Tilting Theory: AGiftof to Mathematics

GABRIELLA D’ESTE

Dedicated to the memory of Sheila Brenner and Michael C. R. Butler

n the introduction to his book Tame Algebras and unexpected reference, as I should have done long ago. One Integral Quadratic Forms, Claus Michael Ringel remarks aim of this article is to fill this gap, at least partially. II that ‘‘some general ideas, which have influenced the In the introduction to the Handbook of Tilting Theory results and the methods presented here, are not available in [2], the authors state the following: official publications, or not even written up.’’ This resonates ‘‘Tilting theory arises as a universal method for con- with my personal experiences with both Places and People,to structing equivalences between categories. Originally use the terminology of FDLIST [20]. This article is the written introduced in the context of categories over version of conversations I’ve had through the years with finite-dimensional algebras, tilting theory is now young colleagues on the unofficial history of tilting theory, its considered an essential tool in the study of many general ideas, unexpected facts, and open problems. areas of mathematics, including finite and algebraic The origin of tilting theory in Italy is one of the best group theory, commutative and noncommutative examples of the complexity of general ideas (and their zigzag algebraic geometry, and algebraic topology.’’ journeys). Menini’s paper [27] describes the role played by Perhaps it is for this reason that the referee of [15] wrote Adalberto Orsatti and his Algebra Team in Padova, and it ‘‘tilting theory is the gift of algebra representation theory to contains an account of important public events and official mathematics.’’ This remark inspired the title of this article. publications, as well as information on private conversa- ‘‘Tilting’’ has an immediate visual connotation, and tions, classical letters, and unexpected connections between readers unfamiliar with the history of the field might distant places and people. The first paper by Italian authors wonder. I refer to the Brenner & Butler quote at the top of on tilting theory is by Menini and Orsatti [28]. In the intro- the Wikipedia article on tilting theory (https://en.wikipe duction of [28], the authors thank Masahisa Sato, Enrico dia.org/wiki/Tilting_theory) for the origin of the term. Gregorio, and me. I regret that I have never expressed offi- People care about tilting modules, because it is often useful cially—in a paper—my thanks (and surprise) for this to compare the categories of modules over A and that of

Ó 2017 The Author(s). This article is an open access publication, Volume 39, Number 3, 2017 77 DOI 10.1007/s00283-017-9733-y modules over the B of a tilting A- during a NATO meeting held in Antwerpen (Belgium) in module T, which sometimes is a better-behaved ring. the period July 20–29, 1987. After some time Sato wrote It is likely that only a reader who has at least some to Orsatti showing an example of a tilting module.’’ passing acquaintance with homological algebra and the I remember very well what happened next. Orsatti told me to representation theory of finite-dimensional algebras will make a copy of the letter received from Japan. My homework best appreciate the nuances of the mathematics described was to look at Happel and Ringel’s example of a tilting here. However, I hope that the story I tell here will offer module described in the last page of [21] and to give a talk some insight into this beautiful subject for most readers. about it. So my unique and small contribution to [28] was just There will be a lot of pictures, as making pictures has a talk. And hence my surprise for those unexpected thanks. taught me much about tilting theory. Here are the lessons I Now I realize that I should have thanked Menini and Orsatti learned by making pictures of tilting-type objects: for the opportunity. While preparing for the talk I was pleas- antly surprised to observe directly that Auslander–Reiten (a) ‘‘Simple’’ and combinatorial objects may have unex- quivers can help one guess and see possible equivalences pected concealed topological properties (see ‘‘Do before constructing a proof of their existence. Indeed tilting Finite-Dimensional Bimodules Have a Topology?’’). equivalences and cotilting dualities also have a combinatorial (b) ‘‘Nonsimple’’ objects may have unexpected concealed nature, inherited and suggested by that of quivers and modules. discrete properties (See ‘‘Do Infinite-Dimensional I was also able to discover the magic power of Auslander’s Modules Need No Topological Tools?’’). formula, which reduces the computation of Ext1–groups to that of Hom–groups. (For details, see [3, Proposition 4.6 and In the following, K denotes an algebraically closed field, Corollary 4.7] or [32, conditions (5) and (6), pages 75–76]). and we assume that all vector spaces and algebras are Consequently we may verify the vanishing of certain Ext1 defined over K. groups by simply looking at the Auslander–Reiten . Without this formula, I would be unable to check that Happel A Letter from Japan and My First Homework on and Ringel’s example is Ext1–self-orthogonal. Tilting Theory Picture 1 below (depicting a very combinatorial object Menini [27, page 11] describes Sato’s contribution to her with several symmetries) illustrates the shape of Happel– joint paper with Orsatti [28]: Ringel’s bimodule T ¼ ATB, where B ¼ EndAT . It is good to ‘‘I would like to recall here that it was Masahisa Sato that consider the whole bimodule ATB, and not only the pointed out to us that tilting modules might provide underlying module AT , in order to visualize equivalences examples ... In fact Orsatti explained this problem to Sato and dualities.

Picture 1

5 a

6 a

2 b 5 b 5 c 4 c

6 b 6 c

1 d 3 d

5 d

2 d 4 d

6 d 6 d

1 e 3 f

2 e 5 e 5 f 4 f

6 e 6 f

78 THE MATHEMATICAL INTELLIGENCER enough to give for free a lot of indirect information in a When drawing figures representing bimodules, we compact way. adopt the following conventions: For example, Picture 1 tells us how the tilting equivalence • Every square indicates an element v of a fixed basis represented by the HomðÞATB; À (between the mod- of the underlying vector space of T. ules generated by the left A–module T and the modules • The index x on the left (resp. y on the right) of a cogenerated by the left B–module DðTBÞ¼HomK ðTB; KÞ) small square corresponding to the vector v indi- acts on some indecomposable modules. For instance, the cates that exv ¼ v ¼ vey, where ei is the path of indecomposable summands of AT are sent via this functor to length zero around the vertex i. the following indecomposable summands of BB : • Following Ringel’s suggestion during my stay in 5 a, 25 b , 54 c , Bielefeld, the small squares have a special posi- 6 6 a 6 a tion, so that they also describe in an obvious way the composition factors of the same module. (See, for instance, [29, 30] and [31, page 126] for 13d 1 e 3 f 254 , 25 bd, 54 dc. descriptions and/or pictures of complicated a modules.) 66 6 a 6 a • We recall that the bimodules associated to ‘‘valued’’ On the other hand, Picture 1 also describes how the arrows in [19] and [18] gave me the idea of adding cotilting duality induced by HomðÞÀ;A TB (between the two indices (on the left and on the right of the modules cogenerated by left A-module T and the modules small squares). In this way we can see the action of cogenerated by the right B-module T ) acts on some inde- left or right multiplication by the primitive idem- composable modules. For instance, the indecomposable potents, corresponding to vertices of some quiver. summands of AT are sent via this functor to the following • Straight lines describe left modules and wavy lines indecomposable summands of BB: describe right modules. Recall that a representa- 5 a 25 b 54 c tion of a quiver consists of vector spaces (one for bdc, , , 6 6 e 6 f each vertex) and of linear maps (one for each ef arrow) eventually satisfying suitable relations.

According to [21], the underlying vector space of T has 13 d 1 3 254 , 25 e , 54 f . dimension 23, whereas A is the algebra given by the ef 66 6 6 Dynkin quiver E6 with ‘‘subspace orientation,’’ that is of the form

What Happened Next: Talk with Big Matrices, More or Less Abstract Cancellations... The analysis of Happel–Ringel’s bimodule was both my first homework on tilting theory, and the subject of my On the other hand, the algebra B ¼ EndAT is isomorphic to first talk on this subject. This was not my first talk in the algebra given by the fully commutative quiver Padova containing some quivers. However, all the quivers I used earlier were much smaller and so much easier to describe without pictures. When I tried to do the same with Happel–Ringel’s example, I realized that even basic techniques of representation theory of finite-dimensional algebras can make otherwise invisible things become visible. For instance, to give a definition of A and B without quivers, I distributed some pages with the largest matrices I ever used, namely 23 Â 23 matrices, which I did not want Now Picture 1 is not as old as my homework. I used to draw at the blackboard. After some time I noticed that I similar pictures to visualize rather small bimodules for the could use more reasonable matrices to describe the K-lin- first time, and more or less by chance. Of course, for big, ear maps from T to T, corresponding to multiplications by complicated bimodules, it would be better to replace a 2- elements of A or B. Indeed, T has the strong property that dimensional ‘‘global’’ visualization (of the main properties the groups of all morphisms between two indecomposable of the left and right underlying modules) by a 3-dimen- summands of AT and TB, respectively, are K-vector spaces sional one, without or with few self-intersections. of dimension at most one. Hence, after cancellation of However, even flat pictures such as Picture 1 are powerful many ‘‘inessential’’ rows and columns, one can see that A

Ó 2017 The Author(s), Volume 39, Number 3, 2017 79 and B are isomorphic to subalgebras of the algebra of all always easier to deal with more than one (but finitely 6  6 matrices of the form: many) elementary properties instead of dealing with just 0 1 one property on classes of modules. K 00000 B C Let us begin with one such longer definition, for the B KK0000C 1 B C classical case. We say that a module M is a classical tilting B C B 00K 000C or cotilting module (more precisely, a 1-tilting or 1-cotilting B C B 00KK00C module), respectively, if the following conditions hold: B C @ 0000K 0 A • The projective (respectively, injective) dimension of M is KKKKKK at most 1. 0 1 • Ext1 ðM; È MÞ¼0 (respectively, Ext1 ðPM; MÞ¼0Þ, KKKKKK R R B C where È M (respectively, PMÞ is any direct sum B 0 K 00K 0 C B C (respectively product) of copies of M. B C B 00K 00K C • There is a short exact sequence of the form and B C: B 000KKKC 0 00 B C 0 À! R À! M À! M À! 0 @ 0000K 0 A (respectively, 0 À! M0 À! M00 À! Q À! 0), where M0 00000K and M00 are direct summands of direct sums (respec- Endomorphism rings of abelian groups were the subject tively, products) of copies of M. of my master’s thesis, ‘‘Abelian Groups Whose Endomor- For finite-dimensional modules over finite-dimensional phism Ring is Locally Compact in the Finite Topology,’’ algebras, the setting studied by Brenner and Butler [5], written under the direction of Adalberto Orsatti. Moreover, the third condition may be replaced [32, page 167] by a some of my first papers dealt with endomorphism rings. numerical condition: However, without using quivers, this previous abstract The number of isomorphism classes of indecom- experience on (usually large) endomorphism rings posable summands of M is equal to the rank of the wouldn’t have been useful to verify that B ¼ EndAT actu- of R. ally had the indicated form. Next consider two short definitions (for the general case). The cancellation of rows and columns was only the first of Given an R-module M and a natural number n [ 0, we many other (more abstract) cancellations made in the sequel, denote by GennðMÞ the class of all modules X such that concerning modules and complexes (see the sections enti- there is an exact sequence of the form tled ‘‘Cancellation of Summands’’ and ‘‘An Example of Mð1ÞÀ!...À! MðnÞÀ!X À! 0; Cancellations’’). For more on the theoretical importance of cancellations in tilting theory, I refer the reader to Ringel’s where the M(i)’s are direct summands of direct sums of lecture [33] on the occasion of the 20th anniversary of the copies of M. Following [4], we say that M is a tilting Department of Mathematics of the University of Padova [see (respectively partial tilting) module of projective dimension the section ‘‘Fully Documented Lectures’’ in Ringel’s home at most n if GennðMÞ is equal to (respectively, is contained page]. Note that the suggestive title of this lecture is ‘‘Tilting in) the orthogonal class Theory: the Art of Losing Modules.’’ \ ? i M ¼ Ker ExtRðM; ÀÞ: Short or Long Definitions With or Without Classes i [ 0 of Modules As in the classical case, the long definition of such M Definitions Long and Short. The most compact and elegant consists of two or three conditions on just two modules, definitions of tilting modules and of partial tilting modules, namely on M and a very special projective or injective natural generalizations of tilting modules, consist of pre- module. cisely one rather short property. For instance, any Here is one of the reasons that it may be difficult to ‘‘classical’’ tilting (resp. cotilting) R–module M (hence also check equalities or inclusions of classes of modules that Happel–Ringel’s module T) has the property that the class play a key role in the characterization of ‘‘nonclassical’’ of all modules generated (respectively, cogenerated) by M tilting and cotilting-type modules M. Even in dealing with 1 coincides with the of ExtRðM; ÀÞ (respectively, algebras of finite representation type, in the ‘‘nonclassical’’ 1 ExtRðÀ; MÞ). This global property is equivalent to a much case, these modules M have the property that one of these longer characterization capturing three discrete properties two classes of modules (namely an orthogonal class) is of two modules, namely of M and of the regular module R closed under direct summands, whereas the other does not (respectively, an injective cogenerator Q). For me it was necessarily have this closure property. To see this, it

1See [4, Proposition 3.6 and Lemma 3.12] for the beautiful technical condition (on the relationship between two classes of modules), which characterizes ‘‘nonclassical’’ tilting or cotilting modules and their generalizations.

80 THE MATHEMATICAL INTELLIGENCER T n suffices to consider the following example. Let R be the n [ 0 Ker ExtRðM; ÀÞ is the class of all injective modules. algebra given by the quiver Let T be a sincere summand of M, obtained from M after cancellation of a projective summand P. Then we deduce from [13, Theorem 11.4] that T is a large partial tilting module. [We will soon give a ‘‘minimal’’ example of this result, where T is

(1) the unique indecomposable injective module that is not projective; (2) a uniserial module such that every simple module has multiplicity one as a composition factor of T, that is a such that the composition of any two arrows is zero. Then sincere module of minimal dimension. we deduce from [14, Example B] that the injective module 4 23 1 1 T is a partial tilting module (of Cancellations have both theoretical and practical impor- 5 4 3 2 tance in tilting theory. It turns out that various types of projective dimension 3) such that Gen3ðT Þ contains the module 1 È 1 but not its summand 1. Moreover both cancellations can help us deal with more abstract partial Gen ðT Þ and the class AddðT Þ, formed by all injective R- tilting objects. For instance, this often happens with partial 3  modules without simple summands, have the same inde- tilting complexes, say T , in the sense of Rickard [34] with composable modules, namely the 4 indecomposable the following property: summands of T. This means that we cannot determine the (a) T  is the projective resolution of a large partial tilting class Gen3ðT Þ by just looking at the Auslander–Reiten quiver. In other words, the operation of making direct sums module T, which is not a tilting module. (the ‘‘only really well-understood construction’’ [36, page By [34], this hypothesis guarantees the existence of a 476]) is not enough to investigate an important class of nonzero right bounded complex (of projective mod- modules generated by a maximal direct summand of a ules) X  with the following property: tilting module of projective dimension 3. Note that T is an (b) X  is not the projective resolution of a module and example of a faithful module that is not tilting. every morphism from T  to any shift of X  is homotopic to zero. Cancellation of Summands. Let us next consider the can- The examples constructed in [15, 16, 17]suggest cellation of an injective nonprojective summand (to obtain that there is no canonical way to obtain X  from a ‘‘large’’ partial tilting module). We obtain the faithful T . Moreover, the same holds by confining our- module T constructed above from the minimal injective selves to complexes T  and X  satisfying (a)and cogenerator DðRRÞ¼HomK ðRR; KÞ by means of cancella- (b), respectively, and with the following additional tion of its injective summand 1 (of projective dimension 3). ‘‘very combinatorial’’ property (in the words of 2 On the other hand, ExtRðT ; 5Þ 6¼ 0 and 5 is the unique [35]): indecomposable module X such that HomRðT ; XÞ¼0. (c) Any nonzero component of the indecomposable sum- Consequently, we have mands of T  and X  is an indecomposable module. \ ? ðÃÞ Ker HomRðT ; ÀÞ T ¼ 0: Indeed, up to shift, the choices of the indecomposable complexes X  satisfying both (b)and(c) may be quite In other words, T is a large partial tilting module in the different. For instance, there may be either zero [15, sense of [13]. Example C (iii) and (iv)], or one [17, Remark after Example Large partial tilting modules are partial tilting modules 1], or infinitely but countably many [17, Remark after satisfying ðÃÞ. Any tilting module is a large partial tilting Example 3], or uncountably many [17, Example 4] choices. module [4, page 371]. Any finitely-generated large partial The following example shows that by deleting some tilting module of projective dimension at most 1 is a tilting components of an indecomposable complex T , with module [6, Theorem 1]. Now Property (Ã) and [13, Lemma properties (a)and(c), we may obtain all the shortest 11.2] imply that any large partial tilting module over a complexes X  with properties (b)and(c), that is, the semiperfect ring, say again T,issincere [24], that is, it has the ‘‘elementary’’ complexes in the sense of [35]oftheform property that HomðP; T Þ 6¼ 0 for every (d)0À! P À! Q À! 0 with P and Q indecomposable P 6¼ 0. Consequently, a module T of finite length is sincere if projective modules. every simple module is a composition factor of T [3, 32]. Now let us look at the cancellation of a projective- An Example of Cancellations. For every even integer injective summand of a tilting module with minimal m [ 2, there is a uniserial nonfaithful injective module T orthogonal class (to obtain a ‘‘large’’ partial tilting mod- (of projective dimension m) such that we obtain all the ule). Let R be a finite-dimensional algebra or, more indecomposable complexes X  with the above proper- generally, a noetherian and semiperfect ring such that ties (b)and(d) by means of various types of every indecomposable injective module has a simple cancellations of some components of T , that is, left socle. Let M be an injective tilting R–module (of projective cancellations, right cancellations, and, sometimes, central dimension [ 1) such that the orthogonal class M? ¼ cancellations also.

Ó 2017 The Author(s), Volume 39, Number 3, 2017 81 Indeed, let A be the Nakayama algebra [26] given by the is, right (respectively, left) multiplications, are described by quiver nice matrices with many entries equal to zero; they seem to be ‘‘continuous’’ extensions of their restriction to C. How- ever, as the following toy example shows, the property of being an indecomposable bimodule is neither hereditary nor left-right symmetric. Toy Example of a Finite-Dimensional Cotilting Bimodule with relation an ÁÁÁÁÁa1 ¼ 0, where 2n ¼ m þ 2. Next, let T denote the injective module of the form [12, Example D] . Let S (respectively, R) be the algebra given by the quiver 2 3 456 . . : n (respectively, 213), and let S CR be an inde- composable cotilting bimodule with 1

4 5 We obtain T from the minimal injective cogenerator C 5 5 S 6 DðAAÞ¼HomK ðAA; KÞ after cancellation of its n À 1 inde- 6 composable projective summands. Thus we deduce from 1 1 respectively, CR 2 the ‘‘Cancellation of Summands’’ section (or from [13, 23 2 Example 11.6]) that T ? is the class of all injective modules and T is a large partial tilting module of projective dimen- of the form sion m. (The letter n in condition (c) of [13, Example 11.6] is a misprint; it should be m.) Moreover the projective Picture 2 42 resolution T  of T satisfies (a) and (c), and the complexes X  satisfying (b) and (d) are of the form 51 0 À! IðiÞÀ!IðjÞÀ!0; where IðÃÞ denotes the indecomposable injective module 52 53 corresponding to the vertex à and i [ j [ 1[16, Proposition 1]. Hence they are exactly the complexes with two 61 nonisomorphic injective components different from zero, obtained from T  after suitable cancellations. 62

Do Finite-Dimensional Bimodules Have a Then C has codimension 2 in its left (respectively, right) Topology? injective envelope Visualizing bimodules as in the ‘‘A Letter from Japan and My First Homework on Tilting Theory’’ section, I could first ‘‘see’’ and 4 4 4 EC 5 5 then prove that even rather small bimodules behave badly with 5 6 6 respect to some quite natural constructions such as embeddings respectively, EC 1 1 1 1 . into bimodules with an underlying left (or right) injective 2 2 2 3 module. Indeed, according to certain results on the socle of E(C)/C [25, Lemma 2.2] and on the modules cogenerated by E(C)/C [25, Propositions 1.7 and 2.1 and Theorem 1.17], it is Moreover E(C) is the support of an indecomposable (re- natural to measure the gap between a cotilting module C and its spectively, a decomposable) bimodule D, containing C as a injective envelope E(C), at least for modules that are finite-di- bimodule, of the form mensional vector spaces. However, in this special situation where the discrete topology seems to be the most natural Picture 3 41 topology, two radically different situations show up. 42 43 Bad Case. It is not always possible to embed a finite-di- mensional cotilting bimodule C in another bimodule D with the property that D, as a left (respectively, right) module is 51 the injective envelope of C [12, Example B (c), (d)]. More- over, no left-right symmetry exists, because only one of the 52 53 constructions may be possible [12, Example A (c), (d)]. 61 Good Case. When such an embedding exists, the structure of D as a right (respectively, left) module seems to be the most obvious one. Multiplications on the opposite side, that 62

82 THE MATHEMATICAL INTELLIGENCER 41

42 51

51 respectively, Picture 4 53.

52

61

62

Remarks on the Action of Primitive Idempotents and Nilpotent Elements in the Good Case. In all the examples constructed in [12], where S CR is a cotilting bimodule such that EðS CÞ admits a structure of SÀR bimodule, containing the cotilting bimodule S CR, we see that the ring R is hereditary. Furthermore, if X is an indecomposable sum- mand of S C and e is a primitive idempotent of R such that 0 0 xe ¼ x for every element x 2 X, then we also have x e ¼ x Let S CR be the cotilting bimodule such that for every element x0 E X . Finally the nilpotent ele- 2 ðS Þ Ã ments of S and R corresponding to arrows act on the and Next, let R elements of EðCÞnC in the easiest possible way, by a kind be the hereditary algebra given by the Dynkin diagram of shift. Indeed, if s 2 S, r 2 R and u, v, w (resp. u, w, w, x) are linearly independent elements of C (resp. of E(C)) such that x E C C, sx v, sw u, vr u, 2 ð Þn ¼ ¼ ¼ Ã then we have xr ¼ w, as illustrated in Picture 5: Finally, let F : R À! R be the obvious ring epimorphism. Then dim ker F ¼ 1 and C, regarded as a S–RÃ bimodule, is contained in the S–RÃ bimodule U, satisfying (i) and (ii), Picture 5 x described in Picture 6. w Picture 6 43 v 42 u 53 41 Roughly speaking, the above picture says that the vectors x and v have similar neighborhoods, and so the whole 52 bimodule looks like a topological object. Remarks on the Action of New Concealed Rings in the Bad 63 Case. In all the examples constructed in [12], where S CR is a cotilting bimodule such that EðS CÞ does not admit a structure of S–R bimodule containing the cotilting bimodule 6 2 S CR, we can show that An Example (of the Bad Case) with Nonhereditary RÃ.Asin (i) The ring R is not hereditary. [12, Example C], let R (respectively, S) be the algebra given (ii) We can find a ring RÃ, a ring epimorphism F : RÃ À! R, and a bimodule S URÃ containing S CRÃ , such that by the quiver with relation ab ¼ 0 (respectively, EðS CÞ¼S U .

An Example (of the Bad Case) with Hereditary RÃ.Asin[12, with relation cd ¼ 0). Next, let C be the Example A], let R (respectively, S) be the algebra given by S R the quiver

Ó 2017 The Author(s), Volume 39, Number 3, 2017 83 cotilting bimodule such that and projective or summands of U [1, Propositions 20.13 and 20.14 and Corollary 20.16] and the rest of the world, that is, Ã Next, let R denote the algebra given by the the nonobvious indecomposable reflexive modules. The cotilting bimodule described in the section entitled quiver with relation aba ¼ 0. Finally, let F : ‘‘Toy Example of a Finite-Dimensional Cotilting Bimodule’’ RÃ À! R be the obvious ring epimorphism. Then admits four indecomposable reflexive left (respectively, dim Ker F ¼ 2 and C regarded as an S–RÃ bimodule is right) modules. Moreover, comparing Auslander-Reiten contained in the S–RÃ bimodule U, satisfying (i) and (ii), quivers (and looking at Picture 2), we see that all of them are described in Picture 7. obvious reflexive modules and the duality D acts as follows:

4 5 1 1 5 2 , , 53, 6 . 6 23 2 6

A similar situation holds for the cotilting module described in the ‘‘An Example (of the Bad Case) with Hereditary RÃ’’ section (respectively, the ‘‘An Example (of the Bad Case) with Nonhereditary RÃ’’ section, where the cotilting duality D acts as follows:

3 4 2 5 6 , 5 , 41, 2 2 1 6 6

3 1 2 4 respectively, 4 2 , 3 , 1 . 1 3 3 1

Let us wrap this section up with two open problems on On the other hand, Happel–Ringel’s cotilting (and tilting) bimodules. bimodule T described in the ‘‘A Letter from Japan and My First Homework on Tilting Theory’’ section easily gives us an Problem 1. Are the conditions of the ‘‘Remarks on the example of a cotilting bimodule admitting nonobvious Action of Primitive Idempotents and Nilpotent Elements in reflexive modules. Indeed, by comparing Auslander–Reiten the Good Case’’ section satisfied by any bimodule as in the quivers (and by looking at Picture 1), it is easy to see that T good case described in the section preamble? admits fourteen indecomposable reflexive left and right 5 modules, respectively. But is the unique indecomposable Problem 2. Are conditions (i) and (ii) of the section entitled 6 ‘‘Remarks on the Action of New Concealed Rings in the Bad projective summand of the left A–module T. Therefore T Case’’ satisfied by any bimodule as in the bad case descri- admits eleven obvious and three nonobvious indecompos- bed in the section preamble? able reflexive left and right modules, respectively. We already described (at the end of the ‘‘A Letter from Japan and Do Infinite-Dimensional Modules Need No My First Homework on Tilting Theory’’ section) how the cotilting duality D acts on the indecomposable summands of Topological Tools? T . Now D acts on the remaining obvious left A-modules, Another reason for surprise at the presence of the kind of A that is, the five indecomposable projective modules that are topology described in the section entitled ‘‘Do Finite-Di- not summands of T , via: mensional Bimodules Have a Topology?’’ (see the remark A after Picture 5) is the absence of topological arguments in 1 d 2 bd 3 d 2 , , 4 , the proof of a result concerning dualities induced by e 6 e f cotilting bimodules of infinite dimension. Before we dis- 6 6 cuss this, we note that a left S-module (respectively, a right 4 dc a , 6 bddc. R-module) M is reflexive with respect to the bimodule S UR 6 f ef (or just U-reflexive or reflexive, for short) if M is canonically isomorphic to its double dual with respect to U, that is, to Finally, D acts on the three indecomposable nonobvious the module DðDðMÞÞ, where D denotes both the con- reflexive left A–modules via: travariant Hom?ðÀ;S URÞ for ? ¼ R; S, and the group DðXÞ is equipped with its module structure (see [1, 254 bdc 3 bd 1 dc , 254 , 254 . Proposition 4.4] or [23, Propositions 3.4 and 3.5]) for any 66 ef ef ef 66 66 left S-module and any right R-module X.

Obvious and Nonobvious Reflexive Modules. Even in Several important results about nonobvious modules can special cases, for instance, given a faithfully balanced be obtained by means of rather technical topological tools bimodule U, there is a big gap between the well-known (see [8, Sections 2 and 3], [7, 9, 10], and the papers cited in indecomposable reflexive modules, which are either [8]). That’s why it was a pleasant surprise to see that a

84 THE MATHEMATICAL INTELLIGENCER discrete bimodule was enough to answer Colpi’s question: indecomposable projective-injective modules. This justifies ‘‘Are reflexive modules closed under submodules?’’ His the thinking that tilting objects exist in nature and are not additional hint was ‘‘You cannot use finite-dimensional artificial. The examples considered in this article (con- algebras and modules!’’ (See [9, Theorem 1] for the nice cerning tilting modules of projective dimension at most one behavior of submodules of reflexive modules over Artin and cotilting modules of injective dimension at most one) algebras.) also suggest that classical tilting and cotilting modules are complicated enough to solve interesting questions, as the Proposition [11, Lemma 2.4 and Theorem 2.5 (ii)]. Reflexive question on reflexive modules presented previously. modules with respect to a cotilting bimodule are not nec- I should confess that I always try to use these mod- essarily closed under submodules. Moreover, even the ules (and in particular the pictures describing them) to well-known reflexive modules with respect to a faithfully visualize more-or-less mysterious equivalences and dual- balanced bimodule S UR, that is, the indecomposable sum- ities. However I always prefer to use the long and mands of both the left (respectively, right) regular module S combinatorial definitions of tilting and cotilting-type (respectively, R) and of the module U are not necessarily objects, that is, the definitions where classes of modules closed under submodules. do not appear directly. I view the existence of com- The next example—more precisely, the next picture— pletely different definitions as a proof of the depth of shows that the above result has a purely combinatorial these objects. Indeed they have both combinatorial and motivation coming from basic linear algebra. Indeed, the functorial properties and it is possible to define them by result follows from the same reason that an infinite-di- means of either set of properties separately. mensional K-vector space cannot be isomorphic to its Through the years, tilting objects became more and double dual, and so it cannot be reflexive with respect to more abstract and were defined over Grothendieck cat- the regular bimodule K KK . (See [8, Proposition 1.8] for a egories and triangulated categories. However I believe general result on direct sums of infinitely many nonzero that the original notion of a tilting module is still very reflexive modules with respect to a cotilting bimodule.) significant. I finally recall that a word coming from physics—reflection—appeared in the title of the first Toy Example of a Finite-Dimensional Cotilting Bimodule as paper [5] by Brenner and Butler on tilting theory. Sheila in the previous Proposition [11, Lemma 2.4 and Theorem 2.5 Brenner told me they extensively discussed options for a (ii)] . With terminology suggested by [22], assume R ¼ S is suggestive name; I believe they succeeded in finding the ‘‘generalized’’ Kronecker algebra given by the quiver one.

ACKNOWLEDGMENTS I would like to thank colleagues in Milan, Ferrara, and Sherbrooke for opportunities to present this work to engaged audiences, and the referees and editors for helpful suggestions.

Then the indecomposable projective nonsimple left (re- Department of Mathematics spectively, right) module is a reflexive module with respect University of Milano to the cotilting bimodule RRR. However, its maximal sub- Via Saldini 50 module, that is, the Jacobson radical of R, generated by the 20133 Milano infinitely many arrows from 1 to 2, is not reflexive with Italy respect to R: e-mail: [email protected]

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