morita.qxp 5/21/97 4:03 PM Page 680 Kiiti Morita 1915–1995 A. V. Arhangel´skii, K. R. Goodearl, and B. Huisgen-Zimmermann iiti Morita, a pioneer in both algebra fluential work in this area and of its continua- and topology, died in Tokyo on Au- tion and ramifications within current research. gust 4, 1995, at the age of eighty. His Following that, we will sketch his impact on last journal publication had appeared topology. only six years earlier. KBorn in Hamamatsu, Japan, on February 11, Morita’s Contributions to Algebra 1915, Morita received his Ph.D. degree from the Unlike most of his prominent Japanese con- University of Osaka in 1950 for a doctoral the- temporaries in mathematics, Morita had not sis in topology. His basic university education supplemented his education through an acade- had been focused on algebra—as a topologist, mic excursion abroad by the time he started Morita was largely self-taught. In 1939 he was publishing groundbreaking concepts and re- appointed assistant at the Tokyo University of sults. According to one of his Ph.D. students in Science and Literature, and after an interlude as algebra, H. Tachikawa, Morita was not connected lecturer/professor at Tokyo Higher Normal with the Nagoya research group, the most active School he was appointed professor at the former Japanese group in his field of algebraic spe- university in 1951, where he taught for twenty- cialty, homological algebra. As a result, signifi- seven years (a period during which the two in- cant developments in the field reached him only stitutions were combined and later relocated with considerable delay. For example, while draft from Tokyo to Tsukuba). Finally, as emeritus copies of H. Cartan and S. Eilenberg’s modern from the University of Tsukuba, he continued his cast of the subject, Homological Algebra [10], activities at Sophia University. were already in circulation in the early 1950s Morita’s focus in teaching was again split be- among the young mathematicians surrounding tween topology and algebra—throughout his re- T. Nakayama, this text did not reach Morita until search career he successfully bounced back and its official publication in 1956. forth between the two subjects. His algebraic con- Morita’s primary source of algebraic inspira- tributions, however, while sparser in number tion was Pontryagin’s duality theory for locally (about 75 percent of his eighty-seven journal compact abelian groups. In that case, duality is publications were on topological subjects), were effected by the contravariant Hom-functor the ones that made his name a household word D = Hom ( , R/Z). In experimentally placing in- Z − within the mathematical community at large. jective modules over more general rings into We will start with a brief account of his most in- the second argument of Hom-functors (injec- tive modules take over the role of divisible A. V. Arhangel´skii is professor of mathematics at Ohio abelian groups in the wider context), Morita no- University and Moscow University, Russia. His e-mail ad- ticed that correlations reminiscent of those es- dress is [email protected]. K. R. Good- tablished by Pontryagin were preserved for cer- earl and B. Huisgen-Zimmermann are professors of mathematics at the University of California at Santa Bar- tain pairs of objects (M,D(M)). In particular, bara. Their e-mail addresses are goodearl@math. this naturally led him to new characterizations ucsb.edu and [email protected], respectively. of quasi-Frobenius rings, namely, rings for which 680 NOTICES OF THE AMS VOLUME 44, NUMBER 6 morita.qxp 5/21/97 4:04 PM Page 681 all free modules are injective. Jointly with his stu- vision algebras); in [27, Section 7] he actually dents Tachikawa and Y. Kawada, Morita exploited proved this result for rings with minimum con- the findings in two articles in 1956–57 [38, 39] dition. and assigned further exploration of this line to One of the main promoters of Morita’s new Tachikawa. The latter continued the study of and fundamental viewpoint was H. Bass, who had annihilator relations for pairs (M,D(M)), ob- been alerted to Morita’s ideas by S. Schanuel at taining, in particular, sharp results for the spe- Columbia. Bass’s mimeographed notes on cial case of rings satisfying the descending chain Morita’s theorems [6], based on lectures he gave condition for one-sided ideals [47]. At that point, at a National Science Foundation summer insti- Morita succeeded in pinning down dualities be- tute at the University of Oregon in the early tween “reasonable” subcategories of module cat- 1960s, were copied, recopied, and rapidly dis- egories in full generality, proving that they are tributed throughout the U.S. and to many Euro- always of the form HomA( ,Q):A-Mod pean mathematics departments. Another publi- − → Mod-B, where B is the endomorphism ring of the cation sped the dissemination of Morita’s ideas A-module Q, and he completely determined in Western Europe, namely, P. Gabriel’s disser- the eligible objects Q inducing such dualities. (We tation of 1962 in which Morita’s equivalence mention that for the case of rings with the de- theorem is re-proved [16]. It is hardly surpris- scending chain condition, G. Azumaya indepen- ing that the benefits of transporting desirable dently obtained results of the same strength, which features from one category to another by means were published a year later than Morita’s in [4].) of well-behaved functors, as advertised by It appears that Morita’s treatment of equiva- Morita’s work, triggered the construction of a net- lences of module categories came as an after- work of such bridges linking and unifying ex- thought to his work on duality. For one thing, tensive parts of algebra. Furthermore, these ben- the title of his famous 1958 paper “Duality for efits have not gone unnoticed in other modules and its applications to the theory of fields, as witnessed by M. Rieffel’s de- rings with minimum condition” [27] made no velopment of Morita equivalence for mention of equivalences. For another, Morita re- C*-algebras and W*-algebras [43, portedly balked at an editor’s suggestion to pub- 44], extended to Banach algebras by lish the material in two separate treatises in N. Grønbæk [17], and even to Pois- order to give the topics of duality and equiva- son manifolds by P. Xu [50]. lence independent emphasis. Let us follow a single exciting Morita’s theorem on equivalence [27, Section thread within the representation the- 3] is probably one of the most frequently used ory of rings that extends Morita’s single results in modern algebra. It states that, equivalences of module cate- given two rings A and B, the categories A-Mod gories to equivalences of and B-Mod are equivalent if and only if B arises “derived categories”. The as the endomorphism ring of an A-module P story begins with cer- which is a direct summand of a power An of the tain reflection functors regular module A, with the property that A in introduced by I. N. turn is a direct summand of a direct power P m Bernstein, I. M. of copies of P. Further, each equivalence be- Gel´fand, and V. A. tween A-Mod and B-Mod is induced by a co- Ponomarev [7] in 1973 that display variant Hom-functor HomA(P, ) with P as − above. In this situation, the rings A and B are exceptionally now referred to as being “Morita equivalent”. (The good behavior symmetry of this relation is an easy exercise.) The relative to cer- identification of Morita equivalent pairs of tain subcate- rings—such as any ring A coupled with the gories of rep- resentations. n n matrix ring Mn(A)—at once allowed cast- × ing a great deal of technical ballast overboard: In a ground- breaking there was no longer any need to verify compu- paper [9], tationally that the corresponding module cate- gories display identical behavior. Indeed, Morita’s motivation and immediate usage was of this ilk: He provided a general reduction principle by showing that any finite-dimensional algebra is Morita equivalent to a “basic” algebra—one which, modulo its Jacobson radical, is a finite di- rect product of division algebras (rather than, as in the general case, of matrix algebras over di- JUNE/JULY 1997 NOTICES OF THE AMS 681 morita.qxp 5/21/97 4:04 PM Page 682 S. Brenner and M. C. R. Butler proved that this Morita’s Contributions to Topology striking behavior is shared by a much larger Morita belongs to the third generation of gen- family of functors dubbed “tilting functors”. eral topologists, with Hausdorff, Vietoris, Alexan- The theory of such functors was clarified and droff, Urysohn, Kuratowski, Sierpinski, and R. L. pushed further by D. Happel, C. M. Ringel [19], Moore making up the first, and Tychonoff, Cˇech, K. Bongartz [8], and Y. Miyashita [23]. In their Hurewicz, and Tumarkin belonging to the sec- present incarnation, tilting functors are covari- ond. Morita’s first topological paper appeared in ant Hom-functors HomA(T, ):A-Mod − → 1940 [24], which means that he started his re- B-Mod, where T replaces the modules P in- search in general topology at roughly the same ducing Morita equivalences by much more gen- time as E. Hewitt, R. Bing, M. G. Katetov, eral objects; at this point, T is required only to H. Dowker, and A. H. Stone. have finite projective dimension, not to admit He contributed to all major directions within any nontrivial higher self-extensions, and to general topology, some of his most important allow for a nice resolution of A in direct sum- topological work being on normality, paracom- mands of finite direct powers of T.