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 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 . In this way, we obtain a sequence () of pairwise non-isomorphic noncommutative schemes which generalize the scheme .
Partially supported by the Dynastia Foundation, by the President of Russian Federation grant MD-288.2007.1, and by the Russian Basic Research Foundation project 05-01-01034.