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COHERENT ALGEBRAS AND NONCOMMUTATIVE PROJECTIVE LINES
, 2007
"... Abstract. A wellknown conjecture says that every onerelator 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 a ..."
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Abstract. A wellknown conjecture says that every onerelator 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
LINEAR EQUATIONS OVER NONCOMMUTATIVE GRADED RINGS
, 2005
"... Abstract. 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 bee ..."
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Abstract. 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. 1.