DEFINITION 10.1
Let

be rings. Let

be a ring homomorphism.
We have already introduced

as a continuous map

. Now that the spaces

carry
structures of locally ringed spaces, we (re)define

as a morphism of locally ringed spaces by defining

as in Example
9.11.
PROOF..
Let us put

and

. The data
is equivalent to a data
which gives rise to a ring homomorphism
Let us take this homomorphism as

.
By the hypothesis of
being a morphism of locally ringed spaces,
is local homomorphism.
That means,
From the definition of

, we have
We have thus proved that
is equal to
as a map
.