Morphism of schemes - Imported Knowlpedia Content

A morphism of schemes $f:X\to Y$ is a morphism of locally ringed spaces. It consists of a continuous map of underlying spaces and a compatible morphism of structure sheaves into a direct image sheaf:

f^\#:\mathcal O_Y\longrightarrow f_*\mathcal O_X.

For every $x\in X$ , the induced map on stalks

f_x^\#:\mathcal O_{Y,f(x)}\longrightarrow\mathcal O_{X,x}

must be a local ring homomorphism: the inverse image of the target's maximal ideal is the source's maximal ideal.

The basic example comes from a ring homomorphism $\varphi:A\to B$ . It induces a morphism of affine schemes

\operatorname{Spec}B\longrightarrow\operatorname{Spec}A, \qquad \mathfrak q\longmapsto\varphi^{-1}(\mathfrak q).

The reversed direction is fundamental: $\operatorname{Spec}$ is contravariant. In particular, a field inclusion $F\hookrightarrow K$ gives $\operatorname{Spec}K\to\operatorname{Spec}F$ .