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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

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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-

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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. It was only pactness, classification of spaces by mappings, at this stage of the development that a deeper dimension theory, homotopy theory, and shape reason for the remarkable preservation prop- theory. However, behind the great variety of top- erties of tilting functors surfaced. In [18] Hap- ics Morita considered, there was also in his topo- pel recognized, for special classes of rings, that logical work the idea of a conceptually unifying they induce equivalences on the level of the de- approach via category theory, an idea that was rived categories of the two module categories still new at the time. This is reflected, for ex- involved. Such derived categories, introduced by ample, in his interest in the stability of proper- J.-L. Verdier [49] after a suggestion of A. Grothen- ties under formation of direct products. On the dieck, can be thought of as devices to bare the other hand, next to his fascination with general homological skeleton of the original abelian cat- schemes, Morita was also a brilliant problem egories. To be slightly more precise: they are ho- solver; this is witnessed by his numerous deep motopy categories of complexes of objects, lo- concrete results in dimension theory, for in- calized at those morphisms which induce stance. isomorphisms on the level of homology, part of Let us mention explicitly some of Morita’s the information they carry being encoded in achievements in topology. His most significant certain distinguished triangles of objects and contributions to the subject can be grouped morphisms. Subsequent work of E. Cline, B. Par- under the following three captions: paracom- shall, and L. Scott [12] represents a push toward pactness and normality, dimension theory, and completing the picture along Morita’s original shape theory. We will provide a few highlights model by establishing a first installment of nec- for each of these. essary and sufficient conditions for the occur- Under the first heading he proved, in partic- rence of certain types of equivalences between ular, that every (regular) Lindelöf space is para- derived categories of module categories. Finally, compact, generalizing an earlier result of this line of insights was topped off by J. Rickard J. Dieudonné, who introduced the fundamental [42] in the late 1980s. On the level of the origi- notion of paracompactness in 1944 and proved nal module categories, Rickard completely char- that every separable metrizable space is para- acterized those equivalences of their derived cat- compact [13]. Moreover, Morita discovered an un- egories which take distinguished triangles to expected link between paracompactness and distinguished triangles. He thus established— normality in terms of the product operation: A in full generality—a Morita theory for derived space X is paracompact if and only if the prod- categories of module categories. uct of X with any (with some) compactification Throughout his career, Morita kept returning of X is normal (this was also proved by to the interplay between category theory and al- H. Tamano [48]). While in general paracom- gebra. In fact, one of the cornerstones of the the- pactness is not preserved in products, Morita es- ory of derived categories, namely, the technique tablished that the product of a paracompact of forming quotient categories modulo localiz- Hausdorff space and a σ-locally compact, para- ing subcategories first systematically explored compact Hausdorff space is paracompact [28]. by Gabriel [16], was thoroughly pursued by The following three problems, posed by Morita throughout the 1970s [30, 31, 32, 36]. Morita in [35] and later circulated as Morita’s Con- In view of this development and others of sim- jectures, strongly influenced the development of ilar impact, respect for Morita grows. His work the field. emerges as not only supplying immensely use- 1. If X Y is normal for any normal space Y, × ful results, but as strongly contributing to our is X necessarily discrete? present mode of thinking about algebraic and 2. If X Y is normal for every normal P-space × geometric structures within categorical settings. Y, must X be metrizable?

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3. Given that X Y is normal for every nor- f 1(y) k +1 for each y in Y, then × | − |≤ mal countably paracompact space Y, does it fol- Ind(Y ) Ind(X)+k. ≤ low that X is metrizable and σ-locally compact? 2. If f is a closed continuous mapping of a non- In fact, it was Morita who introduced the con- empty metrizable space X into a metrizable space cept of a “normal P-space” X and characterized Y and Ind(f 1(y)) k for each y Y , then − ≤ ∈ it by the condition that the product of X with Ind(X) Ind(Y )+k [26]. (This latter implication ≤ every metrizable space is normal [29]. This topic was independently obtained by K. Nagami; see greatly gained in importance and popularity [40].) after M. E. Rudin constructed her famous ex- Under the heading of shape theory, let us ample of a normal space, the product of which briefly touch on the background of this field. with the closed unit interval fails to be normal Shape theory for compact metric spaces was [45]. A positive answer to Morita’s first problem founded by K. Borsuk in 1968 (see [21] for the follows from results of M. Atsuji [3] and M. E. history). Subsequently, the theory was extended Rudin [46], while Z. Balogh [5] recently con- to compact Hausdorff spaces and arbitrary met- firmed the third conjecture. The second con- ric spaces by S. Mardesˇich and J. Segal [22] and jecture finally has been answered in the affir- R. H. Fox [15], respectively. Finally, Mardesˇich [21] mative under the additional set-theoretic and Morita [34, 37] independently pushed it to assumption that V = L (Chiba-Przymusinski- the level of general spaces. Morita’s approach is Rudin [11]). based on P. S. Alexandroff’s idea of the nerve of Another important notion that was intro- a locally finite covering of a space. He assembled duced by Morita in the context of normality of the nerves of all locally finite normal coverings product spaces strongly influenced the subject, of a space and formed an inverse system in the namely, that of an “M-space”. Within the class homotopy category of CW-complexes. This ap- of paracompact spaces, the M-spaces coincide proach exhibits a fruitful interaction among with the p spaces introduced independently by methods of set theory, category theory, and al- gebraic topology. In particular, this approach Arhangel skii [2]; they are characterized by the ´ led Morita to a shape-theoretic version of the existence of perfect mappings onto metrizable Whitehead theorem within the homotopy theory spaces. Among the many excellent papers deal- of CW-complexes [33]. ing with this topic, one finds J. Nagata’s beauti- Probably more than half of the contempo- ful characterization: X is a paracompact M-space rary generation of Japanese topologists consists if and only if X is homeomorphic to a closed sub- of indirect students of Morita, in the sense that space of the product of a metric space and a com- they are using methods he developed and solv- pact Hausdorff space [41]. ing problems he posed. But his influence goes Morita’s contributions to dimension theory far beyond Japan, being particularly strong in the constitute one of the most important parts of his U.S. and in the former Soviet Union, where major scientific inheritance. It is well known that there schools in general topology can be found. Un- are three basic topological definitions of di- doubtedly, he is now considered worldwide as mension, based on different structures: small in- one of the great founders of modern general ductive (ind), large inductive (Ind), and covering topology, and his work serves as a continuing (dim) dimensions. One of the central problems source of inspiration. in the theory is that of establishing relation- One of the authors of this article (A.V.A.) had ships among these three invariants. It is well the honour to be a guest of Professor Morita in known that they coincide for separable metriz- his home in Japan. He remembers him as a very able spaces (Tumarkin and Hurewicz), while the warm, hospitable person, with a lot of humour two inductive dimensions need not coincide for and benevolence, surrounded by several mem- nonseparable metrizable spaces. On the other bers of his family—his wife, Tomiko, and son, hand, Morita [25] and M. Katetov [20] indepen- Jusahiro (a top manager in the Hitachi com- dently established that dim(X) = Ind(X) for ar- pany). Even after the ten years that have passed, bitrary metrizable spaces X. Moreover, Morita ob- the memory of this visit remains fresh. tained the fundamental inequality dim(X) ind(X) in case X is a Lindelöf space References ≤ (see [14] for a discussion). [1] P. S. Alexandrof and B. A. Pasynkov, Introduc- In addition to this comparison of different di- tion to dimension theory (in Russian), Moscow, mensions of a given space, Morita proved the fol- 1973. [2] A. V. Arhangelskii, On a class of spaces con- lowing two mapping results for Ind which play ´ taining all metric and all locally compact spaces, a pivotal role in dimension theory (see [1] and Soviet Math. Dokl. 4 (1963), 1051–1055. [40] for references): [3] M. Atsuji, On normality of the product of two 1. If f is a closed continuous mapping of a spaces, General Topology and Its Relations to metrizable space X onto a metrizable space Y and Modern Analysis and Algebra (Proc. Fourth Prague

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Topology Sympos., Prague, 1976), Part B, 1977, pp. [28] ———, On the product of a normal space with a met- 25–27. ric space, Proc. Japan Acad. 39 (1963), 148–150. G. Azumaya [4] , A duality theory for injective mod- [29] ———, Products of normal spaces with metric ules, Amer. J. Math. 81 (1959), 249–278. spaces, Math. Ann. 154 (1964), 365–382. [5] Z. T. Balogh, Nonshrinking covers and K. Morita’s [30] ———, Localizations in categories of modules. I, third conjecture (abstract), 1995 Annual Spring Math. Zeitschr. 114 (1970), 121–144. Topology Conference, 1995, p. 4. [31] , Localizations in categories of modules. II, [6] H. Bass, The Morita theorems, Mimeographed ——— J. Reine Angew. Math. 242 (1970), 163–169. notes, University of Oregon, 1962. [32] , Localizations in categories of modules. III, [7] I. N. Bernstein, I. M. Gel’fand, and V. A. Pono- ——— Math. Zeitschr. 119 (1971), 313–320. marev, Coxeter functors and Gabriel’s theorem, Russian Math. Surveys 28:2 (1973), 17–32. [33] ———, The Whitehead theorem in shape theory, [8] K. Bongartz, Tilted algebras, Representations of Proc. Japan Acad. 50 (1974), 458–461. Algebras (Puebla, 1980) (M. Auslander and E. Lluis, [34] ———, On shapes of topological spaces, Fund. Math. eds.), Lecture Notes in Math., vol. 903, Springer- 86 (1975), 251–259. Verlag, Berlin, 1981, pp. 26–38. [35] ———, Some problems on normality of product [9] S. Brenner and M. C. R. Butler, Generalizations spaces, General Topology and Its Relation to Mod- of the Bernstein-Gelfand-Ponomarev reflection ern Analysis and Algebra (Proc. Fourth Prague functors, Representation Theory II (Ottawa, 1979) Topological Sympos., Prague, 1976), Part B, Soc. (V. Dlab and P. Gabriel, eds.), Lecture Notes in Czechoslovak Mathematicians and Physicists, Math., vol. 832, Springer-Verlag, Berlin, 1980, pp. Prague, 1977, 296–297. 103–169. [36] ———, Localizations in categories of modules. IV, [10] H. Cartan and S. Eilenberg, Homological algebra, Sci. Rep. Tokyo Kyoiku Daigaku, Sec. A 13 (1977), Princeton Univ. Press, Princeton, NJ, 1956. 153–164. [11] K. Chiba, J. C. Przymusinski, and M. E. Rudin, Nor- [37] , Resolutions of spaces and proper inverse sys- mality of product spaces and Morita’s conjectures, ——— tems in shape theory, Fund. Math. 124 (1984), Topology Appl. 22 (1986), 19–32. [12] E. Cline, B. Parshall, and L. Scott, Derived cat- 263–270. egories and Morita theory, J. Algebra 104 (1986), [38] K. Morita, Y. Kawada, and H. Tachikawa, On 397–409. injective modules, Math. Zeitschr. 68 (1957), [13] J. DieudonnÉ, Une generalisation des espaces com- 217–226. pacts, J. Math. Pures Appl. 23 (1944), 65–76. [39] K. Morita and H. Tachikawa, Character modules, [14] R. Engelking, General topology, Polish Scientific submodules of a free module, and quasi-Frobenius Publishers, Warsaw, 1977. rings, Math. Zeitschr. 65 (1956), 414–428. [15] R. H. Fox, On shape, Fund. Math. 74 (1972), 47–71; [40] J. Nagata, Modern dimension theory, Groningen, 75 (1972), 85. 1965. [16] P. Gabriel, Des categories abeliennes, Bull. Soc. [41] ———, A note on M-spaces and topologically com- Math. France 90 (1962), 323–448. plete spaces, Proc. Japan Acad. 45 (1969), 541–543. [17] N. Grønbæk, Morita equivalence for Banach al- [42] J. Rickard, Morita theory for derived categories, gebras, J. Pure Appl. Algebra 99 (1995), 183–219. J. London Math. Soc. (2) 39 (1989), 436–456. [18] D. Happel, Triangulated categories in the repre- [43] M. A. Rieffel, Morita equivalence for C -alge- sentation theory of finite dimensional algebras, ∗ bras and W -algebras, J. Pure Appl. Algebra 5 London Math. Soc. Lecture Notes, no. 119, Cam- ∗ (1974), 51–96. bridge Univ. Press, Cambridge, 1988. [19] D. Happel and C. M. Ringel, Tilted algebras, Trans. [44] ———, Morita equivalence for operator algebras, Amer. Math. Soc. 274 (1982), 399–443. Proc. Sympos. Pure Math. 38 (1982), 285–298. [45] M. E. Rudin, A normal space X for which X I [20] M. G. Katetov, On the dimension of non-separa- × ble spaces, 1, Czech. Math. J. 2 (1952), 333–368. is not normal, Fund. Math. 73 (1971), 179–186. [21] S. Mardesˇich, Shapes for topological spaces, Gen. [46] ———, κ-Dowker spaces, Czech. Math. J. 28(103) Topology Appl. 3 (1973), 265–282. (1978), 324–326. [22] S. Mardesˇich and J. Segal, Shapes of compacta [47] H. Tachikawa, Duality theorem of character mod- and ANR-systems, Fund. Math. 72 (1971), 41–59. ules for rings with minimum condition, Math. [23] Y. Miyashita, Tilting modules of finite projective Zeitschr. 68 (1958), 479–487. dimension, Math. Zeitschr. 193 (1986), 113–146. [48] H. Tamano, On compactifications, J. Math. Kyoto [24] K. Morita, On uniform spaces and the dimension Univ. 1 (1962), 161–193. of compact spaces, Proc. Phys.-Math. Soc. Japan 22 [49] J.-L. Verdier, Categories derivees, quelques re- (1940), 969–977. sultats (Etat 0), Inst. Hautes Études Sci. Publ. Math. [25] , Normal families and dimension theory for ——— (1963); publ. in Seminaire de Geometrie Alge- metric spaces, Math. Ann. 128 (1954), 350–362. 1 brique du Bois-Marie SGA 4 , Cohomologie Etale, [26] ———, On closed mappings and dimension, Proc. 2 Japan Acad. 32 (1956), 161–165. Lecture Notes in Math., vol. 569, Springer-Verlag, Berlin, 1977, pp. 262–311. [27] ———, Duality for modules and its applications to the theory of rings with minimum condition, Sci. [50] P. Xu, Morita equivalence of Poisson manifolds, Rep. Tokyo Kyoiku Daigaku, Sec. A 6 (1958), Comm. Math. Phys. 142 (1991), 493–509. 83–142.

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