Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4588312 | Journal of Algebra | 2008 | 11 Pages |
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 P1 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 P1. In this way, we obtain a sequence (n⩾2) of pairwise non-isomorphic noncommutative schemes which generalize the scheme .
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