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

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

Noncommutative Grassmannian of codimension two has coherent coordinate ring

A noncommutative Grassmannian NGr(m, n) is introduced by Efimov, Luntz, and Orlov in `Deformation theory of objects in homotopy and derived categories III: Abelian categories' as a noncommutative algebra associated to an exceptional collection of n-m+1 coherent sheaves on P^n. It is a graded Calabi--Yau Z-algebra of dimension n-m+1. We show that this algebra is coherent provided that the codimension d = n-m of the Grassmannian is two. According to op. cit., this gives a t-structure on the derived category of the coherent sheaves on the noncommutative Grassmannian. The proof is quite different from the recent proofs of the coherence of some graded 3-dimensional Calabi--Yau algebras and is based on properties of a PBW-basis of the algebra.Comment: Unfortunately, an error appeared in the Groebner basis calculation. I am grateful to Alexander Efimov who have pointed this out. The algebra menioned in the title consideration is indeed PBW, but the proof of the restricted processing property fails. So, it is still an open problem is this algebra coherent or no