Abstract
The first section of this chapter describes how homological algebra arose by abstraction from algebraic topology and how it has contributed to the knowledge of topology. The other four sections describe applications of the methods and results of homological algebra to other parts of algebra. Most of the material presented in these four sections was not available when this text was first published. Since then homological algebra has indeed found a large number of applications in many different fields, ranging from finite and infinite group theory to representation theory, number theory, algebraic topology, and sheaf theory. Today it is a truly indispensable tool in all these fields. For the purpose of illustrating to the reader the range and depth of these developments, we have selected a number of different topics and describe some of the main applications and results. Naturally, the treatments are somewhat cursory, the intention being to give the flavor of the homological methods rather than the details of the arguments and results.
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Literature
A. Dold: Lectures in Algebraic Topology. New York: Springer-Verlag, 1970.
Literature
P. Hilton: Localization and cohomology of nilpotent groups. Math. Z. 132, 263–286 (1973).
P. Hilton, J. Roitberg: Generalized C-theory and torsion phenomena in nilpotent spaces. Houston J. Math. 2, 525–559 (1976).
J.-P. Serre: Groupes d’homotopie et classes de groupes abéliens. Ann. of Math. 58, 258–294 (1953).
Literature (Books)
R. Bieri: Homological Dimension of Discrete Groups. Queen Mary College Mathematics Notes, 1976.0
K. Brown: Cohomology of Groups. Graduate Texts in Mathematics. New York: Springer-Verlag, 1982.
J.-P. Serre: Cohomologie des Groupes Discrètes. Annals of Mathematical Studies, vol. 70. Princeton, NJ: University Press, 1971.
Literature (Papers)
R.C. Alperin, P.B. Shalen: Linear groups of finite cohomological dimension. Invent. Math. 66, 89–98 (1982).
R. Bieri: Gruppen mit Poincaré-Dualität. Comment. Math. Helv. 47, 373–396 (1972).
M. Bestvina, N. Brady: Morse theory and fmiteness properties of groups. To appear.
R. Bieri, B. Eckmann: Groups with homological duality generalizing Poincaré duality. Invent. Math. 20, 103–124 (1973).
A. Borel, J.-P. Serre: Corners and arithmetic groups. Comment. Math. Helv. 48, 436–491 (1973).
K. Brown, R. Geoghegan: An infinite-dimensional torsion-free FP∞ group. Invent. Math. 77, 367–381 (1984).
B. Eckmann: Poincaré duality groups of dimension 2 are surface groups. In: Combinatorial Group Theory and Topology, Annals of Math. Studies, Princeton, NJ: University Press, 1986.
F.E.A. Johnson, C.T.C. Wall: On groups satisfying Poincaré duality. Ann. of Math. 96, 592–598 (1972).
P.H. Kropholler: Cohomological fmiteness conditions. In: C.M. Campbell et al.: Groups’ 93, Galway/St Andrews, vol. 1. London Math. Soc. Lecture Note Series. Cambridge: Cambridge University Press, 1995.
J.R. Stallings: A finitely presented group whose 3-dimensional integral homology group is not finitely generated. Amer. J. Math. 85, 541–543 (1963).
U. Stammbach: On the weak (homological) dimension of the group algebra of solvable groups. J. London Math. Soc. (2) 2, 567–570 (1970).
Literature (Books)
J.L. Alperin: Local Representation Theory. Cambridge: Cambridge University Press, 1986.
D. Benson: Representation Theory and Cohomology I, II. Cambridge: Cambridge University Press, 1991.
J. Carlson: Modules and Group Algebras. Basel: Birkhäuser Verlag, 1996.
L. Evens: The Cohomology of Groups. Oxford, UK: Clarendon Press, 1991.
Literature (Papers)
J.L. Alperin, L. Evens: Representations, resolutions, and Quillen’s dimension theorem. J. Pure Appl. Alg. 22, 1–9 (1981).
J.L. Alperin, L. Evens: Varieties and elementary abelian subgroups. J. Pure Appl. Algebra 26, 221–227 (1982).
J. Carlson: The varieties and the cohomology ring of a module. J. Algebra 85, 104–143 (1983).
L. Evens: The cohomology ring of a finite group. Trans. Amer. Math. Soc. 101, 224–239 (1961).
J.A. Green: A transfer theorem for modular representations. J. Algebra 1, 73–84 (1964).
O. Manz, U. Stammbach, R. Staszewski: On the Loewy series of the group algebra of groups of small p-length. Comm. Algebra 17, 1249–1274 (1989).
D. Quillen: The spectrum of an equivariant cohomology ring I, II. Ann. of Math. 94, 549–572, 573-602 (1971).
J.-P. Serre: Sur la dimension cohomologique des groupes profinis. Topology 3, 413–420 (1965).
U. Stammbach: On the principal indecomposables of a modular group algebra. J. Pure Appl. Algebra 30, 69–84 (1983).
B.B. Venkov: Cohomology algebras for some classifying spaces (Russian). Dokl. Akad. Nauk SSSR 127, 943–944 (1959).
Literature
A. Borel et al.: Algebraic D-Modules. New York: Academic Press, 1987.
C.W. Curtis, I. Reiner: Methods of Representation Theory, Vol. 1. New York: Wiley, 1961.
B. Eckmann: Homotopie et dualité. Colloque de Topologie Algébrique. Louvain, 1956, pp. 41–53.
P.-P. Grivel: Catégories derivées et foncteurs derivés. In: A. Borel et al.: Algebraic D-Modules. New York: Academic Press, 1987.
P. Gabriel, M. Zisman: Calculus of Fractions and Homotopy Theory. Ergebnisse der Mathematik. Berlin: Springer-Verlag, 1967.
P. Hilton: Homotopy theory of modules and duality. Proceedings of the Mexico Symposium 1958, pp. 273–281.
R. Hartshorne: Residues and Duality. Lecture Notes in Mathematics. Berlin: Springer-Verlag, 1966.
G. Iversen: Cohomology of Sheaves. Universitext. New York: Springer-Verlag, 1980 (Chapter 11).
C.A. Weibel: An Introduction to Homological Algebra. Cambridge Studies in Advanced Mathematics, vol. 38, 1994 (Chapter 10).
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Hilton, P.J., Stammbach, U. (1997). Some Applications and Recent Developments. In: A Course in Homological Algebra. Graduate Texts in Mathematics, vol 4. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-8566-8_11
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