By Dirk Kussin

In those notes the writer investigates noncommutative tender projective curves of genus 0, also referred to as unheard of curves. As a chief outcome he exhibits that every such curve X admits, as much as a few weighting, a projective coordinate algebra that is a now not inevitably commutative graded factorial area R within the feel of Chatters and Jordan. additionally, there's a average bijection among the issues of X and the homogeneous leading beliefs of peak one in R, and those major beliefs are crucial in a robust experience

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2 below). 5). 1. Let R = Π(L, σx ) be the orbit algebra deﬁned by an eﬃcient tubular shift σx at x. Then the prime element πx associated with x is central in R. Proof. Write σ = σx and π = πx . Since for all homogeneous elements r ∈ R of degree n ≥ 0 we have the commutative universal diagram 0 L π the element π is central. σnL e(x) Sx n σ π σ n+1 L 0 rx σr r 0 σL e(x) Sx 0, 44 1. GRADED FACTORIALITY The non-simple bimodule case. 2. Let X be a homogeneous exceptional curve and M = F MG be the underlying tame bimodule which we assume to be of type (2, 2) and to be nonsimple.

Let Sx be a simple sheaf concentrated in the point x ∈ X. Let π x 0 −→ L −→ σ d (L) −→ Sxe −→ 0 be the Sx -universal extension of L. Then the following conditions hold (1) The element πx is normal, that is, Rπx = πx R. (2) Px = Rπx is a homogeneous prime ideal. (3) Px is a completely homogeneous prime ideal (that is, R/Px is a graded domain) if and only if e = 1. Moreover, for any homogeneous prime ideal P of height one there is a point x ∈ X such that P = Px . Because of the last statement and since R is also a noetherian domain, we say that R is graded factorial, in analogy to commutative algebra.

Let U be the subfunctor of (L, −] corresponding to the graded module Ra. There is an epimorphism (L(m), −] −→ U . 1) (n ) t where C is a coproduct i=1 Si i , with (not necessarily non-isomorphic) simple (n ) S1 , . . , St in H0 , concentrated in x1 , . . , xt , respectively. Let Mi i be the graded (n ) 1 i left R-module , L(n)) and Pi = Rπi be the homogeneous prime n≥0 Ext (Si (n ) ideal corresponding to xi . We have Pini ⊂ AnnR (Mi i ). 1) we get an exact sequence 0 −→ (L(m), −] −→ (L, −] −→ Ext1 (C, −].