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Bibliography Bibliography [1] Emil Artin. Galois Theory. Dover, second edition, 1964. [2] Michael Artin. Algebra. Prentice Hall, first edition, 1991. [3] M. F. Atiyah and I. G. Macdonald. Introduction to Commutative Algebra. Addison Wesley, third edition, 1969. [4] Nicolas Bourbaki. Alg`ebre, Chapitres 1-3.El´ements de Math´ematiques. Hermann, 1970. [5] Nicolas Bourbaki. Alg`ebre, Chapitre 10.El´ements de Math´ematiques. Masson, 1980. [6] Nicolas Bourbaki. Alg`ebre, Chapitres 4-7.El´ements de Math´ematiques. Masson, 1981. [7] Nicolas Bourbaki. Alg`ebre Commutative, Chapitres 8-9.El´ements de Math´ematiques. Masson, 1983. [8] Nicolas Bourbaki. Elements of Mathematics. Commutative Algebra, Chapters 1-7. Springer–Verlag, 1989. [9] Henri Cartan and Samuel Eilenberg. Homological Algebra. Princeton Math. Series, No. 19. Princeton University Press, 1956. [10] Jean Dieudonn´e. Panorama des mat´ematiques pures. Le choix bourbachique. Gauthiers-Villars, second edition, 1979. [11] David S. Dummit and Richard M. Foote. Abstract Algebra. Wiley, second edition, 1999. [12] Albert Einstein. Zur Elektrodynamik bewegter K¨orper. Annalen der Physik, 17:891–921, 1905. [13] David Eisenbud. Commutative Algebra With A View Toward Algebraic Geometry. GTM No. 150. Springer–Verlag, first edition, 1995. [14] Jean-Pierre Escofier. Galois Theory. GTM No. 204. Springer Verlag, first edition, 2001. [15] Peter Freyd. Abelian Categories. An Introduction to the theory of functors. Harper and Row, first edition, 1964. [16] Sergei I. Gelfand and Yuri I. Manin. Homological Algebra. Springer, first edition, 1999. [17] Sergei I. Gelfand and Yuri I. Manin. Methods of Homological Algebra. Springer, second edition, 2003. [18] Roger Godement. Topologie Alg´ebrique et Th´eorie des Faisceaux. Hermann, first edition, 1958. Second Printing, 1998. [19] Daniel Gorenstein. Finite Groups. Harper and Row, first edition, 1968. [20] Alexander Grothendieck. Sur quelques points d’alg`ebre homologique. Tˆohoku Mathematical Journal, 9:119–221, 1957. 399 400 BIBLIOGRAPHY [21] Alexander Grothendieck and Jean Dieudonn´e. El´ements de G´eom´etrie Alg´ebrique, IV: Etude Locale des Sch´emas et des Morphismes de Sch´emas (Premi`ere Partie). Inst. Hautes Etudes Sci. Publ. Math., 20:1–259, 1964. [22] Marshall Hall. Theory of Groups. AMS Chelsea, second edition, 1976. [23] David Hilbert. Die Theorie der algebraischen Zahlk¨orper. Jahresberich der DMV, 4:175–546, 1897. [24] Peter J. Hilton and Urs Stammbach. A Course in Homological Algebra. GTM No. 4. Springer, second edition, 1996. [25] G. Hochschild. On the cohomology groups of an associative algebra. Ann. of Math. (2), 46:58–67, 1945. [26] G. Hochschild. On the cohomology theory for associative algebras. Ann. of Math. (2), 47:568–579, 1946. [27] Thomas W. Hungerford. Algebra. GTM No. 73. Springer Verlag, first edition, 1980. [28] Nathan Jacobson. Basic Algebra II. Freeman, first edition, 1980. [29] Nathan Jacobson. Basic Algebra I. Freeman, second edition, 1985. [30] Daniel M. Kan. Adjoint functors. Trans. Amer. Math. Soc., 87:294–329, 1958. [31] Irving Kaplansky. Fields and Rings. The University of Chicago Press, second edition, 1972. [32] Jean Pierre Lafon. Les Formalismes Fondamentaux de l’ Alg`ebre Commutative. Hermann, first edition, 1974. [33] Jean Pierre Lafon. Alg`ebre Commutative. Languages G´eom´etrique et Alg´ebrique. Hermann, first edition, 1977. [34] Serge Lang. Algebra. Addison Wesley, third edition, 1993. [35] Saunders Mac Lane. Categories for the Working Mathematician. GTM No. 5. Springer–Verlag, first edition, 1971. [36] Saunders Mac Lane. Homology. Grundlehren der Math. Wiss., Vol. 114. Springer–Verlag, third edition, 1975. [37] Saunders Mac Lane and Garrett Birkhoff. Algebra. Macmillan, first edition, 1967. [38] M.-P. Malliavin. Alg`ebre Commutative. Applications en G´eom´etrie et Th´eorie des Nombres. Masson, first edition, 1985. [39] H. Matsumura. Commutative Ring Theory. Cambridge University Press, first edition, 1989. [40] Barry Mitchell. The full embedding theorem. Amer. J. Math., 86:619–637, 1964. [41] Patrick Morandi. Field and Galois Theory. GTM No. 167. Springer Verlag, first edition, 1997. [42] John S. Rose. A Course on Group Theory. Cambridge University Press, first edition, 1978. [43] Joseph J. Rotman. The Theory of Groups, An Introduction. Allyn and Bacon, second edition, 1973. [44] Joseph J. Rotman. An Introduction to Homological Algebra. Academic Press, first edition, 1979. [45] Jean-Pierre Serre. Corps Locaux. Hermann, third edition, 1980. [46] Jean-Pierre Serre. Local Algebra. Springer Monographs in Mathematics. Springer, first edition, 2000. BIBLIOGRAPHY 401 [47] B.L. Van Der Waerden. Algebra, Vol. 1. Ungar, seventh edition, 1973. [48] Charles A. Weibel. Introduction to Homological Algebra. Studies in Advanced Mathematics, Vol. 38. Cambridge University Press, first edition, 1994. [49] Ernst Witt. Zyklisch K¨orper und Algebren der Charakteristik p vom Grade pm. J. Reine Angew. Math., 176:126–140, 1937. [50] Oscar Zariski and Pierre Samuel. Commutative Algebra, Vol I. GTM No. 28. Springer Verlag, first edition, 1975. [51] Oscar Zariski and Pierre Samuel. Commutative Algebra, Vol II. GTM No. 29. Springer Verlag, first edition, 1975. [52] Hans J. Zassenhaus. The Theory of Groups. Dover, second edition, 1958. Index M-regular sequence, 325 Artin Schreier I-Global Dimension Theorem, 334 Theorem I, 236, 237 (Co)homological Functors, 271 Theorem II, 236 (N), 344 Artin’s irreducibility criterion, 238–240 (co)homology of algebras, 275 Artin’s theorem, 205, 207, 209, 212, 215, 386 Artin’s theorem of the primitive element, 214 enveloping algebra, 276 Artin, E., 238 Artin–Rees theorem, 182, 183 a-adic completion of A, 371 artinian module, 60 a-adic topology, 371 ascending chain condition for module, 60 abelian category, 366 associated primes of module, 172 AC, 242 atoms, 397 ACC for module, see ascending chain condition for augmentation, 282 module augmentation ideal, 278, 282 acyclic complex, 254 augmentation module, 330 acyclic resolution of modules, 72 augmented ring, 330 additive functor, 69 k-automorphism of field extension, 206 additive group functor, 40 axiom of choice, 10, 129 additive Hilbert 90, 238 A-adic topology, 181 backward image functor, 119 adjoint functors, 283 Baer embedding theorem, 77, 79, 362 admissible monomial, 57 Godement’s proof, 77 admissible word, 49 Baer radical of group, 344 affine group, 28 Baer representation theorem, 75, 77 Akizuki’s structure theorem, 139, 184 Baer’s theorem, 345 false for noncomm. rings, 140 Baer, R., 79 Akizuki, Y., 138 bar resolution, 278 algebraic closure, 242 base extension of module to algebra, 113 algebraic closure of field, 191 Basic Existence Theorem, 242 algebraic element over commutative ring, 59 bounded complex, 254 algebraic geometry, 143 above, 254 algebraically closed, 242 below, 254 algebraically closed field, 191 Brauer, R., 394 algebraically dependent set over commutative ring, bundle homomorphism of vector bundles, 360 59 Burnside basis theorem, 16, 18, 344 algebraically independent set over commutative ring, Burnside dimension, 18 59 Burnside’s Lemma, 397 alternating R-algebra, 98 Burnside, W., 9, 32 amalgamated product of groups, 50 butterfly lemma, see Zassenhaus’ butterfly lemma annihilator of submodule, 60 arrow of category, see morphism of category Cameron-Cohen Inequality, 397 Artin ordering, 100 canonical flasque resolution, 295 402 INDEX 403 canonical map to localization, 118 cohomological spectral sequence, 300 canonical site, 374 cohomologous 2-cocycles on group, 23 Cartan–Eilenberg injective resolution of complex, 312 cohomology casus irreducibilis of cubic, 387 long exact sequence, 221 category, 39 of a cyclic group, 224 examples, 39 of presheaves, 288 category co-over object, 45 of sheaves, 288 category over object, 45 cohomology cofunctor, 41 Cauchy, A., 5 cohomology group of R with coefficient in M, 278 Cayley graph, 349 0-th cohomology group of group Cayley transform, 395 calculation, 30 Cayley–Hamilton theorem, 181 n-th cohomology group of group, 30 center of group, 13 cohomology of groups, 315 central group extension, 26 comaximal ideals, 136 centralizer of group element, 3 comma category, see category over or co-over object chain, 10 commutator group of group, 16 chain detour lemma, 185, 186 commutator of group elements, 13 chain map, 254 compatible filtration and coboundary map, 299 d-chain on manifold, 97 compatible filtration and grading, 299 character, 298 complementary index of spectral sequence, 300 character of representation, 204 completely reducible module, 329 characteristic class of extension of module by module, S-component of submodule, 179 273 composition factors of composition series, 10 characteristic of ring, 353 composition of morphisms between objects of cate- characteristic subgroup of group, 15 gory, 39 Chevalley, 158 composition series for group, 10 Chinese Remainder Theorem, 136, 139 compositum of two fields, 215 classical, 136 conductor, 165 classification of groups of order pq,29 of the integral closure, 165 closed set in Spec, 125 conjugacy class of group element, 3 n-th coboundary group of group, 30 H-conjugacy class of group element, 3 n-th coboundary map, group cohomology, 30 k-conjugate fields, 201 coboundary of 1-cochain on group, 23 conjugation group action, 3 n-th cochain group of group, 30 connecting homomorphism, 221 cochain map, 254 connecting homomorphism in LES for (co)homology, 1-cochain on group, 23 263 2-cochain on group, 22 connecting homomorphism in snake lemma, 81 n-th cocycle group of group, 30 constant presheaf with values in
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