially true. Assume therefore that I = R. We apply the affine Nullstellensatz to the cone of V : C(V ) = V (I) ⊂ k n+1 (cf. c). The statement that V = Vp (I) is empty means exactly that C(V ) contains only the origin in k n+1 and hence that rac(I) is equal to R+ , which proves 1). We now prove 2).

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