Linear equations over noncommutative graded rings

We call a graded connected algebra $R$ effectively coherent, if for every linear equation over $R$ with homogeneous coefficients of degrees at most $d$, the degrees of generators of its module of solutions are bounded by some function $D(d)$. For commutative polynomial rings, this property has been established by Hermann in 1926. We establish the same property for several classes of noncommutative algebras, including the most common class of rings in noncommutative projective geometry, that is, strongly Noetherian rings, which includes Noetherian PI algebras and Sklyanin algebras. We extensively study so--called universally coherent algebras, that is, such that the function $D(d)$ is bounded by 2d for $d >> 0$. For example, finitely presented monomial algebras belong to this class, as well as many algebras with finite Groebner basis of relations.Comment: 22 pages; corrections in Propositions 2.4 and 4.3, typos, et

Koszul algebras associated to graphs

Quadratic algebras associated to graphs have been introduced by I. Gelfand, S. Gelfand, and Retakh in connection with decompositions of noncommutative polynomials. Here we show that, for each graph with rare triangular subgraphs, the corresponding quadratic algebra is a Koszul domain with global dimension equal to the number of vertices of the graph.Comment: 8 page

Coherent algebras and noncommutative projective lines

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.Comment: 10 pages. In this version, Prop. 1.5 extended, few comments added et