Morphism of schemes

In algebraic geometry, a morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety. It is, by definition, a morphism in the category of schemes.

More generally, morphisms p:X →S with various schemes X but fixed scheme S form the category of schemes over S (the slice category of the category of schemes with the base object S.) An object in the category is called an S-scheme and a morphism in the category an S-morphism; explicitly, an S-morphism from p:X →S to q:Y →S is a morphism ƒ:X →Y of schemes such that p = q ∘ ƒ.

By definition, a morphism of schemes is just a morphism of locally ringed spaces.

A scheme, by definition, has an open affine chart and thus a morphism of schemes can also be described in terms of such charts (compare the definition of morphism of varieties). Let ƒ:X→Y be a morphism of schemes. If x is a point of X, since ƒ is continuous, there are open affine subsets U = Spec A of X containing x and V = Spec B of Y such that ƒ(U) ⊂ V. Then ƒ: U → V is a morphism of affine schemes and thus is induced by some ring homomorphism B → A (cf. #Affine case.) In fact, one can use this description to "define" a morphism of schemes; one says that ƒ:X→Y is a morphism of schemes if it is locally induced by ring homomorphisms between coordinate rings of affine charts.

Let ƒ:X→Y be a morphism of schemes with ${\displaystyle \phi :{\mathcal {O}}_{Y}\to f_{*}{\mathcal {O}}_{X}}$. Then, for each point x of X, the homomorphisms on the stalks:

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