`
germ _{y}°f^{#}_{U} :
O_{X}(U) → f_{*}O_{Y}(U) = O_{Y}(V) → O_{Y,y}.
`

`
O _{X,f(y)} → O_{Y,y}.
`

`
φ _{Q} : R_{P} → S_{Q},
`

`
(f, f ^{#}) : (Y, O_{Y}) → (X, O_{X})
`

∅ | |→ | 0 |

{η} | |→ | K |

X | |→ | R, |

O_{X,x} = R, | O_{X,η} = K. |

∅ | |→ | 0 |

{η} | |→ | K |

{η, x_{1}} | |→ | R |

{η, x_{2}} | |→ | R |

Y = {η, x_{1}, x_{2}}
| |→ | R. |

Let's now look at a more interesting class of examples. Start
with a graded ring `S`. This means, among other
things, that we can decompose `S = ⊕ _{d≥0}
S_{d}` as a direct sum of abelian groups

As a set, define `Proj(S)` to be the set of
homogeneous prime ideals `P ⊂ S` that do not
contain the ideal `S _{+}`. Given any homogeneous ideal

If `P` is a homogeneous prime ideal in
`S`, write `T` for the
multiplicative set of all **homogeneous** elements
in `S \ P`. Because we are only inverting
homogeneous elements, the localized ring `T ^{-1}S`
has a natural grading. Write

Phew. Of course, that's exactly the sort of definition we've
seen twice before. It's clear that it gives a sheaf of rings, making
`Proj(S)` into a ringed space.

- The stalk
`O`is naturally isomorphic to the degree zero localization_{X, P}`S`._{(P)} - For any homogeneous element
`f ∈ S`, let_{+}`D`. Each_{+}(f) = Proj(S) \ Z(f)`D`is open in_{+}(f)`Proj(S)`, and the collection of all`D`covers_{+}(f)`Proj(S)`. - For any homogeneous element
`f ∈ S`, let_{+}`S`be the subring of elements of degree zero in the localized ring_{(f)}`S`. There is an isomorphism of locally ringed spaces between_{f}`D`and_{+}(f)`Spec(S`._{(f)})

(i) Any element of the stalk is represented in some
neighborhood as a degree zero fraction, and thus gives rise to an
element of the localized ring. This construction clearly defines a
surjection. The proof of injectivity is then similar to the argument
we gave in the affine case. Note that, as a consequence, we can
conclude that `Proj(S)` is a locally ringed space.

(ii) The `D _{+}(f)` are clearly open. If

(iii) Given any homogeneous ideal `J`, define
`φ(J) = (JS _{f}) ∩ S_{(f)}`. When

`
α : Mor(X, Spec(R)) → Hom(R, Γ(X, O _{X})).
`

`
α(f) : R = O _{Y}(Y) → f_{*}O_{X}(Y) = O_{X}(f^{-1}(Y)) =
O_{X}(X) = Γ(X, O_{X}).
`

`
Mor(X, Spec(Z)) = Hom(Z, Γ(X, O _{X})),
`

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

Kevin R. CoombesDepartment of Biomedical Informatics

The Ohio State University

Columbus, Ohio 43210