Coherent Algebras and Noncommutative Projective Lines

A well-known conjecture says that every one-relator group is coherent. We state and partly prove a similar statement for graded associative algebras. In particular, we show that every Gorenstein algebra A of global dimension 2 is graded coherent. This allows us to define a noncommutative analogue of the projective line P as a noncommutative scheme based on the coherent noncommutative spectrum qgrA of such an algebra A, that is, the category of coherent A-modules modulo the torsion ones. This category is always abelian Ext -finite hereditary with Serre duality, like the category of coherent sheaves on P. In this way, we obtain a sequence P n (n ≥ 2) of pairwise non-isomorphic noncommutative schemes which generalize the scheme P = P 2 .