Gröbner Bases and the Computation of Group Cohomology

140 Pages·2003·0.76 MB·English

by  David J. Green (auth.)

1828
Lecture Notes in Mathematics
Editors:
J.--M.Morel,Cachan
F.Takens,Groningen
B.Teissier,Paris
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David J. Green
Gro¨bner Bases and
the Computation of
Group Cohomology
1 3
Author
DavidJ.Green
FBC-MathematikundNaturwissenschaften
BergischeUniversita¨tWuppertal
Gaussstr.20
42097Wuppertal
Germany
e-mail:[email protected]
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MathematicsSubjectClassification(2000):20J06,16S15,16E05,16Z05,20C05,20D15
ISSN0075-8434
ISBN3-540-20339-7Springer-VerlagBerlinHeidelbergNewYork
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For Birgit, Thomas and Anne
Preface
The motivation for this book is the desire to perform complete calculations
of cohomology rings in the area called the cohomology of finite groups. The
theories presented here belong to that part of computational algebra known
as noncommutative Gr¨obner bases. It happens that existing Gro¨bner basis
methods perform particularly poorly in the conditions imposed by cohomol-
ogy calculations. So new types of Gro¨bner bases had to be developed, in-
formed by practical computability considerations.
Thanks to the work of J. F. Carlson, one can compute the cohomology
ring of a p-group from a sufficiently large initial segment of the minimal pro-
jective resolution. The first new Gro¨bner basis theory – for modules over the
modular group algebra – was developed to construct the minimal resolution
as efficiently as possible. In all probability it applies equally well to finite
dimensional basic algebras.
Carlson’smethodalsoneedstheabilitytomanipulatetherelationsinthe
cohomology ring. As such rings are graded commutative rather than strictly
commutative,itwasnecessarytodeviseatheoryofGr¨obnerbasesforgraded
commutativerings.Thereismorethanonewaytodothis.TheGro¨bnerbases
presented here were designed to resemble the classical commutative case as
closely as possible. Strictly speaking, they are Gro¨bner bases for right ideals
in a more general type of algebra which is here called a Θ-algebra.
Many cohomology computations have been performed using these meth-
ods. In particular, the essential conjecture of Mui and Marx was shown to
be false. The counterexample is the Sylow 2-subgroup of U (4), a group of
3
order 64.
Acknowledgements
I am very grateful to the following people.
Jon F. Carlson aroused my interest in the computer calculation of group
cohomology. All the big computations described here were performed on his
machine toui.
Klaus Lux and Peter Dra¨xler drew my attention to the potential of non-
commutative Gr¨obner basis methods, and in particular to the work of Ed
Green and his collaborators.
VIII Preface
Prof. G. Michler (who suggested computational cohomology to me as
long ago as 1991) allowed me the use of the computers at the Institute for
Experimental Mathematics in Essen during the long developmental phase.
ThisbookismyWuppertalHabilitationsschrift,withminorupdates.But
for the exertions of Erich Ossa and Bjo¨rn Schuster, the number of gram-
matical errors in the German original would have been considerably higher.
Wuppertal, August 2003 David J. Green
Table of Contents
Introduction.................................................. 1
Part I Constructing minimal resolutions
1 Bases for finite-dimensional algebras and modules ........ 13
1.1 Finite-dimensional algebras .............................. 14
1.2 Free right modules...................................... 16
1.3 Implementation ........................................ 17
1.4 The matrix of a general element.......................... 17
1.5 The Jennings ordering .................................. 18
2 The Buchberger Algorithm for modules .................. 21
2.1 The Diamond Lemma for modules........................ 22
2.1.1 Reduction systems................................ 22
2.1.2 Ambiguities and the Diamond Lemma .............. 23
2.2 The Buchberger Algorithm .............................. 26
2.3 Implementation ........................................ 29
2.3.1 The algorithm Incorp.............................. 29
2.3.2 The Buchberger Algorithm ........................ 31
3 Constructing minimal resolutions ......................... 33
3.1 The kernel of a homomorphism........................... 33
3.2 Minimal generating sets ................................. 35
3.2.1 Heady reduction systems .......................... 36
3.2.2 Algorithms for heady reduction systems ............. 37
3.3 Implementation ........................................ 39
3.3.1 The variants of the algorithm Incorp ................ 40
3.3.2 The variants of the Buchberger Algorithm ........... 42
3.4 Computing preimages................................... 45
X Table of Contents
Part II Cohomology ring structure
4 Gro¨bner bases for graded commutative algebras .......... 49
4.1 The structure of Θ-algebras.............................. 50
4.2 Gro¨bner bases for right ideals ............................ 55
4.2.1 Gro¨bner bases and the Division Algorithm........... 55
4.2.2 The Buchberger Algorithm ........................ 56
4.3 The kernel of an algebra homomorphism .................. 59
4.4 Intersections and Annihilators: Gro¨bner bases for modules ... 62
5 The visible ring structure................................. 67
5.1 Basics................................................. 67
5.2 Practical considerations ................................. 68
5.2.1 Lifts of cocycles .................................. 68
5.2.2 Gro¨bner bases and the visible ring structure ......... 69
5.3 Monomial ordering and generator choice................... 72
5.3.1 Nilpotent generators .............................. 72
5.3.2 Regular generators ............................... 74
5.3.3 In summary ..................................... 74
5.4 Calculating products in batches .......................... 75
5.5 Restriction to subgroups ................................ 79
6 The completeness of the presentation..................... 81
6.1 The Koszul complex .................................... 81
6.2 Carlson’s completeness criterion .......................... 83
6.3 Duflot regular sequences................................. 84
6.4 Groups of small rank: Koszul complex and Poincar´e series ... 86
6.5 Identifying subgroups ................................... 89
Part III Experimental results
7 Experimental results...................................... 93
7.1 Cohomology rings of small p-groups....................... 93
7.1.1 The groups of order 81............................ 95
7.1.2 The groups of order 625........................... 96
7.1.3 Two essential classes with nonzero product .......... 96
7.1.4 A 3-group with Cohen–Macaulay defect 2 ........... 98
7.2 Resolutions for larger p-groups ........................... 99
7.3 The period of a periodic module.......................... 100
Table of Contents XI
A Sample cohomology calculations .......................... 101
A.1 The cyclic group of order 2 .............................. 101
A.2 The cyclic group of order 4 .............................. 101
A.3 The Klein 4-group ...................................... 101
A.4 The dihedral group of order 8 ............................ 102
A.5 The quaternion group of order 8.......................... 103
A.6 The Sylow 2-subgroup of U (4)........................... 105
3
A.7 The Sylow 3-subgroup of A ............................. 123
9
A.8 Small Group No. 16 of order 243 ......................... 126
Epilogue...................................................... 131
References.................................................... 133
Index......................................................... 137