Factorization in noncommutative curves

Abstract

A commutative curve ( f 0 ) ∈ k [ x 1 , … , x n ] has many noncommutative models, i.e. f ∈ k 〈 x 1 , … , x n 〉 having f 0 as its image by the canonical epimorphism κ from k 〈 x 1 , … , x n 〉 to k [ x 1 , … , x n ] . In this note we consider the cases, where n = 2 . If the polymomial f 0 has an irreducible factor, g 0 , then in terms of conditions on the noncommutative models of ( f 0 ) , we determine, when g 0 2 is a factor of f 0 . In fact we prove that in case there exists a noncommutative model f of f 0 such that Ext A 1 ( P , Q ) ≠ 0 for all point P , Q ∈ Z ( f 0 ) , where A = k 〈 x , y 〉 / ( f ) , then g 0 2 is a factor of f 0 . We also note that the “converse” result holds. Next we apply the methods from above to show that in case an element f in the free algebra has 2 essential different factorizations f = gh = h 1 g ′ h 2 , where g 0 = g 0 ′ and with g 0 irreducible and prime to h 0 , then Z ( g 0 ) ∩ Z ( ( h 1 ) 0 ) = ∅ , i.e. g 0 and ( h 1 ) 0 do not have a common zero.