Factorization in noncommutative curves

Abstract

AbstractA commutative curve (f0)∈k[x1,…,xn] has many noncommutative models, i.e. f∈k〈x1,…,xn〉 having f0 as its image by the canonical epimorphism κ from k〈x1,…,xn〉 to k[x1,…,xn].In this note we consider the cases, where n=2.If the polymomial f0 has an irreducible factor, g0, then in terms of conditions on the noncommutative models of (f0), we determine, when g02 is a factor of f0.In fact we prove that in case there exists a noncommutative model f of f0 such that ExtA1(P,Q)≠0 for all point P,Q∈Z(f0), where A=k〈x,y〉/(f), then g02 is a factor of f0.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 factorizationsf=gh=h1g′h2,where g0=g0′ and with g0 irreducible and prime to h0, thenZ(g0)∩Z((h1)0)=∅,i.e. g0 and (h1)0 do not have a common zero