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Mathematical Proceedings of the Cambridge Philosophical Society
Article . 1998 . Peer-reviewed
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On the module of effective relations of a standard algebra

Authors: Planas Vilanova, Francesc d'Assís;

On the module of effective relations of a standard algebra

Abstract

Let \(A\) be a commutative ring. The author denotes by a standard \(A\)-algebra a commutative graded \(A\)-algebra \(U=\bigoplus_{n\geq 0}U_n\) with \(U_0= A\) and such that \(U\) is generated as an \(A\)-algebra by the elements of \(U_1\). Let \(\underline x\) be a (possibly infinite) set of generators of the \(A\)-module \(U_1\). Let \(V=A[\underline t]\) be the polynomial ring with as many variables \(\underline t\) (of degree one) as \(\underline x\) has elements and let \(f:V\to U\) be the graded free presentation of \(U\) induced by the \(\underline x\). For \(n\geq 2\), the \(A\)-module \(E(U)_n=\ker f_n/V_1\cdot\ker f_{n-1}\) is called the module of effective \(n\)-relations. In this paper the author gives two descriptions of the \(A\)-module of effective \(n\)-relations. In terms of André-Quillen homology it is \(E(U)_n=H_1(A,U,A_n)\). It turns out that this module does not depend on the chosen \(\underline x\). In terms of Koszul homology the author proves that \(E(U)_n=H_1(\underline x;U)_n\). Using this characterizations, there are established some properties on the module of effective \(n\)-relations and the relation type of a graded algebra. Finally, the author characterizes, in terms of a system of generators, which ideals have module of effective \(n\)-relations zero.

Keywords

Koszul homology, Teoria d', polynomial ring, Homologia, Commutative rings, André-Quillen homology, Homologia, Teoria d', Anells commutatius, module of effective \(n\)-relations, Graded rings, Standard Algebra, (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.), Polynomial rings and ideals; rings of integer-valued polynomials, Classificació AMS::13 Commutative rings and algebras::13D Homological methods, Algebra, Effective Relations, graded algebra, :13 Commutative rings and algebras::13A General commutative ring theory [Classificació AMS], Homological, Classificació AMS::13 Commutative rings and algebras::13A General commutative ring theory, :13 Commutative rings and algebras::13D Homological methods [Classificació AMS], Algebra, Homological

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selected citations
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This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
14
Top 10%
Top 10%
Average
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