Lattices over Orders II
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142 Klaus W. Roggenkamp McGill University, Montreal
Lattices over Orders II
Springer-Verlag Berlin· Heidelberg' NewYork 1970
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C Tide No. 329fl.
Berlin . Heidelberg 19'16. Library of Congress Cat:dog Card Number 71-108334. Printed in Germany.
PREFACE This volume is a continuation of Volume I and reference is made to·the statements in Volume I simply by number without quoting special theorems. I would like to express my grat1tude to Verena HUber-Dyson, who has read these notes carefully and who has made valuable 1mprovements. At this po1nt I have to ment10n my wife Chr1sta, who has pat1ent1y endured all my moods dur1ng the preparat10n of these notes, w1 th the equanim1ty that only a w1fe has, and who has typed most of Vol. I and all of Vol. II for me. There are more distingu1shed people who should have wr1tten these notes; however, "nullus est liber tam malus, ut non aliqua parte pro8it". (Plinius sen,)
CONTENT Chapter VII Modules over orders, one-sided ideals over maximal orders 1 Local equivalence
VI
1
2 Separable orders
VI
8
3 The Krull-Schmidt theorem
VI 14
4 The Jordan-zassenhaus theorem
VI 22
5 Irreducible lattices
VI 29
6 Infinite primes over
VI 38
7 A theorem of Eichler on algebras that are not totally definite quatern10n algebras
VI 44
8 Ideals and norms of ideals
VI 60
Chapter VII. Genera of lattices 1 Preliminaries on genera
VII
1
2 The number of non-isomorphic lattices in a genus
VII
8
3 Embedding theorems for modules in the same genus
VII 19
4 Genera of special types of lattices
VII 30
Chapter VIII. Grothend1eck groups 1 Grothend1eck groups and other groups associated with modules
VIII
1
2 The Wh1tehead group of a ring
VIII
7
3 Grothendieck groups of orders
VIII 20
4 Grothendleck groups and genera
VIII 46
5 Jacob1nskl's cancellation theorem
VIII 55
Chapter IX. Special types of orders 1 Clean orders
IX
2 Hereditary orders
IX 19
3 Grothend1eck rings of finite groups
IX 37
4 D1vis1b1l1ty of latt1ces
IX 65
5 Bass-orders
IX 77
6 Classification of Bass-orders
IX 87
1
Chapter XI The number of indecomposable lattices over orders 1 Orders with an infinite number of non-isomorphic 1ndecomposable latt1ces
X 1
v 2 Separation of the three different cases ..
h
3 The case A • D
4 The case A"
X 18 X 29
= D.. 1
... • D 2 A A 1\ A S The case A = D1 • D • D 2 J 6 Reduction of the proof of (2.1) to the decomposition of matrices
x46 X
51
7 Decomposition of the matrix
of 6,A,I
X
66
of 6,A,II
X 11
8 Decomposition of the matrix
!A
9 Decomposition of the matrix 6,A,III
X
35
