Maurice Auslander 1926-1994

maurice.qxp 5/8/98 1:47 PM Page 450 Maurice Auslander 1926–1994 Maurice Auslander died of cancer on November at the University of Chicago, the University of 18, 1994, in Trondheim, Norway. Michigan, and at the Institute for Advanced Auslander has made fundamental contribu Study in Princeton. He joined Brandeis Univer tions in many central parts of algebra. It would sity in 1957, where he was chairman 1960–1961 be senseless to try to describe his work within and 1976–1978. He held visiting positions in a given specialty or to present it under a par Paris, Urbana, London, Trondheim, Austin, and ticular title. Quite the contrary, he liked to attack Blacksburg. He had fellowships from Sloan, problems by surprise, from ap Guggenheim, and Fulbright, parently nowhere. This ap and a few weeks before his proach resulted in many origi death he was awarded a Senior nal theorems in commutative Humboldt Research Prize. Aus and noncommutative ring the lander was a Fellow of the ory, for orders and Brauer In American Academy of Arts and groups, and in the representa Sciences and a member of the tion theory of Artin algebras as mathematics Royal Norwegian Society of Sci well as in the theory of singu ences and Letters. larities. Among the main char he had a In commutative ring theory, acteristics of his work one will Auslander was immediately at particularly remember the ex sense for tracted by homological meth treme elegance of the methods beauty. ods. The title of his first joint he liked to introduce and de work with Buchsbaum was “Ho velop and also his ability to pre mological dimension in noe sent and explain the crucial therian rings” (1956). It soon points. These qualities were became “Homological dimen also typical of his personality. sion in noetherian rings I” for The importance of his scientific influence is all algebraists, since their collaboration on this clearly established for many years to come. topic was long and fruitful. Progressively, they Maurice Auslander was born on August 3, organized and formalized what became “Ho 1926, in Brooklyn, New York. He received his B.S. mological commutative algebra with a view to in 1949 and his Ph.D. in 1954, both from Co ward algebraic geometry”. The characterization lumbia University. His thesis was within group of regular rings as the rings with finite homo theory and was written under the direction of logical dimension (Auslander-Buchsbaum-Serre) Robert L. Taylor. He spent the years 1953–1957 as well as the factoriality of regular local rings 450 NOTICES OF THE AMS VOLUME 42, NUMBER 4 maurice.qxp 5/8/98 1:47 PM Page 451 his favorite book by Cartan-Eilenberg. The mono graph on stable module theory (with Bridger) is rather typical of this evolution. A central role is played by the transpose of a module, one of Auslander’s favorite tools. Although this work is still influenced by algebraic geometry, through dualities, the representation theory of Artin al gebras is a developing theme. In this monograph are also the roots of Auslander’s successful ver sion of noncommutative Gorenstein rings, giv ing rise to what is now known as Auslander- Gorenstein rings and Auslander regular rings. The new phase for Auslander took a clearer form around 1970, during his visits to Urbana and London, with a decided interest in the mod ule theory of Artin algebras. Modular group rep resentation theory was a first source of inspi ration. Higman had much earlier described the group algebras of finite (representation) type in terms of p-Sylow subgroups. Also the Brauer- Thrall conjectures had their origin from group algebras, the first one having been proven by Roi ter for algebras over a field. Auslander started by homologically characterizing finite type in terms of endomorphism algebras, now called Auslander algebras. Then he proved the first Brauer-Thrall conjecture for left Artin rings, and (Auslander-Buchsbaum) are now classic results. with an elegant application of functors he char The interest of Auslander in this theme culmi acterized infinite type in terms of existence of nated with “Modules over unramified regular indecomposable modules which are not finitely local rings” (1961). This was also the title of his generated. This work, together with the work of talk at the World Congress of 1962. In this work Roiter and Gabriel’s work on representations of he started a research program for thirty years quivers, belongs to the start of the modern phase to come. The list of contributors to this pro of representation theory of Artin algebras, where gram includes the names of his students Pesk Auslander was a main contributor until the end ine and Szpiro and also of Hochster and Fulton of his life. (via the local Riemann-Roch theorem). This pro Starting also in the early seventies, Auslander gram was essentially completed with the an developed with Reiten the theory of almost-split swers of Roberts to the “Homological Conjec sequences, also called Auslander-Reiten se tures”. Ramification theory was, during the same quences, and the related irreducible maps. The years, Auslander’s other favorite topic. He col transpose of a module plays a major role in this laborated there with Buchsbaum, Goldman, and work. Almost-split sequences and the associ Rim. The new methods were first developed for ated Auslander-Reiten quivers are now central noetherian rings. Later on, a general theory of in representation theory. They are important separable algebras over a commutative ring was for classification theorems, provide a useful proposed. The success of this theory and par combinatorial invariant for an algebra, and cri ticularly of the homological difference of an al teria for finite type. They also play an important gebra over a ring is now well established. In role in group representation theory, in particu 1962, Auslander gave a new proof of “the purity lar through the work of Webb and Erdmann. of the branch locus”. It was very much in his By interpreting work of Bernstein-Gelfand- style, completely surprising for the specialists Ponomarev, Auslander (with Platzeck and Reiten) and strikingly beautiful. First he reduced the gave the first module theoretic construction of question to proving that if R is a regular local what is now, through the work of Brenner-But ring and M a reflexive finitely generated R-mod ler and Happel-Ringel, known as tilting theory. ule such that Hom(M,M) is isomorphic to a di A few years ago Auslander and Reiten had also rect sum of copies of M, then M is free. His proof established a close connection between tilting of this last statement is really exhilarating. and the theory of contravariantly finite subcat With his work on coherent functors in the mid- egories, as developed by Auslander-Smalo sixties, Auslander started to develop systemat around 1980. This connection has beautiful ap ically a functorial point of view, in the spirit of plications to quasihereditary algebras, as dis- APRIL 1995 NOTICES OF THE AMS 451 maurice.qxp 5/8/98 1:47 PM Page 452 covered by Ringel, and inspired applications of Maurice Auslander had a warm and sensitive tilting to algebraic groups by Donkin. The con personality. His door was always open to his travariantly finite subcategories also were im friends. He enjoyed discussions, often provoca portant in Auslander’s recent work (with Sol- tive ones, about mathematics and its philosophy berg) on relative homological algebra in in particular. He had considerable administra representation theory. tive talent, doubling the size of the department Even though the representation theory of during his first chairmanship at Brandeis. His in Artin algebras, including the interplay with group terests outside mathematics included art, poetry representation theory, was Auslander’s main and music, and he enjoyed playing the violin. He field of research during the last 20–25 years, he was himself “un homme libre”, free of all influ always strived to carry his ideas and proofs to ences, and wanted others to be the same. He their utmost generality. would pin down cliches on Through these efforts he the spot. In mathematics obtained interesting appli he had a sense for beauty; cations to the theory of he truly enjoyed some re commutative and noncom- sults and their proofs. He mutative noetherian rings. had a special interest and In the mid-seventies, before concern for young re the theory of almost-split searchers, including his sequences was generally ac thirty or so doctoral stu cepted in representation dents, who loved and ad theory of Artin algebras, he mired him. He would fre generalized the basic exis quently visit their offices tence theorem to a setting or wake them up by phone including lattices over or calls in the early mornings, ders and (maximal) Cohen- inquiring about the latest Macaulay modules over progress. One of his stu complete local Cohen- dents once complained that Macaulay rings. Auslander He was himself Maurice did the easy work. several years later discov “un homme libre” When things were getting ered connections with sin harder, he would leave the gularity theory, thus open free of all subject. He answered good- ing up a new, exciting area heartedly that the student of research. He showed that influences, and was welcome to do the his methods were particu same. larly well suited in dimen wanted others to Maurice loved to travel. sion two, in fact in some On his travels he attracted sense better than for Artin be the same. students and collaborators algebras. He saw that the for himself and visitors to Auslander-Reiten quivers Brandeis. Through his ex for two-dimensional in tensive travels and collab variant rings are isomorphic to the corre orations, he had a positive influence on the de sponding McKay quivers, which for the special velopment of algebra in many foreign countries.

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