By Daniel Perrin

ISBN-10: 1848000561

ISBN-13: 9781848000568

Aimed basically at graduate scholars and starting researchers, this booklet presents an advent to algebraic geometry that's relatively appropriate for people with no prior touch with the topic and assumes purely the traditional history of undergraduate algebra. it really is constructed from a masters path given on the Université Paris-Sud, Orsay, and focusses on projective algebraic geometry over an algebraically closed base field.

The booklet starts off 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 contemporary algebraic geometry: size; singularities; sheaves; types; and cohomology. The remedy makes use of as little commutative algebra as attainable by means of quoting with no facts (or proving simply in specific circumstances) theorems whose facts isn't helpful in perform, the concern being to increase an realizing of the phenomena instead of a mastery of the method. a variety of routines is supplied for every subject mentioned, and a range of difficulties and examination papers are accumulated in an appendix to supply fabric for additional research.

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**Additional resources for Algebraic Geometry: An Introduction (Universitext)**

**Example text**

B) It is always possible to consider a given sheaf to be a sheaf of functions (in a non-natural way) (cf. 1). This remark enables us to restrict ourselves to sheaves of functions if necessary. c) If F is a presheaf on X (for the sake of simplicity we assume it is a presheaf of functions from X to K), we can embed it in a canonical way in a sheaf F + called the associated sheaf of F. To do this we localise the condition for being an element of F(U ) in the following way. For any open set U in X we set F + (U ) = {f : U −→ K | ∀ x ∈ U, ∃ V open, such that x ∈ V ⊂ U, and g ∈ F(V ) such that f |V = g}.

4 in the general case, a little care is needed. The equality of the sections si on the intersection then only means that there is a natural number N such that fiN fjN (ai fjn −aj fin ) = 0. We then write f m = j bj fjn+N and a = j aj bj fjN . c) Calculating Γ (U, OV ) for a non-standard open set is harder. Consider for example U = k 2 − {(0, 0)} (cf. 2). 3 Aﬃne varieties We have equipped any aﬃne algebraic set V with the structure of a ringed space (V, OV ) by taking the sheaf of regular functions OV deﬁned above.

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).

### Algebraic Geometry: An Introduction (Universitext) by Daniel Perrin

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Categories: Algebraic Geometry