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Gerhard Hochschild (1915–2010) Walter Ferrer Santos and Martin Moskowitz, Coordinating Editors

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Gerhard Hochschild (1915–2010) Walter Ferrer Santos and Martin Moskowitz, Coordinating Editors Gerhard Hochschild (1915–2010) Walter Ferrer Santos and Martin Moskowitz, Coordinating Editors In the mid-1940s and early 1950s, homological and Dickson (1906); and later Brauer (1928), Baer algebra was ripe for formalization, or to put it (1934), and Hall (1938). In 1941 H. Hopf gave an in Hochschild’s own words—see his now classic explicit formula for the second homology group in review of the book Homological Algebra by Car- terms of a description of the first homotopy group tan and Eilenberg ([1])— “The appearance of this using generators and relations which, according book must mean that the experimental phase of to Mac Lane, provided the justification for the homological algebra is now surpassed.” study of group cohomology. The actual definition From its topological origins in the work of Rie- of homology and cohomology of a group first mann (1857), Betti (1871), and Poincaré (1895) on appears in the famous papers by Eilenberg and the “homology numbers” and after 1925 with the Mac Lane: first in the 1943 announcement [2] and work of E. Noether showing that the homology of later in 1947 in [3] in full detail. More or less a space was better viewed as a group and not as simultaneously and independently, Freudenthal Betti numbers and torsion coefficients, the subject considered the same concepts in the Netherlands of homology of a space became more and more (c. 1944) and, in a different guise, Hopf was doing algebraic. A further step was taken with the ap- the same in Switzerland. In his 1944 paper, instead pearance of the concept of a chain complex in 1929 of using explicit complexes as in [3], Hopf uses as formulated by Mayer and in a different guise free resolutions to define homology groups with by de Rham in his 1931 thesis. In 1935 Hurewicz integral coefficients, the concept of free resolution showed that the homology of an aspherical space having been used in algebra since its introduction depended only on its first homotopy group, hence by Hilbert in his famous 1890 paper on invariant defining implicitly the (co)homology of a group. In theory. For the above historical comments the au- parallel, and purely from the viewpoint of algebra, thors followed closely the information appearing the low-dimensional cohomology of a group had in Chapter 28, History of Homological Algebra by been considered earlier. For example, it appeared C. A. Weibel in [13]. implicitly in Hilbert’s Theorem 90 in 1897—known While still a graduate student at Princeton, also to Kummer before—as well as in the study Hochschild submitted a paper for publication of so-called factor sets used to classify group dealing with the study of the behavior of Lie extensions. It also occurred in other contexts—in algebras and associative algebras with respect to the early work of Hölder (1893), Schur (1904), derivations. In the Introduction to [4]—which was published in 1942 shortly after he was drafted Walter Ferrer Santos is professor of mathematics at the Univ. de la República, Montevideo, Uruguay. He finished into the army at the end of 1941—he states: his thesis under Hochschild’s direction in 1980. His email “These ‘generalized derivations’…were found to address is [email protected]. be significant for the structure of an algebra. In fact Martin Moskowitz is professor emeritus of mathematics we shall obtain a characterization of semisimple at the City University of New York Graduate Center. He Lie algebras and semisimple associative algebras finished his thesis under Hochschild’s direction in 1964. in terms of these generalized derivations.” His email address is [email protected]. Gerhard’s dissertation committee, cochaired by All photographs, unless otherwise noted, are courtesy of Chevalley and Lefschetz, reports: “The thesis deals the Hochschild family. with certain important problems in Lie algebras 1078 Notices of the AMS Volume 58,Number8 and related questions in associative algebras. It mathematics graduate student at UIUC—and his contains in particular a highly interesting charac- daughter and son were born in 1955 and 1957, terization of semisimple Lie algebras in terms of respectively. Some relevant aspects of his mathe- the operation of formal derivation. The thesis is matical work during this period are described in highly worthy of publication as it contains many the following works. new results in addition to those indicated above. In the contribution by John T. Tate, “Memories Furthermore, Hochschild set the problem himself of Hochschild, with a Letter from Serre”, the and also did the research in an essentially inde- author writes about his pendent way.”1 In modern terms Gerhard proved personal and mathematical in his thesis that an associative algebra has the interaction with Gerhard in property that all its derivations are inner if and the genesis of the cohomo- only if it is separable. The method he uses is logical perspective of class basically homological, and with the tool of a sep- field theory. He also adjoins arability idempotent, constructs from the given a letter by Serre describ- derivation the element that is used to characterize ing the process that led it as a conjugation. In these terms he was dealing to the construction of the with what is now called the first Hochschild co- so-called Hochschild-Serre homology group and proving its triviality. It still spectral sequence. took Gerhard three years to publish a proof of the There are many mathemati- natural extension of his thesis results to a general cal objects carrying Gerhard’s cohomology theorem, and for that it was necessary name. In another contribu- first to construct an adequate cohomology theory. tion to this memorial, Andy With that purpose in mind he generalized the com- Magid, in “The Hochschild- plexes used in group cohomology to define what Mostow group”, reviews this is now called Hochschild cohomology. Putting it in concept (named in that fash- Gerhard Hochschild at his his own words: “The cohomology of an associative ion by Lubotzky in 1979), daughter Ann’s wedding, algebra is concerned with the m-linear mappings which appeared initially un- 1978. of an algebra into a two-sided module…A linear der the name of the “group of R(G) R(G) mapping…analogous to the coboundary opera- proper automorphisms of ”, being the G tor of combinatorial topology and leading to the algebra of representative functions of the group . notion of ‘cohomology group’ has been defined This construction appeared in a series of papers by by Eilenberg and Mac Lane (unpublished)…The these authors published between 1957 and 1969. present paper is concerned primarily with the G.HochschildandG.D.Mostowtogetherwrote connections between the structure of an algebra seventeen papers in different subjects. This very and vanishing of its cohomology groups”—see the rich mathematical collaboration started while both Introduction to [5]. All these aspects of Hochschild were members of the Institute for Advanced Study cohomology, as well as its applications and many in the academic year 1956–1957. Dan Mostow, other topics, are treated in detail in the contri- in “Gerhard Hochschild as Collaborator”, writes a bution by M. Gerstenhaber, “Hochschild’s Work very personal and illustrative note on their per- on Cohomology”, that appears later in this arti- sonal and mathematical interaction. One of its cle. In that contribution the author also describes landmarks is the important paper on the invariant the particular circumstances in which these first theory of unipotent groups [10]. Another is their papers were written. Particularly interesting are theorem on faithful representations of real and his considerations concerning the back-and-forth complex connected Lie groups which is a global interactions between the results of [5] and of extension of Ado’s theorem on Lie algebras (see the [3]—e.g., concerning the concept of dimension contribution to this memorial by M. Moskowitz). shifting. The subject of representative functions is also Paul Bateman contributed a note, “Gerhard dealtwithintheinterview that Pierre Cartier Hochschild at Urbana (1948–58)”, about the par- gave to one of the authors, which appears later. ticipation of Hochschild in the early development In particular, Cartier mentions the influence that of the Department of Mathematics at the University his view of Tannaka duality and the concept of Hopf algebras had in this line of research. This of Illinois at Urbana-Champaign in the late 1940s is also explicitly spelled out in the introduction and 1950s. That was an extremely fruitful period to [6], in which Hochschild writes: “We are con- of Gerhard’s professional as well as personal life. cerned with the analogues for Lie algebras of the He married Ruth Heinsheimer in 1950—then a problems centered around the Tannaka duality 1We thank the librarians of the Seeley G. Mudd Man- theorem…The analogue of the Tannaka theorem uscript Library, Princeton, NJ, and Hochschild’s family for semisimple Lie algebras was established by for providing access to Hochschild’s records at Princeton Harish-Chandra in 1950. More recently P. Cartier University. has sketched (without proofs) a general duality September 2011 Notices of the AMS 1079 theory for algebraic groups and Lie algebras which went together with, but often transcended, his absorbs Harish-Chandra’s result.” mathematical influence. These two notes are by Cartier—as well as Tate in his article—also N. Nahlus, his last student (who finished in 1986), comments on the not-too-well-known interaction “Gerhard Hochschild as My Advisor and Friend”, of Gerhard with the Bourbaki group, which began and M. Moskowitz, who finished in 1964, “Some in the early 1950s when he attended three of the Reminiscences”. group congresses.2 Finally, with the help of Hochschild’s family The crucial institutional role he played in the files, the science writer and Gerhard’s son-in-law, development of mathematics at U.
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