We're on the verge of being able to define an affine scheme.
It's more than the set `Spec(R)`; it's even more than
the Zariski topology on this set; it's all that plus a structure sheaf.

`
s : U → ∐ _{P∈U} R_{P}
`

- for all
`P ∈ U`, one has`s(P) ∈ R`; and_{P} - for all
`P ∈ U`, there exists an open neighborhood`P ∈ V ⊂ U`and elements`f, g ∈ R`such that for all`Q ∈ V`, one has`f ∉Q`and`s(Q) = g/f ∈ R`._{Q}

`
f ^{M} = ∑ e_{Q} h_{Q}.
`

`
f ^{M} g = ∑ e_{Q} h_{Q} g = ∑ 0 = 0.
`

- The sets
`D(f`cover_{i})`D(f)`; - The fractions
`g`define sections in_{i}/f_{i}^{mi}∈ R_{fi}`O`that agree when restricted to the intersections_{X}(D(f_{i}))`D(f`for different_{i}) ∩ D(f_{j})`i`and`j`.

`
[g _{i} / f_{i}] =
[g_{j} / f_{j}]
∈ R_{fi fj}.
`

`
(f _{i} f_{j})^{N} (g_{i} f_{j} - g_{j} f_{i}) = 0 ∈ R.
`

`
[g _{i} f_{i}^{N}] [f_{j}^{N+1}] - [g_{j} f_{j}^{N}] [f_{i}^{N+1}] = 0.
`

`
f ^{T} = ∑ e_{i} f_{i}.
`

`
g = ∑ e _{i} g_{i}.
`

`
g f _{j} = ∑ e_{i} g_{i} f_{j} = ∑ e_{i} f_{i} g_{j} = f^{T} g_{j} ∈ R.
`

Before stating some of the important corollaries of this result, I need to introduce yet another piece of notation.

Now let's look at a pair of rings and a homomorphism
`φ : R → S` between them. Since we're
supposed to think of elements of the rings as functions on the
corresponding spectra, we can at least hope that
`φ` is related in some sensible way to a map on
spectra that goes in the other direction (`f : Spec(S) →
Spec(R)`). After a moment's reflection, we can see what
this map should be on the level of sets: Given a prime ideal
`P` in `S`, its inverse image
`φ ^{-1}(P)` is a prime ideal in

So far, we've defined a map on spectra, coming from a
homomorphism, that reflects the topological part of the
structure. What can we expect on the sheaf-theoretic part? Well, let's
write `X=Spec(R)` and `Y=Spec(S)`,
and keep the rest of our notation the same, so that `φ :
R → S` is a ring homomorphism that defines a
continuous function `f : Y → X`. Take an open
subset `U ⊂ X` and a section `s
∈ O _{X}(U)`. Write, for the moment,

`
s°f : V → ∐ _{P∈U} R_{P}.
`

`
"φ"
°s°f : V → ∐ _{Q∈V} S_{Q}.
`

There appear to be some technical difficulties with using this
construction to understand how the structure sheaves on the two
spectra are related. First, we haven't explained what to do with
points `P ∈ U` that are not in the image of
`f`. Second, we've only described what's going on
with sections on open sets of `Y` of the form
`V = f ^{-1}(U)`, and not on general open sets.

`
f _{*}F(U) = F(f^{-1}(U))
`

Now we can correctly interpret the construction we started
describing earlier. Given a ring homomorphism `φ : R
→ S`, it not only determines a continuous function on
spectra `f : Y = Spec(S) → X = Spec(R)`, but
it also defines a morphism of sheaves

`
f ^{#} : O_{X} → f_{*}O_{Y}
`

Comments on this web site should be addressed to the author:

Kevin R. CoombesDepartment of Biomedical Informatics

The Ohio State University

Columbus, Ohio 43210