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§2. Morphisms Into Projective Spaces 387

example, a point x ∈ X is a prime ideal in any affine open Spec(R) containing x.
Prime ideals have invariants of dimension, height, and depth, which carry over to the
points on schemes. Ideal sheaves determine closed subschemes as an ideal I ⊂ R
determines Spec(R/I) a closed subset of Spec(R). There are Noetherian conditions
and the concepts of irreducibility and Krull dimension.

For basic (commutative) ring theory, we recommend the following book: H. Mat-
sumura, Commutative Algebra, Second Edition, Benjamin, 1980. It covers those top-
ics most relavent to local scheme theory.
For general scheme theory in English we prefer: D. Mumford, Red Book,SLN
1358. We will make reference to this book though this chapter.
There is still just one very basic reference of scheme theory, namely EGA,
Grothendieck, A. and J. Dieudonn´ e, El´ ements de G´ eom´ etrie Alg´ ebrique, Publ. Math.
I.H.E.S., 4, 8, 11, 17, 20, 24, 28, 32 (1961–1967), and Springer-Verlag, Berlin, 1971.
The serious student can only look forward to many hours of encounter with this book.

§2. Morphisms Into Projective Spaces Determined byLine
Bundles, Divisors, and Linear Systems
In the previous section, we have characterized the morphisms Hom (l/rg/sp) (X, Spec(R))
from a local ringed space X into an affine scheme Spec(R) with a natural isomor-
phism

Hom (rg) (R, (X, O X )) → Hom (l/rg/sp) (X, Spec(R)).
n
Now, we try to characterize the set of morphisms Hom (Sh/R) (X/R, P ) from a
R
n
scheme X over R to the scheme given by n-dimensional projective space P also
R
defined over a ring R.
n
(2.1) Remark. By using local isomorphisms with O , we can extend the assertion
X
of (1.2) to a locally free O X -sheaf E of finite rank, that is, a vector bundle. For a
section s ∈ (X, E), the set X s of points x ∈ X where s(x) = 0isanopenset which
in turn is a subset of the closed set of points x ∈ X where s x = 0. Consider the
case where E = L a line bundle, that is, an O X -sheaf locally isomorphic to O X and
s ∈ (X, L). Then, for each t ∈ (X s , L), we can form t/s ∈ (X s , O X ) which is
uniquely described by the relation (t/s).s = t in (X s , L).
n
Returning to the description of Hom (Sh/R) (X/R, P ), we will define P n =
R R
Proj(R[y 0 ,..., y n ]) and its canonical line bundle O(1) which is generated by sec-
n
n
tions y 0 ,..., y n ∈ (P , O(1)) as follows. The scheme P is covered by n +1 open
R R
n
sets W i where y i (x) = 0 and as affine subschemes of P they are given by
R
W i = Spec(R[y 0 /y i ,..., y n /y i ]).
n
The covering condition P = W 0 ∪ ··· ∪ W n implies that the sections generate the
R
line bundle, which for the line bundle O(1) means that at least one section is nonzero
at each x of the space.