By Daniel Perrin
Aimed basically at graduate scholars and starting researchers, this ebook offers an advent to algebraic geometry that's really compatible for people with no prior touch with the topic and assumes in basic terms the normal heritage of undergraduate algebra. it's built 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 begins with easily-formulated issues of non-trivial options – for instance, Bézout’s theorem and the matter of rational curves – and makes use of those difficulties to introduce the elemental instruments of recent algebraic geometry: measurement; singularities; sheaves; kinds; and cohomology. The therapy makes use of as little commutative algebra as attainable through quoting with out evidence (or proving in simple terms in distinctive circumstances) theorems whose evidence isn't worthy in perform, the concern being to increase an figuring out of the phenomena instead of a mastery of the process. a variety of routines is supplied for every subject mentioned, and a range of difficulties and examination papers are amassed in an appendix to supply fabric for extra learn.
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Additional info for Algebraic Geometry: An Introduction (Universitext)
B) For any open set U of Y we deﬁne a homomorphism of rings ϕ∗U : Γ (U, OY ) −→ Γ (ϕ−1 U, OX ) by setting ϕ∗U (g) = gϕ. These homomorphisms are compatible with restrictions. In other words they satisfy the condition rϕ−1 V,ϕ−1 U ϕ∗U = ϕ∗V rV,U . For ringed spaces whose sheaves are not simply sheaves of functions a morphism consists not only of the data of a continuous map ϕ but also a collection of homomorphisms ϕ∗U satisfying the above compatibility conditions (cf. 1). 2 The structural sheaf of an aﬃne algebraic set 41 2 The structural sheaf of an aﬃne algebraic set Let V ⊂ k n be an aﬃne algebraic set.
3. Let V be an aﬃne algebraic set and consider f ∈ Γ (V ). The open set D(f ) equipped with the restriction of the sheaf OV to D(f ) is an aﬃne algebraic variety. Proof. Assume that V is embedded in k n : set I = I(V ) and let F be a polynomial whose restriction to V is f . Our aim is to show that D(f ) is isomorphic to an aﬃne algebraic set. The trick is to look for this set in k n+1 : we consider the map ϕ : (x1 , . . , xn ) −→ (x1 , . . , xn , 1/f (x1 , . . , xn )) sending D(f ) into k n+1 .
Indeed, if ρ(g/f n ) = 0, then g(x) = 0 on D(f ) and hence f g = 0 on V , which implies that g/f n is zero in the localised ring (cf. Summary loc. ). We then have the following deﬁnition. 3. Let V be an aﬃne algebraic set and consider a non-zero f ∈ Γ (V ). We set Γ (D(f ), OV ) = Γ (V )f (identiﬁed with a subring of the ring of k-valued functions on D(f ) via ρ). By this method we deﬁne a sheaf of rings on V called the sheaf of regular functions. In the special case where Γ (V ) is an integral domain, the ring Γ (V )f is a subring of the ﬁeld of fractions K(V ) of Γ (V ) (the ﬁeld of rational functions on V , cf.
Algebraic Geometry: An Introduction (Universitext) by Daniel Perrin