MEMOIRS

74 4. COMMUTATIVITY AND MULTIPLICITY FREENESS(2’) R is almost commutative.(3) For each homogeneous prime ideal P in R of height one the localization RP0is a local (equivalently, a semiperfect) ring.Proof. Recall Proposition 4.2.2 and Proposition 2.2.15.□Problem 4.3.9. Let X be a homogeneous exceptional curve. Is the categoryH =coh(X) uniquely determined (up to equivalence) by the function field k(X)?This is only clear for commutative function fields.In the context of this section Problem 2.3.10 becomes interesting again:Problem 4.3.10. Find a formula for the skewness s(X) intermsofthemultiplicityfunction e. From such a formula Theorem 4.3.1 should be derived as aspecial case.Remark 4.3.11. In case s(X) = 2 the existence of a rational point x such thate(x) =s(X) follows directly from Theorem 4.3.1.Remark 4.3.12. The function field and the multiplicities are also related by afundamental exact sequence. For this sequence one has to consider the Grothendieckcategory Qcoh(X) = ModZ (R)Mod Z 0 (R),the quotient category modulo the Serre subcategoryformed by the Z-graded torsion modules.The injective hull Q of the line bundle L is a generic sheaf (corresponding tothe generic Λ-module, where Λ is the associated bimodule algebra). Moreover, Qis the injective hull of each line bundle and the endomorphism ring End(Q) isthefunction field k(X). (Compare [56, Lemma 14].)For each x ∈ X denote by Sxω the Prüfer sheaf, which is the direct limit of allS x(n) (the indecomposable sheaf of length n concentrated in x).There is the short exact sequence in Qcoh(X)0 −→ L −→ Q −→ ⊕ ⊕Sx ω −→ 0x∈X e(x)involving the multiplicities e(x). This sequence already appeared in [90, Prop. 5.2].