DERIVED CATEGORIES AND ALGEBRAIC GROUPSlJ2. Leonard L. Scott

Transcription

1 Contemporary Mathematics Volume 82 (1989) DERIVED CATEGORIES AND ALGEBRAIC GROUPSlJ2 Leonard L. Scott ABSTRACT. The notion of a derived category challenges our classical concept of structure in algebra. Nevertheless, these categories have played a remarkable role in the proof of major results in representation theory. In this paper, which is partly a philosophical essay, I discuss these categories in the context of the general history of algebra and representation theory, as well as efforts of CPS to understand them better algebraically in their work on the Lusztig conjecture. Because we mathematicians have the capability of giving logically irrefutable proofs of our theorems, it is often assumed that our subject is objective, permanent, and in particular not subject to the great philosophical debates of other academic disciplines. Those of us who have been doing mathematical research for many years know this is not true. First of all, though the logical validity of a theorem is indeed hardly ever a matter of serious long-term debate, the issue of what constitutes a good theorem, or what constitutes a good area of mathematics in which to prove a theorem, is very much subject to debate and can easily change with time. And perhaps even more remarkable than the things we debate are the things we do not debate and ignore--unconscious and unchallenged assumptions about the way mathematics is and should be. In this lecture I wish to focus on and reexamine just such an unconscious assumption. Specifically I want to reexamine the way we think about "structure" in algebra.. '1980 Mathematics Subject Classification (1985 Revision) Primary 20G05; Secondary 01-02, 03A05, 18E30, 20-03, 2OC20. 2 This research was supported in part by NSF American Mathematical Society /89 $ $.25 per page

2 128 LEONARDL.SCOTT We have largely taken what the philosophers would call an "atomistic" view of the structure of algebraic systems. That is, we try to determine how the system is built from its simplest units, and then determine more precisely what these simplest units look like. For example, in the representation theory over the complex numbers of a finite group one has -Theorem: Every representation module M over B: for a finite group G is a direct sum M = Sl e... e Sn of irreducible submodules. (Also, the number of times an isomorphic copy of any given irreducible module appears in such a decomposition does not depend on the exact choice of the decomposition.) So this is how the general module over c for G is built from simpler pieces, the "irreducible modules". One can now ask what we know about these simpler pieces. A great deal can usually be said for any given finite group G. There is a well developed "character theory" of finite groups which is usually capable of producing at least a complete list of all the irreducible modules in the form of a "character table". Such a table gives much information, such as the dimension, concerning the representation, though the table does not quite give the representations explicity. A tremendous illustration of what is possible in this direction is the "Atlas of finite groups" compiled by John Conway and his associates. This "Atlas" gives, among other things, the character tables of all 26 'sporadic' simple groups, including the group called the "monster", which has order approximately 105' (something like ). It is of course a very slow and painful process to determine the character table of any one individual group. Fortunately, many groups, especially the finite versions of the classical groups which are the focus of this conference, fall into a few well-defined families. Professor Lusztig and his colleagues have

3 DERIVEDCATEGORIESANDALGEBRAICGROUPS 129 made great progress in understanding the irreducible characters of these groups. Another much older success story I would like to mention at this point is the finite dimensional representation theory of complex semisimple Lie algebras over c. There is a complete analogue of Maschke's theorem for such representations and information equivalent to a character table is contained in the famous formulas of Weyl and Kostant. See Humphreys' book on Lie algebras. This theory is essentially equivalent to a corresponding one for connected compact Lie groups, such as the classical orthogonal or unitary groups. These representation theories I have just described, over c, for finite groups and for finite-dimensional representations of semisimple Lie algebras, have been applied in many diverse areas, including coding theory and theoretical physics, and may be regarded as highly successful. One good thing about a successful theory is that we then have a model on which to build even more ambitious theories. And one bad thing is that we almost always tend to take the model too literally and try to imitate it just a little too closely. The next steps in the finite group and Lie algebra theories I have been discussing are the following A theory of finite dimensional representation modules of finite groups over finite fields, and A theory of infinite dimensional representation modules for semisimple Lie algebras over If. There is no doubt that each of these ties in with other central issues. For example, a good theory for the finite groups over finite fields is very important for the theory of their maximal subgroups presently being studied by Gary Seitz and others. The difficulties involved in approaching both theories are quite similar. Most of Maschke's theorem fails, and the irreducible representations are largely unknown, though great progress has recently been made in the Lie algebra case, and I will return to this in a moment.

4 130 LEONARDL.SCOTT The part of Maschke's theorem that is retained is essentially the Jordan-Holder-Schreier Theorem: Let R be a ring and M an R-module (a context which applies to both the group and Lie algebra situation) of finite length--that is, M has a composition series": a chain of submodules 0 s Ml E M Mn = M with each Mi+l/Mi an irreducible R-module (called a composition factor). Then the number of composition factors isomorphic to any given irreducible R-module is independent of the choice of composition series. Thus there are still "atomic" building blocks, the composition factors, but we have no prescribed way to build the module M itself from them. Also, we do not even know the possible order in which the composition factors may appear. Over time workers in representation theory have come to visualize such modules in terms of the order of factors in one or more composition series. Here are some examples of these pictures for various matrix rings. (The field of coefficients is unimportant. Also, one could easily give similar examples in the finite group or Lie algebra context we have been discussing.) [i f H] on [i] is viewed as izi [i f H] on [i] is viewed as 'w' Perhaps this is a good time to mention an example of a ring R for which Maschke's theorem does not hold, yet where we all recognize that the representation theory is successful. This is the ring "[Xl of polynomials in one indeterminant. Its representation theory is just the general theory of a single linear transformation, and the information needed to describe an R-module up to isomorphism is precisely given by assigning a matrix to X in terms of "Jordan blocks". For example

5 DERIVEDCATEGORIESANDALGEBRAICGROUPS 131 Xc-, a a 1 ;: P is viewed as (al e z e b 0 a p 10 " I/ b 0 0 P 1 This example is encouraging, and makes one feel comfortable about general modules of finite length. Perhaps, however, it makes us too comfortable. Once Maschke's theorem fails for a ring, there is at least one other reasonable way to think about the "structure" of its modules which is not in the "atomic" tradition. Instead of looking "inside" the module for better understood pieces, one can look "outside" and resolve the module in terms of a complex of other modules. Thus we know that every R-module has a resolution by projective R-modules, unique up to a homotopy equivalence of complexes. Under reasonable hypotheses (for example, if R is a finite dimensional algebra) the indecomposable projective modules are in l-l correspondence with the irreducible R-modules. In some sense it is just as reasonable to think of a module as built from the "outside" in terms of projective (or injective) indecomposable modules as from the inside in terms of irreducible modules. In practice, however, the complexes are more difficult to work with or visualize, and we have tended to keep to our atomistic "inside" view, just thinking of the complexes as potentially useful extra structure. This is, consciously or unconsciously, the way researchers in representation theory have been thinking about their subject. Recently, however, we have been forced to regard the complexes themselves much more seriously. This new thinking is forced upon us by the way the recent progress in the infinite dimensional representation theory of Lie algebras took place. A conjecture of Professors Kazhdan and Lusztig was proved by Kashiwara-Brylinkski, and also by Beilenson-Bernstein, using many

6 132 LEONARDL.SCOTT recently developed methods from topology and analysis as well as algebra. From the point of view of algebra, the most interesting ingredient is the very profound way in which complexes enter. One starts by studying certain Lie algebra modules, but eventually they correspond to comnlexes of objects in another category. So all at once we have to take complexes seriously, as objects in themselves worthy of study. A context for such a serious study, appropriate to the above correspondence, is the theory of derived categories. Let me give a brief introduction. The material below is taken from my article to appear in the proceedings of the '86 conference at Arcata, California on the representation theory of finite groups. More detail has been included than is strictly necessary to understand the examples and discussion which follow. The main notions to understand are quasi-isomorphism of complexes, triangles, and translations. A auick course in derived cateaories Let A be an abelian category, such as the category of modules for a ring or the category of rational modules for an algebraic group, or the category of sheaves for a topological space. We shall also assume than A has enough injectives. (This is just a a little more common for the categories which interest us than the property of having enough projectives. The hypothesis is not necessary for parts of the theory, while other parts are best expressed assuming enough projectives, which is certainly also an interesting hypothesis.) The objects of our derived categories will be complexes K' of A, except that quasi-isomorphic complexes are effectively identified: A quasi-isomorphism of complexes is a map inducing an isomorphism on homology, and these maps are regarded as invertible in the derived category. Derived categories come in various sizes and shapes, the most common called D+(A), D-(R), Db(A) and D(R). These arise

7 DERIVED CATEGORIES AND ALGEBRAIC GROUPS 133 from complexes of A whose cohomology is bounded below, bounded above, bounded, and perhaps unbounded in either direction, respectively. quasi-isomorphic If K' represents an object in Db(A), it is to a complex I' of injectives which is bounded below. In this way D+(A) is equivalent to the category of injective complexes bounded below, with the maps being homotopy classes of maps of complexes. (Maps back at the derived category level are the compositons with maps of complexes obtained by throwing in formal inverses of quasi-isomorphisms, very much like localizing a ring. Actually, one only needs compositions of length two, and the inverse may appear on the left or the right. See Hartshorne's book cited below.) The reader undoubtedly learned at an early age at least a basic form of the above quasi-isomorphism to an injective complex: The case where K is a complex consisting of a single object A of A, viewed as a complex concentrated in degree 0. A quasi-isomorphsim A + I' is essentially an injective resolution of A. Such material is presented in a homological algebra course just prior to defining the notion of right derived functors R"F of a given functor F: A + %, where 53 is also an abelian category. Namely, one sets R"F(A) = H"(F(I'), and argues that this cohomology is well-defined in %. In fact, one shows that the homotopy class of I', and thus of the complex F(I'), does not depend on the chosen resolution.. From the present point of view, we think of this first of all as being a consequence of the equivalence of the derived category Db(A) with that of the homotopy category of injective complexes bounded below. (Two injective resolutions of A are isomorphic in the derived category to A, thus to each other, and thus are homotopy equivalent. The beauty of thinking of it this way is that the argument applies

8 134 LEONARDL.SCOTT verbatim with A replaced by a general K'.) And second, we think of the conclusion not in terms of cohomology, but instead as saying that F(1') is well defined in D+(B). The derived category notion of a derived functor can now be described: This functor is denoted IRF, and assigns to K' in D+(/?) the object in D+(8) which is the complex F(I'), if, as above I' is an injective complex isomorphic in D+(A) to K'. The old notion of derived functor on A is recovered by the equation R"F(A) = Hn(lRF(A)). Thus, as in the theory of topological spaces, the individual cohomology terms are just key invariants of something with much more substance, the complex lrf(a) in D+(B). (Note that a map of complexes preserving homology is by definition an isomorphism in the derived category: in the topological parallel, this is a well-known theorem in the homotopy category of simply connected CW complexes, using integral homology.) There is of course also a nice formal aspect of the definition: Since frf(a) is a complex, and it makes sense to apply a derived functor now to a complex, it at least makes sense to compose the derived functors IRF and RG, if G: % + 'c is an additive functor on abelian categories with enough injectives. (Often the result is the derived functor lr(gof) of the composite. The hypotheses of the Grothendieck spectral sequence imply this, and the resulting derived functor conclusion is more desirable than the existence of a spectral sequence--which it implies--if there are futher compositions to be considered.) Finally, it also makes sense to talk about the derived functor, of a functor on complexes not obtained term-by-term. For example HomA(K',-) has a derived functor in the sense of the above procedure. (We will not use this generalization in this paper.) The derived category has two other structural features we must discuss, translation and triangles. Translation is the

9 DERIVEDCATEGORIESANDALGEBRAICGROUPS 135 invertible functor T on D+(A), or Db(A), etc., which sends K' to the complex K'[l], where K[m]n=Km+n. In other words T shifts K' to the left by one degree. Triangles, or more precisely, "distinguished triangles", are the substitute in a derived category for exact sequences in an abelian category. An example of a triangle is the seqence of maps K' f L' + C'(f) -+ T(K') obtained naturally from the mapping cone C'(f) of a map f: K' + L' of complexes. All other examples are isomorphic in the derived category to this one, and that is the usual definition. Rather than recalling the definition of a mapping cone, however, I will give one further important example of a triangle: Let O+A+B+C+O be am exact sequence in A. Form the complex C' which is A in degree -1, B in degree 0, zero in other degrees, and which has the above map A -t B as its differential in degree -1. This complex is clearly isomorphic in the derived category to C, that is, to the complex which is C in degree 0 and zero elsewhere. However, for this realization C' of C, we have an obvious map C' -I T(A). In this way the above exact sequence in A can naturally be completed to a triangle in the derived category. (It is also true that any exact sequence of complexes in the usual sense can be completed to a triangle, but the construction is more complicated. The abelian category C(A) of complexes is an integral part of the theory, though it tends to remain in the background.) So the notion of triangle generalizes and substitutes for that of an exact sequence. At this point we encounter a "fearful symmetry," which challenges our concept of structure in algebra: If the translation operator T is applied to a triangle, the

10 136 LEONARDL.SCOTT result is, except for a sign change in one of the maps, again a triangle. So, suppose you have a triangle A + B -+ C -t T(A) of objects in A or the derived category. Thinking in terms of the above exact sequence example, we would like to think of this as displaying B structurally in terms of two pieces, A and C, with C on top of A. However, there is also a triangle B+C+ T(A) -+ T(B), so we must also think of C in terms of two pieces B and T(A), with T(A) on top of B. In the first instance C was a piece of B, and in the second, B is a piece of C. And where A was a bottom piece in the first instance, its shift plays the role of top piece in the second. This is all somewhat upsetting to our usual Jordan-Holder view of the structure of things! Nevertheless, in practice, one can use much of the old intuition in all of these situations, and there are technical substitutes in derived categories (and more generally, in "triangulated categories," which formalize the additional translation and triangle structures we have been discussing) for the isomorphism theorems basic to the structure-theorectic way of thinking. For the explicit axioms for triangulated categories, see Hartshorne's book Residues and Dualitv. Examwle of a nontrivial eauivalence of derived cateaories The derived category equivalences which occur in Kashiwara-Brylinski proof are quite sophisticated, involving topological stratifications and analytic considerations (analytic D-modules), and the authors quoted results proved using highly developed theories from algebraic geometry (Weil conjectures). Fortunately, there are much simpler examples which occur in the "tilting" theory of finite-dimensional hereditary algebras. It was D. Happel who realized that the correct interpretation of tilting was in terms of the derived category. Let A be the ring of 2 x 2 lower triangular matrices over a field. Then A has two irreducible modules, which we denote a and b. The columns of A give the projective indecomposable

11 DERIVEDCATEGORIESANDALGEBRAICGROUPS 137 a modules, which we can take to be b and a module This b' particular algebra A is so small that the three modules listed above are in fact all the indecomposable modules for A. I will now describe an equivalence of the derived category (Db say) of A-modules with itself, or at least I will give the equivalence on a and b (which in fact determines the equivalence uniquely): a corresponds to ;: b corresponds to a[--11, a complex which is a in degree 1 and 0 elsewhere. One could check directly that this leads to an equivalence of derived categories, though this is a consequence of Happel's tilting theory. "tilting module" T is ES a. The functor above is RHom(T,-), where the Note how far it is from being induced by an equivalence of categories! The irreducible module a corresponds again to a module, meaning a complex concentrated in degree 0, but not an irreducible module! The object corresponding to b appears to be in some sense to be irreducible, but it it not located in degree O! The reader can check as an interesting exercise that the module ;: corresponds to b. Consequently, the three complexes corresponding to a, b, and b namely F9 a[-11, and b, together with all the direct sums of these complexes, abelian subcategory of the derived category, exact sequences. system of Beilenson-Bernstein-Deligne form an using triangles for This abelian category satisfies the axiom in Asterisque 100 for perverse sheaves, with respect to a suitable "stratification" its derived category. Such purely algebraic stratifications have been intensely studied by Ed Cline, Brian Parshall, and myself, and I will just refer you to our paper "Algebraic stratification representation categories" for details. Instead, I would like to of of

12 138 LEONARDLSCOTT discuss a class of algebras found in the context of that work for which this stratification also works. This class of algebras is sufficiently broad to describe, up to Morita equivalence (equivalence of categories) all representations which enter into the Kazhdan-Lusztig conjecture and a well-known analogue of Professor Lusztig's for algebraic groups in characteristic p. (This analogue, if true, would tell us much about the irreducible modules for finite groups over finite fields.) This material is taken from my article "Simulating algebraic geometry with algebra I: The algebraic theory of derived categories", which will appear in the proceedings of the Arcata '86 conference on representations of finite groups. Quasi-hereditarv alaebras We recursively define a class of algebras A. We call them "quasi-hereditary" because the axioms describe properties almost immediate for hereditary algebras (in which every left ideal is projective): Any A in A has a nonzero ideal J such that 1) As a left A-module, J is projective. 2) HomA(J,A/J) = 0, and J(radA)J=O. (That is, the composition factors in J/rad J do not appear in A/J, and rad J = (rad A)J is an A/J module. 3) If J f A, then the factor ring A/J belongs to A. In the hereditary case, one can just take J to be the socle of A. Using Theorem (3.1) in the CPS paper mentioned above, one can show these algebras all have finite global dimension. The cited theorem gives a derived category embedding Db(A/J-mod) -t Db(A-mod), which implies the Ext groups of A/J modules can be computed in A-mod. This can also be used to show that the axioms have left-right symmetry. (Taking duals, we get a similar statement about Ext for right modules. Now the issue is showing Ext$A/J,S) = 0 for S any simple right A-module which is not an

13 DERIVED CATEGORIESANDALGEBRAIC GROUPS 139 A/J module. But clearly Exti(S*,(A/J)*) is a summand of Exti(J/radJ,(A/J)*) = Exti(J,(A/J)*), using the derived category * embedding for left A/J-modules and the injectivity of (A/J). The latter Ext group is of course zero by the projectivity Notice the mering imposed by the above axioms. level the stratification in the left ideal structure which is This reflects at a ring-theoretic of their module derived categories. To illustrate the left ideal structure, and the difference between hereditary and quasi-hereditary of J.) in simple cases, we give below the decomposition of the left regular representation in schematic form of two algebras. modules, labeled a and b, of dimension one. Each has exactly two irreducible a [I [b] b e b [I [I Hereditary $asi-hzreditary The classical Schur algebras described in Green's Springer Lecture Notes turn out to be examples of quasi-hereditary algebras. The Schur algebras have been generalized independently by Donkin and CPS, using categories of modules with suitable bounds on the dominant weights they involve, to all semisimple algebraic groups, in arbitrary characteristic. It seems likely that similar Lie algebra categories can be treated in the same way. Concludina remarks. We have seen recently that a very non-classical view of structure in algebra not only makes sense, but was even an essential ingredient in the proof of the important Kazhdan-Lusztig conjecture regarding certain infinite dimensional representations of semisimple Lie algebras over (f. This proof used much additional machinery such as stratified topological spaces. There is, however, an analogous and equally interesting conjecture of Professor Lusztig for algebraic groups in characteristic p, where much of the machinery just mentioned is not available. This conjecture is very important for the

14 140 LEONARDL.SCOTT representation theory of finite groups over finite fields. We have begun the investigation of a class of algebras where some parts of the Lie algebra proof can be understood--especially those involving stratified spaces, derived categories, and perverse sheaves--in a purely algebraic way. We hope to deepen and extend this algebraic framework so that eventually it is possible to prove the Lusztig conjecture for characteristic p algebraic groups within it, and in the process obtain a more algebraic proof of the original Lie algebra conjecture.

15 DERIVEDCATEGORIESANDALGEBRAICGROUPS 141 Some referee [Bon1 K. Bongartz, Tilted algebras, Springer-Verlag Lecture Notes in Math. 903 (1982), [BrBl S. Brenner and M. C. R. Butler, Generalizations of the Bernstein-Gelfand-Ponomarov reflection functors, Proc. Int. Conf. on Representations of Algebras, LNM 832, Springer-Verlag 1980, [BBDI A. Beilinson, J. Bernstein, and P. Deligne, Analyse et topologie sur les espaces singuliers, Asterisque 100, sot. math. France (1982). [=I J. Brylinski and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. math. 64 (1981), [CPSl] E. Cline, B. Parshall, and L. Scott, Derived categories and Morita theory, J. Algebra 104 (1986), [CPS2] E. Cline, B. Parshall, and L. Scott, Algebraic stratification in representation categories, to appear in J. Algebra. [Don1 S. Donkin, On Schur algebras and related algebras,i,ii preprints. The first appears in J. Algebra 104 (1986), ; the second is to appear in J. Algebra. 161 J. A. Green, Polynomial representations of Gl,, Springer-Verlag Lecture Notes in Math. 830 (1980). [Harl R. Hartshorne, Residues and dualitv, Springer-Verlag Lecture Notes in Math. 20 (1966).

16 142 LEONARD L. SCOTT [Hall 13. Happel, Triangulated categories in representation theory of finite dimensional algebras, preprint, January [Ha21 D. Happel, On the derived category of a finite-dimensional algebra, preprint (apparently a revision of Hal). [HaRl D. Happel and C. Ringel, Tilted algebras, Trans. Amer. Math. Sot. 274 (2)(1982), [KLll D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke Algebras, Invent. Math. 53 (1979), [KL21 D. Kazhdan and G. Lusztig, Schubert varieties and Poincare duality, Proc. Symp. Pure Math. 36, AMS 1980, [Ll G. Lusztig, Some problems in the representation theory of finite Chevalley groups, Proc. Symp. Pure Math. 37, AMS 1980, [Mel Z. Mebkhout, Une equivalence de categories et une autre equivalence de categories, Compo. Math. 51 (1984), 51-62; [Sl L. Scott, Representations in characteristic p, Proc. Wmp. Pure Math. 37, AMS (1980) Department of Mathematics University of Virginia Charlottesville, VA U.S.A. May 1987