Lecture notes in mathematics

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 1828 Editors:

J. M Morel, Cachan

F Takens, Groningen

B Teissier, Paris

Berlin Heidelberg New York Hong Kong London Milan Paris

Tokyo

David J Green

Gr¨obner Bases and the Computation of Group Cohomology

1 3

David J Green

FB C - Mathematik und Naturwissenschaften

Bergische Universit¨at Wuppertal

Gaussstr.20

42097 Wuppertal

Germany

e-mail: green@math.uni-wuppertal.de

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Mathematics Subject Classification (2000):20J06, 16S15, 16E05, 16Z05, 20C05, 20D15 ISSN0075-8434

ISBN3-540-20339-7 Springer-Verlag Berlin Heidelberg New York

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For Birgit, Thomas and Anne

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

basic algebra, 4

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, 3

– two-speed, 4, 24

Duflot regular element, 74

Duflot regular sequence, 85

– complete, 85

Duflot’s theorem, 84–85, 132

elimination dimension, 60 essential class, 73 essential conjecture, 1, 98 essential generator, 73 essential ideal, 94 Gr¨obner basis, 3

– for a kG-module, 25 – for a right ideal in a Θ-algebra, 55

– minimal, 25, 55 – preimage, 34 Hall–Senior number, 98 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, 3 ordering – elimination, 33, 60 – graded lexicographical, 52 – Jennings, 19

– negative, 4

138 Index

– reverse length-lexicographic, 14

– useable, 14, 16, 51

p-rank, 5

Poincar´e series, 87

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

system of parameters – method for choosing one, 85, 132

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 1 step, 58, 65 Type 2 step, 58, 65 weakly reducible, 56 weakly resolvable, 57

y-degree, 73