On the module of effective relations of a standard algebra
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| Title: | On the module of effective relations of a standard algebra |
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| Authors: | Planas Vilanova, Francesc d'Assís |
| Contributors: | Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya. EGSA - Equacions Diferencials, Geometria, Sistemes Dinàmics i de Control, i Aplicacions |
| Source: | Recercat. Dipósit de la Recerca de Catalunya instname UPCommons. Portal del coneixement obert de la UPC Universitat Politècnica de Catalunya (UPC) |
| Publisher Information: | Cambridge University Press (CUP), 1998. |
| Publication Year: | 1998 |
| Subject Terms: | Koszul homology, Teoria d', polynomial ring, Homologia, Commutative rings, André-Quillen homology, Homologia, Teoria d', Anells commutatius, module of effective \(n\)-relations, 01 natural sciences, 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, 0101 mathematics, 13 Commutative rings and algebras::13D Homological methods [Classificació AMS], Algebra, Homological |
| Description: | 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. |
| Document Type: | Article Other literature type |
| File Description: | application/pdf; application/xml |
| Language: | English |
| ISSN: | 1469-8064 0305-0041 |
| DOI: | 10.1017/s030500419800262x |
| Access URL: | https://upcommons.upc.edu/bitstream/2117/908/1/9604planas.pdf http://hdl.handle.net/2117/908 https://hdl.handle.net/2117/908 https://zbmath.org/1207977 https://doi.org/10.1017/s030500419800262x https://ui.adsabs.harvard.edu/abs/1998MPCPS.124..215P/abstract https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/on-the-module-of-effective-relations-of-a-standard-algebra/4B2F385DA262897B6B32AAEE59212D07 |
| Rights: | Cambridge Core User Agreement CC BY NC ND |
| Accession Number: | edsair.doi.dedup.....3fd0945e7c2c49be80a2b59a552e21bf |
| Database: | OpenAIRE |
Nájsť tento článok vo Web of Science | 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. |
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| ISSN: | 14698064 03050041 |
| DOI: | 10.1017/s030500419800262x |