Elsevier

Journal of Algebra

Volume 319, Issue 8, 15 April 2008, Pages 3280-3290
Journal of Algebra

Coherent algebras and noncommutative projective lines

https://doi.org/10.1016/j.jalgebra.2007.07.010Get rights and content
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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 Pn1 (n2) of pairwise non-isomorphic noncommutative schemes which generalize the scheme P1=P21.

Keywords

Coherent ring
Graded algebra
Noncommutative scheme

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1

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.