Coherent algebras and noncommutative projective lines

Abstract

A well-known conjecture says that every one-relator group is coherent. We state and partly prove an analogous 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 $\PP^1$ as a noncommutative scheme based on the coherent noncommutative spectrum $\cohp A$ 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 $\PP^1$. In this way, we obtain a sequence $\PP^1_n $ ($n\ge 2$) of pairwise non-isomorphic noncommutative schemes which generalize the scheme $\PP^1 = \PP^1_2$.