C.: I.M. HOMOLOGY (Google eBook)

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Springer Science & Business Media, Feb 15, 1995 - Mathematics - 422 pages
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In presenting this treatment of homological algebra, it is a pleasure to acknowledge the help and encouragement which I have had from all sides. Homological algebra arose from many sources in algebra and topology. Decisive examples came from the study of group extensions and their factor sets, a subject I learned in joint work with OTTO SCHIL LING. A further development of homological ideas, with a view to their topological applications, came in my long collaboration with SAMUEL ElLENBERG; to both collaborators, especial thanks. For many years the Air Force Office of Scientific Research supported my research projects on various subjects now summarized here; it is a pleasure to acknowledge their lively understanding of basic science. Both REINHOLD BAER and JOSEF SCHMID read and commented on my entire manuscript; their advice has led to many improvements. ANDERS KOCK and JACQUES RIGUET have read the entire galley proof and caught many slips and obscurities. Among the others whose sug gestions have served me well, I note FRANK ADAMS, LOUIS AUSLANDER, WILFRED COCKCROFT, ALBRECHT DOLD, GEOFFREY HORROCKS, FRIED RICH KASCH, JOHANN LEICHT, ARUNAS LIULEVICIUS, JOHN MOORE, DIE TER PUPPE, JOSEPH YAO, and a number of my current students at the University of Chicago - not to m~ntion the auditors of my lectures at Chicago, Heidelberg, Bonn, Frankfurt, and Aarhus. My wife, DOROTHY, has cheerfully typed more versions of more chapters than she would like to count. Messrs.
  

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Contents

II
8
III
9
IV
13
V
15
VI
19
VII
21
VIII
25
IX
28
LXVI
206
LXVII
210
LXVIII
211
LXIX
215
LXX
218
LXXI
220
LXXII
224
LXXIII
227

X
34
XI
35
XII
39
XIII
42
XIV
44
XV
49
XVI
51
XVII
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XVIII
57
XIX
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XX
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XXII
67
XXIII
72
XXIV
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XXV
82
XXVI
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XXVII
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XXVIII
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XXIX
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XXX
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XXXI
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XXXII
103
XXXIII
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XXXIV
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XXXV
108
XXXVI
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XXXVII
114
XXXVIII
120
XXXIX
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XL
124
XLI
129
XLII
131
XLIII
134
XLIV
138
XLV
141
XLVI
142
XLVII
146
XLVIII
148
XLIX
150
L
154
LI
159
LII
163
LIII
166
LIV
170
LV
173
LVI
175
LVII
177
LVIII
181
LIX
184
LX
187
LXI
189
LXII
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LXIII
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LXIV
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LXV
204
LXXIV
228
LXXV
233
LXXVI
236
LXXVII
238
LXXVIII
244
LXXIX
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LXXX
249
LXXXI
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LXXXII
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LXXXIII
260
LXXXIV
262
LXXXV
265
LXXXVI
270
LXXXVII
273
LXXXVIII
278
LXXXIX
280
XC
283
XCI
288
XCII
290
XCIII
293
XCIV
295
XCV
298
XCVI
301
XCVII
303
XCVIII
308
XCIX
311
C
315
CI
318
CIII
322
CIV
326
CV
332
CVI
336
CVII
340
CVIII
342
CIX
345
CX
347
CXI
351
CXII
355
CXIII
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CXIV
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CXV
361
CXVI
364
CXVII
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CXVIII
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CXIX
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CXX
379
CXXI
386
CXXII
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CXXIII
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CXXIV
397
CXXV
400
CXXVI
404
CXXVII
413
CXXVIII
415
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Page 22 - The transpose of the product of two matrices is the product of their transposes in reverse order, ie, (AB) T = BTAT The concept of matrix multiplication assists in the solution of simultaneous linear equations.
Page 21 - Since a module is projective if and only if it is a direct summand of a free module, we have: Corollary 1.6.5.
Page 2 - jj which is exact in the sense that the kernel of each homomorphism is exactly the image of the preceding. Let E...
Page 408 - Relations between the fundamental group and the second Betti group. Lectures in Topology. Ann Arbor, pp.
Page 408 - Infinite abelian groups and homological methods', Ann. of Math. 69 (1959), 366-391.

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About the author (1995)

Biography of Saunders Mac Lane

Saunders Mac Lane was born on August 4, 1909 in Connecticut. He studied at Yale University and then at the University of Chicago and at Göttingen, where he received the D.Phil. in 1934. He has tought at Harvard, Cornell and the University of Chicago.

Mac Lane's initial research was in logic and in algebraic number theory (valuation theory). With Samuel Eilenberg he published fifteen papers on algebraic topology. A number of them involved the initial steps in the cohomology of groups and in other aspects of homological algebra - as well as the discovery of category theory. His famous and undergraduate textbook Survey of modern algebra, written jointly with G. Birkhoff, has remained in print for over 50 years. Mac Lane is also the author of several other highly successful books.