Coherent algebras and noncommutative projective lines

Abstract

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 1 as a noncommutative scheme based on the coherent noncommutative spectrum qgr A 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 1 . In this way, we obtain a sequence P n 1 ( n ⩾ 2 ) of pairwise non-isomorphic noncommutative schemes which generalize the scheme P 1 = P 2 1 .