Selected Title s i n Thi s Serie s

91 J . P . May et al. , Equivarian t homotop y an d cohomolog y theory , dedicate d t o th e memor y of Rober t J . Piacenza , 199 6

90 Joh n Roe, Inde x theory , coars e geometry , an d topolog y o f manifolds , 199 6

89 Cliffor d Henr y Taubes , Metrics , connection s an d gluin g theorems , 199 6

88 Crai g Huneke, Tigh t closur e an d it s applications , 199 6

87 Joh n Erik Fornaess, Dynamic s i n severa l comple x variables , 199 6

86 Sori n Popa, Classificatio n o f subfactor s an d thei r endomorphisms , 199 5

85 Michi o Jimbo and Tetsuji Miwa, Algebrai c analysi s o f solvabl e lattic e models , 199 4

84 Hug h L. Montgomery, Te n lecture s o n th e interfac e betwee n analyti c numbe r theor y an d

harmonic analysis , 199 4

83 Carlo s E. Kenig , Harmoni c analysi s technique s fo r secon d orde r ellipti c boundar y valu e

problems, 199 4

82 Susa n Montgomery, Hop f algebra s an d thei r action s o n rings , 199 3

81 Steve n G . Krantz, Geometri c analysi s an d functio n spaces , 199 3

80 Vaugha n F. R. Jones , Subfactor s an d knots , 199 1

79 Michae l Frazier , Bjorn Jawerth, an d Guido Weiss , Littlewood-Pale y theor y an d th e stud y

of functio n spaces , 199 1

78 Edwar d Formanek, Th e polynomia l identitie s an d variant s o f n x n matrices , 199 1

77 Michae l Christ , Lecture s o n singula r integra l operators , 199 0

76 Klau s Schmidt , Algebrai c idea s i n ergodi c theory , 199 0

75 F . Thoma s Farrel l and L. Edwin Jones, Classica l aspherica l manifolds , 199 0

74 Lawrenc e C . Evans, Wea k convergenc e method s fo r nonlinea r partia l differentia l

equations, 199 0

73 Walte r A . Strauss , Nonlinea r wav e equations , 198 9

72 Pete r Orlik , Introductio n t o arrangements , 198 9

71 Harr y Dym , J contractiv e matri x functions , reproducin g kerne l Hilber t space s an d

interpolation, 198 9

70 Richar d F . Gundy , Som e topic s i n probabilit y an d analysis , 198 9

69 Fran k D . Grosshans , Gian-Carl o Rota , an d Joe l A . Stein , Invarian t theor y an d

superalgebras, 198 7

68 J . Willia m Helton , Josep h A . Ball , Charle s R . Johnson , an d Joh n N . Palmer ,

Operator theory , analyti c functions , matrices , an d electrica l engineering , 198 7

67 Haral d Upmeier , Jorda n algebra s i n analysis , operato r theory , an d quantu m mechanics ,

1987

66 G . Andrews , ^-Series : Thei r developmen t an d applicatio n i n analysis , numbe r theory ,

combinatorics, physic s an d compute r algebra , 198 6

65 Pau l H . Rabinowitz , Minima x method s i n critica l poin t theor y wit h application s t o

differential equations , 198 6

64 Donal d S . Passman, Grou p rings , crosse d product s an d Galoi s theory , 198 6

63 Walte r Rudin , Ne w construction s o f function s holomorphi c i n th e uni t bal l o f C n, 198 6

62 Bel a Bollobas, Extrema l grap h theor y wit h emphasi s o n probabilisti c methods , 198 6

61 Mogen s Flensted-Jensen, Analysi s o n non-Riemannia n symmetri c spaces , 198 6

60 Gille s Pisier , Factorizatio n o f linea r operator s an d geometr y o f Banac h spaces , 198 6

59 Roge r Howe and Allen Moy, Harish-Chandr a homomorphism s fo r p-adi c groups , 198 5

58 H . Blaine Lawson, Jr., Th e theor y o f gaug e fields i n fou r dimensions , 198 5

57 Jerr y L. Kazdan, Prescribin g th e curvatur e o f a Riemannia n manifold , 198 5

56 Har i Bercovici, Cipria n Foia§, and Carl Pearcy, Dua l algebra s wit h application s t o invarian t subspaces an d dilatio n theory , 198 5

(Continued in the back of this publication)

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Conference Boar d o f the Mathematica l Science s

CBMS Regional Conference Serie s in Mathematic s

Number 2 0

Introduction to Some Methods of

Algebraic K-Theoiy

Hyman Bas s

Published fo r th e Conference Boar d of the Mathematica l Science s

by th e American Mathematica l Societ y

Providence, Rhod e Islan d with suppor t from th e

National Scienc e Foundatio n

http://dx.doi.org/10.1090/cbms/020

Expository Lecture s from th e CBM S Regiona l Conferenc e

held a t Colorad o Stat e Universit y August 24-28 , 197 3

2000 Mathematics Subject Classification. Primary 18F25 ; Secondary 20G35, 18F30.

Library o f Congres s Cataloging-in-Publieatio n D a t a

Bass, Hyman , 1932 -Introduction t o som e method s o f algebrai c if-theory . (Regional conferenc e serie s i n mathematics , no . 20 ) "Expository lecture s fro m th e CBM S regiona l conferenc e hel d a t Colorad o Stat e University ,

August 24-28 , 1973. " Includes bibliographica l references . 1. K-theory . I . Conferenc e Boar d o f th e Mathematica l Sciences . II . Title . III . Series .

QA1.R33 no . 2 0 [QA169 ] 512'.5 5 73-1992 5 ISBN 0-8218-1670- 5

CIP

Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s actin g for them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f th e sourc e i s given .

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© 197 4 b y th e America n Mathematica l Society . Al l right s reserved . The America n Mathematica l Societ y retain s al l right s

except thos e grante d t o th e Unite d State s Government . Printed i n th e Unite d State s o f America .

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Visit th e AM S hom e pag e a t URL : http://www.ams.org /

11 1 0 9 8 7 6 5 0 5 0 4 0 3 0 2 0 1 0 0

Contents

Chapter I: Introduction 1 1. Elementar y matrice s 4 2. Stabilit y 7 3. K X(A [t]) an d unipotents 1 2 4. Grothendiec k and Whitehead groups of categories with exact sequences 1 4 5. A Grothendieck style definition o f K X(A) 1 8 6. Homology , mapping cones, and Euler characteristics 2 0 7. Resolutio n theorems 2 3 8. Projectiv e resolutions and regular rings 2 7 9. Hilbert s syzygy theorem 3 2

Appendix 1 3 6

Chapter II: Localisation, Divissage and Applications 4 1

10. Th e localisation sequence 4 2 11. Categorie s of nilpotent endomorphisms 4 6

12. K x(A[t,rl\) 4 9 13. Devissag e 5 3 14. Som e calculations with the localisation sequence and K Y -Laurent theorem ... . 5 6

15. Computation s with the Mayer-Vietoris sequence 6 3

References 6 7

v

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References

[Ba 1 ] H . Bass, Algebraic K-theory, Benjamin , Ne w York, 1968 . M R 40 #2736 .

[Ba 2] , Some problems in "classical" algebraic K-theory, Proc . Battelle Con-

ference o n Algebrai c AT-Theory , Lecture Notes i n Math. , vol. 341, Springer-Verlag, Berlin

and Ne w York, 1973 .

[B + M ] H . Bass and M. P. Murthy, Grothendieck groups and Picard groups of abelian

group rings, Ann. o f Math . (2) 8 6 (1967), 16-73 . M R 36 #2671 .

[Be] H . Behr, Uber die endliche Definierbarkeit verallgemeinerter Einheitengruppen. II ,

Invent. Math . 4 (1967) , 265-274 . M R 40 #2682 .

[B + H-C ] A . Borel and Harish-Chandra , Arithmetic subgroups of algebraic groups,

Ann. of Math. (2) 7 5 (1962),485~535. M R 26 #5081 .

[C 4* F] L . Claborn an d R . Fossum, Generalisations of the notion of class group,

Illinois J . Math . 1 2 (1968), 228-253. M R 37 #200 .

[Gold] O . Goldman, Hilbert rings and Hilberfs Nullstellensatz, Math . Z. 5 4 (1951),

136-140. M R 13,427 .

[Ha] G . Harder, Minkowskische Reduktionstheorie uber Funktionenkorpern, Invent .

Math. 7 (1969), 33-54 . M R 44 #1667 .

[Hig] G . Higman, The units of group rings, Proc. London Math . Soc. (2) 4 6 (1940) , 231-248. M R 2,5 .

[Hor] G . Horrocks, Projective modules over an extension of a local ring, Proc. Londo n Math. Soc. (3) 1 4 (1964), 714-718 . M R 30 #121.

[Lan] S . Lang, Diophantine geometry, Interscienc e Tract s i n Pure an d Appl . Math. ,

no. 11 , Interscience, New York, 1962 . M R 26 #119 .

[Mat] H . Matsumura, Commutative algebra, Benjamin , Ne w York, 1970 . M R 42 #1813.

[Mil] J . Milnor , Introduction to algebraic K-theory, Ann . o f Math . Studies , no. 72 , Princeton, N . J., 1971 .

[Nag] H . Nagao, On GL(2, K[x]), J . Inst . Poly tech. Osaka Cit y Univ . Ser . A 1 0

(1959), 117-121 . M R 22 #5684.

[O'M] O . T. O'Meara , On the finite generation of linear groups over Hasse domains,

J. Rein e Angew. Math . 21 7 (1965), 79-108. M R 31 #3513.

67

68 HYMAN BAS S

[Roq] P . Roquette , Some fundamental theorems on abelian function fields, Proc .

Internat. Congres s Math. 1958 , Cambridge Univ . Press, New York, 1960 , pp. 322-329 .

MR 22 #9492 .

[Sel] A . Selberg , On discontinuous groups in higher-dimensional symmetric spaces,

Contributions t o Functio n Theor y (Internat . Colloq . Functio n Theory , Bombay , 1960) , Tata

Institute o f Fundamenta l Research , Bombay , 1960 , pp. 147-164 . M R 2 4 #A188 .

[Ser] J.-P . Serre , Arbres, amalgames, SL2, Asterisqu e 4 6 (1977) .

[Ses] C . S. Seshadri , Triviality of vector bundles over the affine space K 2, Proc . Nat. Acad. Sci . U.S.A. 4 4 (1958) , 456-458. M R 21 #1318 .

[Soub] J . P . Soublin, Anneaux et modules coherents, J . Algebr a 1 5 (1970), 455-472 . MR 41 #5422 .

[Va] L . N. VaserStem, On the stabilization of the general linear group over a ring,

Mat, Sb . 79 (121) (1969) , 405-424 = Math . USSR Sb . 8 (1969), 383-400 . M R 42 #1911 .

For a good introductio n t o th e curren t researc h literature i n algebrai c ^-theory th e

reader i s advised t o consul t th e conferenc e proceeding s i n the Springe r Lectur e Note s in Math-

ematics, volumes 341 , 342, 343 (1973) , 55 1 (1977) , and 85 4 (1980) .

Selected Title s i n Thi s Serie s (Continued from the front of this publication)

55 Willia m Arveson , Te n lecture s o n operato r algebras , 198 4

54 Willia m Fulton, Introductio n t o intersectio n theor y i n algebrai c geometry , 198 4

53 Wilhel m Klingenberg , Close d geodesie s o n Riemannia n manifolds , 198 3

52 Tsit-Yue n Lam , Orderings , valuation s an d quadrati c forms , 198 3

51 Masamich i Takesaki , Structur e o f factor s an d automorphis m groups , 198 3

50 Jame s Eells and Luc Lemaire, Selecte d topic s i n harmoni c maps , 198 3

49 Joh n M. Franks , Homolog y an d dynamica l systems , 198 2

48 W . Stephe n Wilson , Brown-Peterso n homology : a n introductio n an d sampler , 198 2

47 Jac k K. Hale , Topic s i n dynami c bifurcatio n theory , 198 1

46 Edwar d G . Effros , Dimension s an d C*-algebras , 198 1

45 Ronal d L. Graham, Rudiment s o f Ramse y theory , 198 1

44 Philli p A. Griffiths , A n introductio n t o th e theor y o f specia l divisor s o n algebrai c curves ,

1980

43 Willia m Jaco, Lecture s o n three-manifol d topology , 198 0

42 Jea n Dieudonne , Specia l function s an d linea r representation s o f Li e groups , 198 0

41 D . J. Newman , Approximatio n wit h rationa l functions , 197 9

40 Jea n Mawhin, Topologica l degre e method s i n nonlinea r boundar y valu e problems , 197 9

39 Georg e Lusztig , Representation s o f finite Chevalle y groups , 197 8

38 Charle s Conley , Isolate d invarian t set s an d th e Mors e index , 197 8

37 Masayosh i Nagata , Polynomia l ring s an d afnn e spaces , 197 8

36 Car l M. Pearcy , Som e recen t development s i n operato r theory , 197 8

35 R . Bowen , O n Axio m A difTeomorphisms , 197 8

34 L . Auslander, Lectur e note s o n nil-thet a functions , 197 7

33 G . Glauberman , Factorization s i n loca l subgroup s o f finit e groups , 197 7

32 W . M. Schmidt , Smal l fractiona l part s o f polynomials , 197 7

31 R . R . Coifma n an d G. Weiss , Transferenc e method s i n analysis , 197 7

30 A . Pelczyriski , Banac h space s o f analyti c function s an d absolutel y summin g operators ,

1977

29 A . Weinstein , Lecture s o n symplecti c manifolds , 197 7

28 T . A . Chapman , Lecture s o n Hilber t cub e manifolds , 197 6

27 H . Blaine Lawson, Jr. , Th e quantitativ e theor y o f foliations , 197 7

26 I . Reiner , Clas s group s an d Picar d group s o f grou p ring s an d orders , 197 6

25 K . W. Gruenberg , Relatio n module s o f finite groups , 197 6

24 M . Hochster , Topic s i n th e homologica l theor y o f module s ove r commutativ e rings , 197 5

23 M . E . Rudin, Lecture s o n se t theoreti c topology , 197 5

22 O . T. O'Meara , Lecture s o n linea r groups , 197 4

21 W . Stoll , Holomorphi c function s o f finite orde r i n severa l comple x variables , 197 4

20 H . Bass , Introductio n t o som e method s o f algebrai c K- theory, 197 4

19 B . Sz.-Nagy , Unitar y dilation s o f Hilber t spac e operator s an d relate d topics , 197 4

18 A . Friedman, Differentia l games , 197 4

17 L . Nirenberg, Lecture s o n linea r partia l differentia l equations , 197 3

16 J . L . Taylor, Measur e algebras , 197 3

15 R . G . Douglas , Banac h algebr a technique s i n th e theor y o f Toeplit z operators , 197 3

14 S . Helgason , Analysi s o n Li e group s an d homogeneou s spaces , 197 2

13 M . Rabin, Automat a o n infinit e object s an d Church' s problem , 197 2

(See th e AM S catalo g fo r earlie r titles )