4 INTRODUCTION

The following theorem provides an explicit one-to-one correspondence between

points of X and homogeneous prime ideals of height one in Π(L, σ), given by forming

universal extensions.

Theorem. Let R = Π(L, σ) with σ being eﬃcient. Let Sx be a simple sheaf

concentrated in the point x ∈ X. Let

0 −→ L

πx

−→

σd(L)

−→ Sx

e

−→ 0

be the Sx-universal extension of L. Then the element πx is normal in R, that is,

Rπx = πxR. Furthermore, Px = Rπx is a homogeneous prime ideal of height one.

Moreover, for any homogeneous prime ideal P ⊂ R of height one there is a

unique point x ∈ X such that P = Px.

In this way X becomes the projective prime spectrum of R. See 1.2.3 and 1.5.1

for the complete statements.

Since a commutative noetherian domain is factorial if and only if each prime

ideal of height one is principal, we say that a noetherian graded domain R, not

necessarily commutative, is a (noncommutative) graded factorial domain if each

homogeneous prime ideal of height one is principal, generated by a normal element.

This is a graded version of a concept introduced by Chatters and Jordan [13].

Corollary. Each homogeneous exceptional curve admits a projective coordi-

nate algebra which is graded factorial.

The following results clarify the role of the multiplicities e(x). The conclusion

is that they measure noncommutativity (“skewness”) in several senses:

Theorem. The function field of X is commutative if and only if all multiplic-

ities are equal to one.

See 4.3.1 for the complete statement; the commutative function fields are ex-

plicitly determined. Moreover:

• The multiplicities e(x) are bounded from above by the square root s(X)

of the dimension of the function field over its centre. More precisely, if

e∗(x) denotes the square root of the dimension of End(Sx) over its centre,

then always e(x) · e∗(x) ≤ s(X), and equality holds for all points x except

finitely many (2.2.13 and 2.3.5).

• In the graded factorial algebra R we have unique factorization in the sense

that each normal homogeneous element is an (essentially unique) prod-

uct of prime elements (which are by definition homogeneous generators

of prime ideals of height one). In contrast to the commutative case, a

prime element πx may factorize into a product of several irreducible ele-

ments. The number of these factors is essentially given by e(x) (see 1.6.5

and 1.6.6).

• We describe the localization RP at a prime ideal P . It turns out that RP

is a local ring if and only if the corresponding multiplicity e(x) is one;

otherwise RP is not even semiperfect (2.2.15).

Another surprising phenomenon due to noncommutativity is the occurrence of

so-called ghost automorphisms. Denote by Aut(X) the group of all (isomorphism

classes of) automorphisms of the category H fixing the structure sheaf L. Let

R = Π(L, σ) be the orbit algebra formed with respect to an eﬃcient automorphism

σ. Every prime element πy ∈ R (that is, a normal element generating the prime