By Dirk Kussin
In those notes the writer investigates noncommutative tender projective curves of genus 0, often known as unprecedented curves. As a chief end result he exhibits that every such curve X admits, as much as a few weighting, a projective coordinate algebra that's a now not inevitably commutative graded factorial area R within the experience of Chatters and Jordan. additionally, there's a normal bijection among the issues of X and the homogeneous major beliefs of peak one in R, and those best beliefs are important in a robust feel
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Additional resources for Noncommutative curves of genus zero: related to finite dimensional algebras
Then f is called 1-irreducible, if whenever f = gh with morphisms g and h between line bundles, then g or h is an isomorphism. The following facts are obvious: (1) Each non-zero map between line bundles has a factorization into 1-irreducible maps. (2) A morphism between line bundles is 1-irreducible if and only if its cokernel is a simple object. (3) Each simple object is cokernel of a 1-irreducible map. Moreover, one of the line bundles can be chosen arbitrarily. (4) If u : L −→ L(n) is a 1-irreducible map, then it is an irreducible element in R.
Let M be the graded left R-module Ext1H (S, L(n)). Then P = AnnR (M ). n≥0 Proof. Let r ∈ Rn , r = 0. The S-universal extension induces a commutative exact diagram 0 π L L(d) Se 0 X Se 0. r 0 L(n) If r ∈ AnnR (M ) then the lower sequence splits and r ∈ P follows immediately. For the converse, we show more generally the next proposition. 5. Let S be a simple sheaf concentrated in x and P = Rπ be the corresponding homogeneous prime ideal. For each n ∈ N let S (n) be the indecomposable sheaf of length n with socle S.
The prime ideals in this ring are in one-to-one correspondence with the homogeneous prime ideals in RX since there is a central unit of degree one in RX . 6. 3 one can assume either α = 1F or δ = 0. If char k = 0 then one can assume δ = 0. Proof. 21] to the case of two variables. 23] to show that δ is an inner derivation, and hence as graded algebras Π(L, σx ) F [X; Y, 1F , 0]. 23] the ring F [Z, 1F , δ] is a simple ring. 5, X consists of precisely one point. But since k is inﬁnite there are inﬁnitely many points by [89, Thm.
Noncommutative curves of genus zero: related to finite dimensional algebras by Dirk Kussin