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Collected papers of John Milnor. V: Algebra. Edited by Hyman Bass and T. Y. Lam. (English) Zbl 1208.01050

Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4876-0/hbk). xii, 425 p. (2010).
The algebra volume of John Milnor’s collected papers is divided into three parts, each grouping a certain direction of his algebraic or algebra-flavored research. Each group of papers is preceded by an introduction, succinctly presenting the content of the papers, as well as some of the history and development of the problems addressed in Milnor’s papers.
The first part, called “Algebras and groups” consists of J. W. Milnor and J. C. Moore [Ann. Math. (2) 81, 211–264 (1965; Zbl 0163.28202)], as well as the 1959 preprint of the same title that eventually became the 1965 paper, [Relativité, groupes et topologie II, Les Houches Éc. d’Été Phys. Théor., Sess. 40, 1983, 1007–1057 (1984; Zbl 0594.22009)], the lecture notes for Mathematics 402 at Princeton University in Spring, 1963, titled “The representation rings of some classical groups”, and [J. Differ. Geom. 2, 447–449 (1968; Zbl 0176.29803)].
The second part, “The congruence subgroup problem”, consists of a 1965 preprint written with H. Bass and titled “On unimodular groups over function fields”, H. Bass, J. W. Milnor, and J.-P. Serre [Publ. Math., Inst. Hautes Étud. Sci. 33, 59–137 (1967; Zbl 0174.05203)], as well as the errata to it in [Publ. Math., Inst. Hautes Étud. Sci. 44, 241–244 (1974; Zbl 0297.20054)], and his paper [Enseign. Math., II. Sér. 29, 281–322 (1983; Zbl 0557.10031)]. As an afterword to the second part, a paper by G. Prasad and A. S. Rapinchuk looks at how the 1967 paper mentioned above, as well as a paper of J.-P. Serre on the same subject “fit in, and can be derived from, the more general results and techniques available today”.
The last part, titled “Algebraic K-theory and quadratic forms”, contains [Invent. Math. 8, 83–97 (1969; Zbl 0177.05204)], [Invent. Math. 9, 318–344 (1970; Zbl 0199.55501)], a 1970 preprint titled “Symmetric inner product spaces over a Dedekind domain”, and [Prospects Math., Ann. Math. Stud. 70, 59–75 (1971; Zbl 0265.10013)]. The third part ends with an informative afterword by A. Merkurjev, titled “Developments in algebraic K-theory and quadratic forms after the work of Milnor”.

MSC:

01A75 Collected or selected works; reprintings or translations of classics
11E70 \(K\)-theory of quadratic and Hermitian forms
11E81 Algebraic theory of quadratic forms; Witt groups and rings
19D45 Higher symbols, Milnor \(K\)-theory
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20M05 Free semigroups, generators and relations, word problems

Keywords:

John Milnor

Biographic References:

Milnor, John
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