Abstract. Quadratic algebras associated to pseudo-roots of noncommutative polynomials have been introduced by I. Gelfand, Retakh, and Wilson in connection with studying the decompositions of noncommutative polynomials. Later they (with S. Gelfand and Serconek) shown that the Hilbert series of these algebras and their quadratic duals satisfy the necessary condition for Koszulity. It is proved in this note that these algebras are Koszul. 1.

. It is a well-known fact that if K is a eld, then the Hilbert series of a quotient of the polynomial ring K[x1 ; : : : ; xn ] by a homogeneous ideal is of the form q(t) (1 t) n ; we call the polynomial q(t) the Hilbert numerator of the quotient algebra. We will generalize this concept to a class of non-nitely generated, graded, commutative algebras, which are endowed with a surjective \co-ltration" of nitely generated algebras. Then, although the Hilbert series themselves can not be dened (since the subvector -spaces involved have innite dimension), we get a sequence of Hilbert numerators qn (t), which we show converge to a power series in Z[[t]]. This power series we call the (generalized) Hilbert numerator of the non-nitely generated algebra. The question of determining when this power series is in fact a polynomial is the topic of the last part of this article. We show that quotients of the ring R 0 by homogeneous ideals that are generated by nitely many monomials hav...

Let F be an arbitrary field and GF = Gal(F /F) be the Galois group of its separable algebraic closure F over it. Two conjectures about the homological properties of the group GF are widely known. First of them, the Bloch–Kato conjecture, states that for a prime number l for which F contains the l-roots of unity the cohomology algebra

INTRODUCTION 1 0.1 Introduction De occulto orbis terrarum situ interrogasti, et si tanta monstrorum essent genera credenda 1 : to the rhetorical question which open Adhelm's Liber monstrorum de diversis generibus [A] we are trying here to give a positive answer by studying Grobner Fan and Universal Bases in the non-commutative case. Our starting point was the question whether the well-known result that the Grobner Fan is finite in the commutative case, was generalizable to the noncommutative case. The answer, of course, is negative and is presented in x 0.4. Building the example presented here, we were attracted by other fascinating examples. This suggested us the idea of preparing this teratological Kunstkammer where we are exhibiting: ffl an example of a principal ideal whose reduced Grobner basis is infinite (x 0.3) -- the example discussed in x<F12.3

We prove some simple properties of the power series ring R = K[[x 1 ; x 2 ; x 3 ; : : : ]] on a countably infinite number of variables over a field K, and of the subring R 0 generated by all homogeneous elements in R. By means of a certain decreasing filtration of ideals, which are kernels of the "truncation homomorphisms" ae n : R 0 ! K[x 1 ; : : : ; x n ], we endow R 0 with a topology, and show that with respect to this topology, homogeneous, finitely generated ideals are closed. We also show that the truncation homomorphisms "commute in a filtered sense" with the formation of greatest common divisors (and least common multiples): for any homogeneous f; g 2 R 0 , there exists an N such that for n ? N , gcd(ae n (f); ae n (g)) = ae n (gcd(f; g)). Key words: Grobner bases, initial ideals, Hilbert series, unique factorisation domain, greatest common divisor, least common multiple, distributive lattices, modular law, separated filtrations, topological rings, closed ideals 1 Intr...

Dedicated to Jean-Louis Loday, on the occasion of his sixtieth birthday 1 Abstract. In this article we give a conceptual definition of Manin products in any category endowed with two coherent monoidal products. This construction can be applied to associative algebras, non-symmetric operads, operads, colored operads, and properads presented by generators and relations. These two products, called black and white, are dual to each other under Koszul duality functor. We study their properties and compute several examples of black and white products for operads. These products allow us to define natural operations on the chain complex defining cohomology theories. With these operations, we are able to prove that Deligne’s conjecture holds for a general class of operads and is not specific to the case of associative algebras. Finally, we prove generalized versions of a few conjectures raised by M. Aguiar and J.-L. Loday related to the Koszul property of operads defined by black products. These operads provide infinitely many examples for this generalized Deligne’s conjecture.

Graded traces and irreducible representations of Aut(A(Γ)) acting on graded A(Γ) and A(Γ)!

1.1. Let F be an arbitrary field and GF = Gal(F /F) be the Galois group of its (separable) algebraic closure F over it. Two conjectures about the homological properties of the group GF are widely known. First of them, the Milnor–Kato conjecture, claims that for any prime number l ̸ = char F the algebra of Galois cohomology with

Let R =⊕i≥0Ri be a quadratic standard graded K-algebra. Backelin has shown that R is Koszul provided dim R2 ≤ 2. One may wonder whether, under the same assumption, R is defined by a Gröbner basis of quadrics. In other words, one may ask whether an ideal I in a polynomial ring S generated by a space of quadrics of codimension ≤2 always has a Gröbner basis of quadrics. We will prove that this is indeed the case with, essentially, one exception given by the ideal I =�x2�xy�y2 − xz � yz � ⊂K�x � y � z�. We show also that if R is a generic quadratic algebra with dim R2 < dim R1 then R is defined by a Gröbner basis of quadrics. © 2000 Academic Press 1.

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