A Royal Road to Algebraic Geometry
This book is about modern algebraic geometry. The title A Royal Road to Algebraic Geometry is inspired by the famous anecdote about the king asking Euclid if there really existed no simpler way for learning geometry, than to read all of his work Elements.
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The Category of Schemes
In this chapter we introduce the categories of preschemes and schemes, and explore some of their basic properties.
10.1 First Approximation: The Category of Ringed Spaces A ringed space is a pair (X, OX ) consisting of a topological space X and a sheaf OX of Comm on X, defined for all non empty open subsets of X. By abuse of notation the pair (X, OX ) is also denoted by X. The topological space is referred to as the underlying topological space, while the sheaf OX is called the structure sheaf of X. A morphism from the ringed space (X, OX ) to the ringed space (Y, OY ) (f, θ) : (X, OX ) −→ (Y, OY ) is a pair consisting of a continuous mapping f : X −→ Y and a homomorphism of sheaves of Comm, θ : OY −→ f∗ (OX ). It is easily verified that the ringed spaces form a category, which we denote by Rs. Note that whenever (X, OX ) is a ringed space, and f : X −→ Y is a continuous mapping, then Y = (Y, f∗ (OX )) is a ringed space and the pair (f, id) is a morphism from X to Y . The most common ringed spaces are topological spaces X with various kinds of function sheaves, which usually take their values in a field K. Frequently the field is either R or C. The sheaf OX may be the sheaf of all continuous functions on the respective open subsets, or when X looks locally like an open subset of Rn or Cn we may consider functions which are n times differentiable, or algebraic functions when X is an algebraic variety over the field K, and so on. A. Holme, A Royal Road to Algebraic Geometry, c Springer-Verlag Berlin Heidelberg 2012 DOI 10.1007/978-3-642-19225-8 10,
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The Category of Schemes
If (X, OX ) and (Y, OY ) are the ringed spaces obtained by taking the sheaves of continuous functions (say to R or to C) on the two topological spaces X and Y , and if f : X −→ Y is a continuous mapping, then we obtained a morphism from the continuous mapping f by composition with the restriction of f by defining θ : OY −→ f∗ (OX ), as follows: For all open U ⊂ Y there are homomorphisms θU : OY (U ) −→ f∗ (OX )(U ) = OX (f −1 (U )) f|f −1 (U )
ϕ
ϕ|U
(U −→ K) → (θU (ϕ) : f −1 (U ) −→ U −→ K), where K is R, C or for that matter, any ring. Similarly, if the topological spaces have more structure, like being differentiable manifolds, algebraic varieties etc., then this also works if we use morphisms in the category to which X and Y belong, instead of just continuous mappings. The details of these considerations are left to the reader. Another type of ringed spaces is obtained by taking a topological space X and letting OX be the sheaf associated to the presheaf defined by O (U ) = A, where A is a fixed ring. The sheaf OX so defined is referred to as the constant sheaf of A on X. Clearly Spec(A) which we have defined above is a ringed space. Moreover, if ϕ : A −→ B is a homomorphism of Comm, then we obtain a morphism of ringed spaces Spec(ϕ) : Spec(B) −→ Spec(A) as follows: The mapping of topological spaces f : Spec(B) −→ Spec(A) is given by q → ϕ−1 (q). As is easily seen, we then have f −1 (D(a)) = D((ϕ(a)B), hence
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