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E-grāmata: Metacyclic Groups And The D(2) Problem

(Univ College London, Uk)
  • Formāts: 372 pages
  • Izdošanas datums: 04-Jan-2021
  • Izdevniecība: World Scientific Publishing Co Pte Ltd
  • Valoda: eng
  • ISBN-13: 9789811222771
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Metacyclic Groups And The D(2) ProblemPalielināt
  • Formāts: 372 pages
  • Izdošanas datums: 04-Jan-2021
  • Izdevniecība: World Scientific Publishing Co Pte Ltd
  • Valoda: eng
  • ISBN-13: 9789811222771
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"The D(2) problem is a fundamental problem in low dimensional topology. In broad terms, it asks when a three-dimensional space can be continuously deformed into a two-dimensional space without changing the essential algebraic properties of the spaces involved. The problem is parametrized by the fundamental group of the spaces involved; that is, each group G has its own D(2) problem whose difficulty varies considerably with the individual nature of G. This book solves the D(2) problem for a large, possibly infinite, number of finite metacyclic groups G(p, q). Prior to this the author had solved the D(2) problem for the groups G(p, 2). However, for q > 2, the only previously known solutions were for the groups G(7, 3), G(5, 4) and G(7, 6), all done by difficult direct calculation by two of the author's students, Jonathan Remez (2011) and Jason Vittis (2019). The method employed is heavily algebraic and involves precise analysis of the integral representation theory of G(p, q). Some noteworthy features are a new cancellation theory of modules (Chapters 10 and 11) and a simplified treatment (Chapters 5 and 12) of the author's theory of Swan homomorphisms"--
Preface v
Introduction 1(16)
Chapter One Projective modules and class groups
17(26)
§1 Exact sequences and projective modules
17(4)
§2 The Grothendieck group of a ring
21(2)
§3 The reduced Grothendieck group
23(1)
§4 Projective modules over R{G)
24(4)
§5 Steinitz' Theorem
28(2)
§6 Artinian rings are weakly Euclidean
30(2)
§7 Milnor's fibre square description of projective modules
32(5)
§8 The Milnor exact sequence
37(2)
§9 Projective modules over Z[ CP]
39(4)
Chapter Two Homological algebra
43(26)
§10 Cochain complexes and cohomology
43(3)
§11 The cohomology theory of modules
46(2)
§12 The exact sequences in cohomology
48(5)
§13 Module extensions
53(3)
§14 The group structure on Ext1
56(2)
§15 The cohomological interpretation of Ext1
58(7)
§16 The exact sequences of Ext1
65(1)
§17 Example, the cyclic group of order m
66(3)
Chapter Three The derived module category
69(24)
§18 The derived module category
70(5)
§19 Stable equivalence and projective equivalence
75(2)
§20 Syzygies and generalized syzygies
77(3)
§21 The corepresentation theorem for Ext1
80(5)
§22 De-stabilization
85(3)
§23 The dual to Schanuel's Lemma
88(1)
§24 Endomorphism rings
89(4)
Chapter Four Extension and restriction of scalars
93(26)
§25 Extension and restriction of scalars
93(3)
§26 Transversals and cocycles
96(2)
§27 Wreath products and group extensions
98(2)
§28 Induced representations
100(1)
§29 Lattices and representations
101(2)
§30 Induced modules
103(4)
§31 Duality
107(2)
§32 The Eckmann-Shapiro Theorem
109(2)
§33 Syzygies and lattices
111(2)
§34 The Eckmann-Shapiro relations in cohomology
113(2)
§35 Frobenius reciprocity
115(4)
Chapter Five Swan homomorphisms
119(36)
§36 Swan's projectivity criterion
119(3)
§37 Tame modules
122(4)
§38 The Swan homomorphism
126(4)
§39 Invariance properties of the Swan homomorphism
130(1)
§40 The relation between consecutive Swan homomorphisms
131(4)
§41 The dual Swan mapping
135(5)
§42 The stability group of a lattice
140(5)
§43 A criterion for monogenicity
145(2)
§44 Swan homomorphisms in their original context
147(1)
§45 The Swan homomorphism for cyclic groups
148(3)
§46 Cancellation over Z[ Cm]
151(4)
Chapter Six Modules over quasi-triangular algebras
155(18)
§47 Modules over Tq (F)
155(2)
§48 Stable classification of projective modules over Tq(A, I)
157(3)
§49 A lifting theorem
160(4)
§50 Modules over Mq(A)
164(1)
§51 Classification of projective modules over Tq(A, π)
165(3)
§52 Properties of the row modules R(i)
168(2)
§53 Duality properties of the modules R(i)
170(3)
Chapter Seven A fibre product decomposition
173(32)
§54 A fibre product decomposition
173(2)
§55 The discriminant of an associative algebra
175(2)
§56 The discriminant of a cyclic algebra
177(3)
§57 The discriminant of a quasi-triangular algebra
180(2)
§58 A quasi-triangular representation of G(p, q)
182(4)
§59 The isomorphism Cq(I*, θ) ~= Tq(A, π)
186(2)
§60 A worked example
188(4)
§61 Lifting units to cyclotomic rings
192(2)
§62 Liftable subgroups of {F*p ×...&time; F*p}q
194(8)
§63 p-adic analogues
202(3)
Chapter Eight Galois modules
205(16)
§64 Galois modules
205(4)
§65 The Galois action of y
209(2)
§66 R(1) and R(q) as Galois modules
211(1)
§67 The theorem of Auslander-Rim
212(3)
§68 Galois module description of the row modules
215(6)
Chapter Nine The sequencing theorem
221(18)
§69 Basic identities
221(3)
§70 Cohomological calculations
224(5)
§71 Decomposing the augmentation ideal of A
229(1)
§72 The basic sequence
230(2)
§73 The derived sequences
232(2)
§74 Computing the sequencing permutation
234(3)
§75 An element of D2fe+i(Z)
237(2)
Chapter Ten A cancellation theorem for extensions
239(14)
§76 Modules
239(6)
§77 A strong cancellation semigroup
245(3)
§78 A-D-modules for metacyclic groups
248(1)
§79 A cancellation theorem for extensions
249(4)
Chapter Eleven Cancellation of quasi-Swan modules
253(24)
§80 The Endper(K)-module structure on Ext1 (Q,K)
253(3)
§81 Nondegenerate modules of type R-Z[ C,q]
256(4)
§82 Nondegenerate modules of type R-IQ
260(1)
§83 A reduction theorem
261(5)
§84 An isomorphism theorem
266(3)
§85 Rearranging the extension parameters
269(3)
§86 A stability theorem
272(2)
§87 Straightness of degenerate modules
274(3)
Chapter Twelve Swan homomorphisms for metacyclic groups
277(22)
§88 Tq(A,π) is full
278(3)
§89 A model for STq
281(1)
§90 The identity Im(SR(K)) = Im(Stq)
282(3)
§91 A simplification
285(1)
§92 The first modification
286(2)
§93 The second modification
288(2)
§94 Fullness of R{k)
290(1)
§95 Changing rings from Z[ Cq] to Z[ G(p,q)]
291(2)
§96 Fullness of [ y - 1)
293(2)
§97 Fullness of R(k) + [ y - 1)
295(4)
Chapter Thirteen An obstruction to monogenicity
299(16)
§98 Obstructions to strong monogenicity
299(2)
§99 The unit group of Z[ y]/(y6 - 1)
301(2)
§100 The unit group of F2[ C6]
303(1)
§101 Projective modules over Z[ C12]
304(3)
§102 R(l) is not strongly monogenic over Z[ G(13,12)]
307(8)
Chapter Fourteen The D(2) property
315(12)
§103 Proof of Theorem IV
315(1)
§104 Proof of Theorem V
316(5)
§105 Proof of Theorems VI, VII and VIII
321(1)
§106 The hypotheses K0(Z[ Cq]) = 0 and Inj(p,q)
322(5)
Appendix A Examples 327(10)
Appendix B Class field theory and condition Inj(p, q) 337(6)
Appendix C A sufficient condition for the D(2) property 343(8)
References 351(4)
Index 355
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
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