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
−→ Sx
−→ 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
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