Lecture Notes in Mathematics 1828 Editors J M Morel, Cachan F Takens, Groningen B Teissier, Paris 3 Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo David J Green Gröbner Bases and the C[.]
Lecture Notes in Mathematics Editors: J. M Morel, Cachan F Takens, Groningen B Teissier, Paris 1828 Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo David J Green Grăobner Bases and the Computation of Group Cohomology 13 Author David J Green FB C - Mathematik und Naturwissenschaften Bergische Universităat Wuppertal Gaussstr 20 42097 Wuppertal Germany e-mail: green@math.uni-wuppertal.de Cataloging-in-Publication Data applied for Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de Mathematics Subject Classification (2000): 20J06, 16S15, 16E05, 16Z05, 20C05, 20D15 ISSN 0075-8434 ISBN 3-540-20339-7 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Violations are liable for prosecution under the German Copyright Law Springer-Verlag is a part of SpringerScience+Business Media springeronline.com c Springer-Verlag Berlin Heidelberg 2003 Printed in Germany The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typesetting: Camera-ready TEX output by the authors SPIN: 10964583 41/3142/du - 543210 - Printed on acid-free paper For Birgit, Thomas and Anne Index algorithm – Buchberger, 27, 31 – ElimBuchberger, 35 – ElimIncorp, 34, 40 – Expand, 27 – HeadyBuchberger, 39 – HeadyExpand, 38 – HeadyIncorp, 37, 41 – Incorp, 26, 30 – KernelInterwoven, 43 – Lift, 46 associated polynomial algebra, 51 elimination dimension, 60 essential class, 73 essential conjecture, 1, 98 essential generator, 73 essential ideal, 94 Gră obner basis, – for a kG-module, 25 – for a right ideal in a Θ-algebra, 55 – minimal, 25, 55 – preimage, 34 Hall–Senior number, 98 basic algebra, Buchberger algorithm, 3, 27, 58, 65 Buchberger criterion, 57, 64 Carlson’s Koszul condition, 83 Cohen–Macaulay defect, 98, 132 Cohen–Macaulay ring, 96 completely reduced form, 56 completeness criterion – Benson’s, 132 – Carlson’s, 2, 83–84 Condition G, 83 Condition R, 83 critical pair, 50 depth, 84 depth-essential conjecture, 95–96 Diag package, 8, 93 Diamond lemma – Bergman’s, – two-speed, 4, 24 Duflot regular element, 74 Duflot regular sequence, 85 – complete, 85 Duflot’s theorem, 84–85, 132 inclusion ambiguity, 23 – inadmissible, 37 – resolvable, 23 – resolvable relative to ≤, 23 irreducible, 22 Jennings ordering, 19 Jennings PC-generators, 18 Jennings presentation, 19 Koszul complex, 81–82 Koszul condition, see Carlson’s Koszul condition minimal resolution – key task for construction of, 4, 33 monomial ideal, 52 normal form, ordering – elimination, 33, 60 – graded lexicographical, 52 – Jennings, 19 – negative, 138 Index – reverse length-lexicographic, 14 – useable, 14, 16, 51 system of parameters – method for choosing one, 85, 132 p-rank, Poincar´e series, 87 T -reduction, 22 – elementary, 22 – – heady, 36 – – radical, 36 – radical, 36 T -reduction site, 22 Θ-algebra, 49, 51 toppling, 23 – radically resolvable, 36 – radically resolvable relative to ≤, 36 – resolvable, 23 – resolvable relative to ≤, 23 Type step, 58, 65 Type step, 58, 65 r-dimension, 74 reduced form, 55 reduction-finite, 3, 22 reduction system, 22 – heady, 36 reduction-unique, 22 regular element, 84 regular sequence, 84 regularity conjecture, 132 S-polynomial – in a module over a Θ-algebra, 64 – in a Θ-algebra, 56 – of inclusion ambiguity, 23 – of toppling, 23 Small Groups library, 89 weakly reducible, 56 weakly resolvable, 57 y-degree, 73 ... of SpringerScience+Business Media springeronline.com c Springer-Verlag Berlin Heidelberg 2003 Printed in Germany The use of general descriptive names, registered names, trademarks, etc in this... theorem, 84–85, 132 inclusion ambiguity, 23 – inadmissible, 37 – resolvable, 23 – resolvable relative to ≤, 23 irreducible, 22 Jennings ordering, 19 Jennings PC-generators, 18 Jennings presentation,... available in the Internet at http://dnb.ddb.de Mathematics Subject Classification (2000): 20J06, 16S15, 16E05, 16Z05, 20C05, 20D15 ISSN 0075-8434 ISBN 3-540-20339-7 Springer-Verlag Berlin Heidelberg