By Daniel Perrin
Aimed essentially at graduate scholars and starting researchers, this ebook presents an creation to algebraic geometry that's quite appropriate for people with no past touch with the topic and assumes basically the normal heritage of undergraduate algebra. it's constructed 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 starts off with easily-formulated issues of non-trivial ideas – 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: measurement; singularities; sheaves; kinds; and cohomology. The therapy makes use of as little commutative algebra as attainable through quoting with no evidence (or proving simply in particular circumstances) theorems whose evidence isn't really precious in perform, the concern being to increase an knowing of the phenomena instead of a mastery of the process. various routines is supplied for every subject mentioned, and a range of difficulties and examination papers are accrued in an appendix to supply fabric for extra examine.
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Additional info for Algebraic Geometry: An Introduction (Universitext)
When U is equal to X, we call the corresponding sections global sections. b. General sheaves We will need more general sheaves than function sheaves. d. 3. Let X be a topological space. A presheaf on X is given by the following data: • • For every open set U in X, a set F(U ); For every pair of open sets U and V with V ⊂ U , a map rV,U : F(U ) → F(V ) called the restriction map, such that the two following conditions are satisﬁed: i) If W ⊂ V ⊂ U , then rW,U = rW,V rV,U , ii) We have rU,U = IdF (U ) ; We set rV,U (f ) = f |V .
Such an ideal is said to be homogeneous. Proof. It is clear that 2) implies 1). Conversely, assume that I is generated by homogeneous elements Gi of degrees αi . Consider F = F0 + · · · + Fr ∈ I, where Fi is homogeneous of degree i. By induction, it will be enough to show Ui Gi , and on identifying terms of that Fr ∈ I. But we can write F = highest degree, we get Fr = Ui,r−αi Gi , so Fr is contained in I. 3. Let R be a graded k-algebra and let I be a homogeneous ideal of R. Let S be the quotient k-algebra S = R/I and p the canonical projection.
Ym ) ∈ I(W ) and x ∈ V . We calculate F (ϕ(x)) = F (θ(η1 ), . . , θ(ηm ))(x). Since θ is a morphism of algebras, F (θ(η1 ), . . , θ(ηm )) = θ(F (η1 , . . , ηm )), and since F (η1 , . . , ηm ) is the image in Γ (W ) of F (Y1 , . . , Ym ) ∈ I(W ), it vanishes and we are done. 8. Let ϕ : V → W be a morphism. Then ϕ is an isomorphism if and only if ϕ∗ is an isomorphism. It follows that V and W are isomorphic if and only if their algebras Γ (V ) and Γ (W ) are isomorphic. 9. The morphism ϕ : k → V = V (Y 2 − X 3 ) given by ϕ(t) = (t2 , t3 ) is not an isomorphism.
Algebraic Geometry: An Introduction (Universitext) by Daniel Perrin