Namely, let
be affine schemes.
Assume that morphisms
and
are given. Then
we have
(If the morphisms and homomorphisms involved are clear from the context, we often abbreviate the above equation as:
by the abuse of language.)
So far, we have not developed enough theory of a general schemes except for the affine case. In local theories, the affine case suffices and the generalization to general schemes is fairly easy. Due to the lack of time, we omit detailed arguments. For more detailed account, see EGA or Iitaka [11]
Note that the universality of the fiber product may be interpreted as the following way.
for any field