
handle: 2117/908
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.
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
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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