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J. reine angew. Math. 391 (1988), 85 - 99 Journal fur die reine und angewandte Mathematik 0 Walter de Gruyter Berlin New York 1988 Finite dimensional algebras and highest weight categories ‘) By E. Cline at Worcester, B. Pm-shall at Urbana and Charlottesville and L. Scott at Charlottesville This paper continues the program begun by us in [8j2), [9] (see also [15], [18]) in which the authors have begun to exploit in the modular representation theory of semisimple algebraic groups some of the powerful techniques of the theory of derived categories. As noted in the above references, the inspiration for this work comes both from geometry, in the form of the classic algebraic work of Bernstein-Beilinson-Deligne [l] on singular spaces and perverse sheaves, and from the tilting theory of finite dimensional algebras [2], [3], [13], [14]. The present paper broadens and extends this connection with finite dimensional algebra representation theory into a central theme. We begin in Section 1 by completing the results of [9], $ 1, which dealt with “recollement” of triangulated categories in the sense of [l]. We apply this work in Section 2 to the situation of module categories. While [9], 9 3, treats the case of the natural exact functor Db(mod-B) -+ Db(mod-A) of derived categories arising when B is a quotient ring of A, we consider in this paper the “dual” situation in which A is a centralizer ring A = End(eB) g eBe, e E B an idempotent. We remark that our interest in this situation was first kindled by Green’s treatment of the Schur algebra in [12], 9 6. Also, it turns out to lit very well with the stratification theory begun in [9]. In Section 3, we define the unifying concept of a highest weight category. Although we obtain this notion by abstracting from the classical representation theory of semisimple groups (or Lie algebras), other examples given in [16], ?$j5, 6, indicate that such categories arise in many (perhaps surprising) situations, including quiver algebras and constructible and perverse sheaves. Theorems 3. 4 and 3. 6 relate the theory of highest weight categories to the theory of quasi-hereditary algebras (intro- duced in [18]). Th ese results especially appear to provide a strong link between the representation theory of finite dimensional algebras and that of semisimple groups and Lie algebras. Theorems 3. 5 and 3. 9 indicate how the recollement set-up works for the derived categories associated to highest weight categories. In particular, Theorem 3. 9 ‘) Research supported in part by N.S.F. ‘) This paper extended some results of Happel [14] and broadened their scope into the beginnings of a Morita theory for derived categories. We record here that further notable progress in this direction has recently been made by Rickard [17]. See also [ 161, 4 3.

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Page 1: Finite dimensional algebras and highest weight categories ‘)people.virginia.edu/~lls2l/finite_dimensional.pdf · Cline, Parshall and Scott, Finite dimensional algebras and highest

J. reine angew. Math. 391 (1988), 85 - 99 Journal fur die reine und angewandte Mathematik 0 Walter de Gruyter Berlin New York 1988

Finite dimensional algebras and highest weight categories ‘)

By E. Cline at Worcester, B. Pm-shall at Urbana and Charlottesville and L. Scott at Charlottesville

This paper continues the program begun by us in [8j2), [9] (see also [15], [18]) in which the authors have begun to exploit in the modular representation theory of semisimple algebraic groups some of the powerful techniques of the theory of derived categories. As noted in the above references, the inspiration for this work comes both from geometry, in the form of the classic algebraic work of Bernstein-Beilinson-Deligne [l] on singular spaces and perverse sheaves, and from the tilting theory of finite dimensional algebras [2], [3], [13], [14]. The present paper broadens and extends this connection with finite dimensional algebra representation theory into a central theme.

We begin in Section 1 by completing the results of [9], $ 1, which dealt with “recollement” of triangulated categories in the sense of [l]. We apply this work in Section 2 to the situation of module categories. While [9], 9 3, treats the case of the natural exact functor Db(mod-B) -+ Db(mod-A) of derived categories arising when B is a quotient ring of A, we consider in this paper the “dual” situation in which A is a centralizer ring A = End(eB) g eBe, e E B an idempotent. We remark that our interest in this situation was first kindled by Green’s treatment of the Schur algebra in [12], 9 6. Also, it turns out to lit very well with the stratification theory begun in [9].

In Section 3, we define the unifying concept of a highest weight category. Although we obtain this notion by abstracting from the classical representation theory of semisimple groups (or Lie algebras), other examples given in [16], ?$j 5, 6, indicate that such categories arise in many (perhaps surprising) situations, including quiver algebras and constructible and perverse sheaves. Theorems 3. 4 and 3. 6 relate the theory of highest weight categories to the theory of quasi-hereditary algebras (intro- duced in [18]). Th ese results especially appear to provide a strong link between the representation theory of finite dimensional algebras and that of semisimple groups and Lie algebras. Theorems 3. 5 and 3. 9 indicate how the recollement set-up works for the derived categories associated to highest weight categories. In particular, Theorem 3. 9

‘) Research supported in part by N.S.F. ‘) This paper extended some results of Happel [14] and broadened their scope into the beginnings of a

Morita theory for derived categories. We record here that further notable progress in this direction has recently been made by Rickard [17]. See also [ 161, 4 3.

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86 Cline, Par-shall and Scott, Finite dimensional algebras and highest weight categories

completes (and considerably extends) our previous results [9], Thms. 4.4, 4.7, obtained for representation categories of algebraic groups.

As shown in [16], 4 5, suitable categories of perverse sheaves in the sense of [l] may be viewed as highest weight categories. Such categories of perverse sheaves have already figured prominently in the solution of the Kazhdan-Lusztig conjecture for Verma modules of complex semisimple Lie algebras (see [18]). We intend eventually to bring these matters to bear on the long outstanding problem of giving a modular Weyl character formula for semisimple groups (on which an analogous conjecture has been formulated by Lusztig).

We take this opportunity to thank the referee for various comments improving the exposition, and, in particular, for suggesting a shortening of the proof of Lemma 3. 4.

6 1. Recollement

For an abelian category d, let Db(&‘) (respectively, D+ (~2’) D- (~4)) be the associated derived category of bounded (resp., bounded below, bounded above) complexes in d. As a general reference for triangulated categories (of which derived categories are particular examples), we refer to [ 191.

We consider triangulated categories g’, 9, a”, together with exact functors 9’ i, 9 * 9” which satisfy the “recollement” conditions Cl], 1.4. 3. l-l. 4. 3. 5. Recall that this means the following conditions are satisfied:

(a) i, = i! admits an exact left adjoint i* and an exact right adjoint i!;

(b) j* =j! admits an exact right adjoint j, and an exact left adjoint j!;

(c) j* i, = 0 (and h ence, by adjoint properties, i*j! = 0 and i!j, = 0);

(d) given X in 2, there exist (necessarily unique) distinguished triangles

i!i!X+X-+j,j*X+ and j,j!X+X+i,i*X--+

in 9;

(4 i,, j.+, j! are full embeddings.

For specific applications (cf. Section 2), it is important to know that the axioms (a)-(e) above can be obtained from only partial information. For example, as shown in [9], Thm. 2. 1, this occurs whenever we are given a full embedding i,: 9’ -+ 2 of triangulated categories which admits exact left and right adjoints i* and i!. In this case, 9” is realized as the quotient category a/& of 9 by the strict image 6 of i,, and j* is the corresponding quotient morphism. In the following result, we show that the above set-up follows given a “quotient morphism” j *: 9 -+ 9” admitting suitable adjoints.

Theorem 1. 1. Let j ‘*: 93 + 9” be an exact jiinctor of triangulated categories. Assume that j* admits an exact right adjoint j,: 9” * 3 and an exact left adjoint j!: 9” -+ 93, and that both j*, j! are full embeddings.

Let $3’ be the full, triangulated subcategory of 9 whose objects consist of those objects X satisfying j*(X) E 0, and let i,: 9’ --+ 9 be the inclusion functor. Then i, admits

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an exact left adjoint i *: 9 --+ 9’ and an exact right adjoint i!: 9 + 9’ SO that

(9, 9, 9, i*, i,, i!, j!, j*, j,) satisfies the recollement axioms (a)-(e) above.

Proof. It is immediate, from the exactness of j *, that 23’ is a triangulated (in fact, Cpaisse) subcategory of 9. To define i!, we consider, for an object X in 9, the

adjunction morphism X + j, j*X, and form the corresponding distinguished triangle Z-+X-+j,j*X+. Then j*ZrO, so we put i!X=ZEOb(S). Given a morphism g: X + Y, there exists a unique morphism i’(g): i!X + i! Y so that (i’(g), g, j, j*(g)) is a morphism of the distinguished triangle i!X + X --t j, j*X --+ to the distinguished triangle i! (Y) + Y + j, j* (Y) -+ because Horn (i!X, j, W) = 0 for an object W in 23” (cf. [1], Prop. 1. 1.9). This defines i! as a functor from 23 to 3’.

To see that i! is exact, let X -+ Y -+ Z -+ X [l] -+ be a distinguished triangle. Apply the “9-lemma” of [l], Prop. 1. 1.11, (suitably shifted) to extend the commutative square

to a commutative diagram

X’ > x + j,j*X -

Y' - Y + j,j*Y -

X’C11 ’ xc11 + j,j*X Cl1 -

in which the rows and columns are distinguished triangles. Note the natural identification of X ’ -+ Y’ with i!X -+ i! Y Thus, Z’ belongs to a’, Z” belongs to F, and it follows easily that Z’ -+ Z identifies with i!Z -+ Z. Now Y’ --+ Z’ identifies with i! Y --+i!Z, and the exactness of i! follows.

Because Horn, (i,X, j, Y) = 0 for X E Ob (W), YE Ob(S’), it is clear that i! is right adjoint to i,. By definition, the adjunction morphism Id,, --+ i!i, is an isomorphism.

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A similar argument (or duality), using the adjunction map j! j* + Id,,,, establishes the existence of the required exact left adjoint i* to i,. By construction, the axioms (a)-(e) above are satisfied, completing the proof. 0

For future reference, we record the following useful observation.

Remark 1.2. Let g be an abelian subcategory of a triangulated category 9. Let 9’ = gB be the full triangulated subcategory of CS generated by 6? (in the sense that 9’ is the smallest strict triangulated subcategory of 9 containing a). Assume that the inclusion g c 6S is induced by an exact functor i, : Db(9#) + 5S which factors through 3’ and which is a full embedding. Then i, induces an equivalence Db(9?)~9’ = CSB of triangulated categories. 0

Implicit in the following example, and in many arguments below, is the fact that a full embedding i, : Db(mod-B) + Db(mod-A) of derived categories induces an isomor- phism Ext; (M, N) g Exta (M, N) of Ext-groups for arbitrary B-modules A4 and N.

Example 1. 3. Let B = A/J be a quotient algebra of a finite dimensional algebra A defined over a field k. Assume that (a) Exti (BA, DA) = 0 for n >O, and (b) B has finite projective dimension as a right A-module. Then, by [9], Thm. 3. 1, the natural exact functor i, : Db (mod-B) -+ D’(mod-A) induced by the inclusion mod-B c mod-A is a full embedding. In particular, this implies that mod-B is a thick subcategory (i.e., is closed under extensions) of mod-A. Also, it is easy to see that mod-B generates the strict, triangulated subcategory Dk,,-, (mod-A) of Db(mod-A) consisting of objects which have cohomology in the subcategory mod-B of mod-A. Therefore, by (1. 2), we have that Db(mod-B) is equivalent to Dk,,-,(mod-A). 0

8 2. Application to rings

In this section, we apply Theorem 1. 1 to derived categories arising from module categories for rings and algebras. Thus, let A be a ring with an idempotent e and consider the exact, additive functor

j* : mod-A + mod-eAe,

MwMe

Then j* admits a right adjoint j, defined by j, N = Hom,JA, N) g HomeAe(Ae, N) as well as a left adjoint j! defined by j! N = N OeAe A z N OeAe eA.

We next consider the corresponding derived functors. By abuse of notation, we often denote the derived functor R+ F : Df (&) -+ D+ (S#) of a functor F: S# -+ B by the same symbol F.

Proposition 2. 1. Let A be a ring, e E A an idempotent. Put 9 = D’(mod-A), 9” = D+ (mod-eAe), and let j*: 23 -+ 9” (respectively, j*: 9” -+ 9) also denote the derived functor induced by j* (resp., j,) above. Then (j*, j,) is an adjoint pair such that the adjunction map j*j, + Id,,, is an isomorphism. In particular, j, is a full embedding, so that Extz,, (M, N) E Horn”, (j, M, j, N) for all n 2 0 and all eAe-modules M, N. If 9 (resp., 9”) is replaced by D- (mod-A) (;esp., D- (mod-eAe)) and j, by j!, then, in a similar

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way, (j!, j*) is an adjoint pair such that the adjunction map Id,,, -+ j*j! is an isomorphism (and j! is a fill embedding).

Proof. Because the functor M H Me of module categories is exact as a functor on abelian categories, the derived functor j*: 9 --+ 9” is defined by j*(X’)” = X”e for each complex X*=(X”) in 9. Exactness also implies that the adjoint functor j, : mod-eAe + mod-A (respectively, j! : mod-eAe --+ mod-A) takes injective (resp., projec- tive) eAe-modules to injective (resp., projective) A-modules. Since Df (mod-eAe) (resp., D-(mod-eAe)) is equivalent to the homotopy category K+(y) (resp., K-(g)), where #J (resp., ??) is the full subcategory of mod-eAe consisting of injective (resp., projective) eAe-modules, the proposition follows immediately from the standard identities:

HomeA, (Ae, M) e E Horn,,, (eAe, M) z M,

wf &e eA)e E M OeAe eAe E M,

which are valid for an arbitrary eAe-module M. 0

Example 2. 2. Let k be an algebraically closed field. For positive integers r and n, we let A= S,(n, r) be the Schur algebra corresponding to rational representations of GL, (k) which are homogeneous of degree r (cf. [12]). F or r 5 n, there is an idempotent e such that eAe g kS,. Therefore, (2. 1) yields a method of calculating cohomology of the symmetric group S, in terms of the hypercohomology of the algebra A. While in positive characteristic p, the group algebra kS, is of infinite global dimension when plr, the Schur algebra A has finite global dimension [lo] ( see also the discussion in [9], 4. 5d)!

In what follows we seek conditions analogous to [9], Theorem 3. 1, which yield the recollement of Section 1 for the bounded derived categories of mod-A and mod-eAe, where e is an idempotent of A. In the following first illustration, let B = A/AeA, the largest quotient ring B of A satisfying Be = 0. Thus, mod-B consists of all right A- modules M satisfying Me = 0. Note that because M H Me is an exact functor, mod-B is a thick subcategory of mod-A. Hence, we can consider the full, triangulated subcategory Dk,,-, (mod-A) of Db (mod-A) consisting of those complexes with cohomology in mod-B.

Theorem 2.3. Let A be a ring and e an idempotent such that eAe has finite right global dimension. Put B = A/AeA, and set 9 = Db(mod-A), 6V = Db(mod-eAe). Then the adjoint functors (j!, j*, j,) of (2. 1) on the module categories mod-A and mod-eAe induce adjoint finctors (j!, j*, j,) on 9 and 9’ which satisfy the hypotheses of (1. l), thus yielding recollement in which i, 9’ identifies with the relative derived subcategory

D&,,-,(mod-A) of 9.

Proof. Because eAe has finite right global dimension, the functors j!, j, of Proposition 2. 1 map Db(mod-eAe) to Db(mod-A). Thus, the triple (j!, j*, j,) as functors on Db(mod-A) and Db(mod-eAe) satisfy the conditions of Theorem 1. 1. It remains to identify 9’. Since a complex in Db(mod-eAe) is isomorphic to the zero complex if and only if it is acyclic, it follows from the exactness of the functor M H Me that a complex X’ in Db(mod-A) satisfies j* (X’) g 0 if and only if X’ has cohomology groups satisfying H”(X’)e = 0, that is, if and only if X’ belongs to the relative derived category

Dk,,-,(mod-A). 0

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Remark 2.4. In at least one natural application to quasi-hereditary algebras, we will be able to identify DL,,-,(mod-A) as Db(mod-B), cf. 9 3. Another example of this, inspired by [l 11, is the following result.

Corollary 2.5. Let A be a finite-dimensional algebra with idempotent e. Put f = 1 -e and assume that eAf = 0, so that both eAe and fAf are quotient algebras of A. Then A has finite global dimension if and only if both eAe and fAf have finite global dimension. In particular, if A has finite global dimension, then the conclusion of (2. 3) holds, and we have recollement:

Db(mod-eAe) s Db(mod-A) s Db(mod-fAf).

Proof. First observe that the condition eAf =0 implies that fAf + fAe = fA is a two-sided ideal J of A satisfying A/J E eAe. Clearly, Horn, (JA, eAe) = 0. Assume that A has finite (left or right) global dimension. Since J is a projective right ideal, [9], 3.6, implies that the recollement set-up of 9 1 holds. Thus, eAe has finite (left or right) global dimension by [9], 3.3, 3.4. A similar argument implies fAf has finite global dimension. Conversely, if eAe and fAf have finite global dimensions, A does also (cf. [ll], Cor. 3. 6, for example). The recollement assertion follows from (1. 3) and (2. 3) since

AfA = fA. 0

One can also generalize the “trivial extension” process used in [l l] for constructing triangular matrix rings to a procedure whereby essentially all quasi- hereditary algebras can be recursively constructed (cf. [16], Theorem 4. 6).

9 3. Highest weight categories

Let %? be an abelian category over a field k. (Much of the development below appears to carry over, mutatis mutandis, when k is replaced by a commutative Artinian ring, though we have not pursued this.) Thus, Horn sets are k-vector spaces such that morphisms in either variable induce k-linear maps. We say that %? is locally artinian if it admits arbitrary directed unions of subobjects, and if every object is a union of its subobjects of finite length. In addition, we assume that %? contains enough injectives and that it satisfies the Grothendieck condition: B n (u A,) = u (B n A,) for a subobject B and family of subobjects {A@} of an object X. A composition factor S of an object A in V is, by definition, a composition factor of a subobject of finite length. The multiplicity (possibly infinite) of S in A, denoted [A: S], is defined to be the supremum of the multiplicity of S in all subobjects of A of finite length.

We say that a poset A is interval-finite provided that, for every ,u 5 1 in /1, the “interval” [pJ]={zd lp&S;z} is finite. We now present the following basic definition.

Definition 3. 1. A locally artinian category %? over k as above is called a highest weight category if there exists an interval-finite poset A (the “weights” of %?) satisfying the following conditions :

(a) There is a complete collection {S(n)}n,, of non-isomorphic simple objects of %? indexed by the set A.

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(b) There is a collection {A(,?)},.. of objects of %7 and, for each 2, an embedding S(Qc A(A) such that all composition factors S(p) of A(i)/S(i) satisfy ~<2. For 2, /J E A, we have that dim, Hom,(A(Q A(p)), and [A(A): S(p)] are finite.

(c) Each simple object S(2) has an injective envelope Z(2) in Q?. Also, Z(i) has a “good filtration” which begins with A(1) -- namely, an increasing (finite or infinite) filtration 0 = F,(2) c F, (2) c ... such that

(i) F, (2) z A(i);

(ii) for n>l, F,(A)/F,-,(A)EA(~) for some p=p(n)>&

(iii) for a given p E A, p(n) = p for only finitely many n;

(iv) u Fi(i)= Z(i). 0

If %’ is a highest weight category, let $7’ be the full subcategory of %? consisting of all objects of finite length. Observe that qJ determines %? by taking direct limits, though Vf rarely contains enough injectives.

We record several immediate consequences of the above axioms for a highest weight category $2 in the following result.

Lemma 3. 2. Let $9 be a highest weight category with poset A of weights. Let il, p E A. Then:

(a) S(2) is the socle of A(A).

(b) If either Extk(A(y), A(I)) or Ext&(S(p), A(1)) is nonzero, then necessarily ,a>). Zf Exti(S(1), S(p)) + 0, then i and p are strictly comparable (i.e., either I > ,a or ,a > 2).

(c) Zf M, N are objects in %? of finite length, then Hom,(M, N) and Exti (M, N) are finite dimensional.

(d) The filtration {F,,(I)) in axiom 3. 1 (c) can be chosen to satisfy the additional condition:

(v) for all i, j>O, if,a(i)<p(j) then i<j.

(Remark: Several of the results above involving Ext’ will be extended to Ext” in . Lemma 3. 8 below.)

Proof. Clearly, (a) is immediate from axiom 3. 1 (c). If Exti(S(z), A(i)) =I= 0, then Horn, (S(r), 1(2)/A(A)) =+ 0. By 3. 1 (c) (iv), it follows that Hom,(S(z), Fj(~)/F, (A)) =+= 0 for some j > 1, and from this and (a) above that for some i 2 1 we have Fi(1)/Fi-,(il) g A(z). Now suppose that Extk(A(p), A(1)) + 0. Th en for some composition factor S(z) of A@) we find that Exti(S(z), A(i))=+0 so that p 2 z ~1. This completes the proof of the first part of (b). If Ext&(S@), S(p)) +O, then Hom,(S(i), ~(,u)/S(~))+O. If Horn, (S(i), A(p),%(p)) + 0, then il< p. Otherwise, Horn, (S(i), Z(p)/A(p)) + 0. Thus, for some j> 1, Hom,(S(A), Fj(p)/A(p))+ 0. Since S(r) is the socle of A(z), it follows easily from this that A(i) g Fi(cL)/Fi-, (p) for some i > 1. Hence, 2 > ,u, completing the proof of (b). Clearly, (b) implies (d).

To prove (c), note that Ext&(A@)/S(Q A(I))=0 by (b), so that Hom,(S(A), A(A)) is a homomorphic (even isomorphic) image of Hom,(A(il), A(i)) and hence is finite dimensional by axiom 3. l(b). Thus, the (isomorphic) submodule Hom,(S(i), S(2)) of

HOme (S (4, A (4) is finite dimensional. It follows easily by dimension shifting from

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3. 1 (c) that Extk (S(i), A(p)) is finite dimensional. Finally, the long exact sequence of Ext&(S(1), -) applied to the exact sequence 0 + S(p) --+ A(p) --+ A(p)/?(p) + 0 shows that Extk(S(A), S(p)) is finite dimensional, completing the proof of (c). 0

We now briefly present several examples of highest weight categories. More details, as well as many more examples, can be found in [16], 9 6.

Examples 3.3. (a) Let A be the k-algebra of n x n upper triangular matrices, and put %‘=mod-A. Let A = (1, 2, . . . , n}, and for 1 iisn, let A(i)=S(i) be the irreducible (one-dimensional) right A-module whose injective envelope I(i) has dimen- sion n - i + 1. It is readily verified that Z(i)/S(i) g I(i + 1) for i < n. It follows easily that %7 is a highest weight category.

(b) More generally, let R be a hereditary algebra over a field k. Let J be the socle of RR. Then J is a two-sided ideal of R, projective as a left-module. Thus, Ext;(J, R/J) = 0, so that by [9], Thm. 3. 1, R/J is also an hereditary algebra. Partially order the simple left R-modules compatibly with the socle series {soc,(,R)} as follows. Given a simple module S, there exists a unique integer n(S) such that S is isomorphic to a summand of soc,(,R). Given two simple modules S, and S,, put S, > S, provided n(S,)< n(S,). It follows that the composition factors S’ of the radical of the projective cover P(S) of a simple module S satisfy S’ > S. Now let q = mod-R. The simple objects in % are the linear duals S* of the simple left R-modules, and inherit their partial ordering. Putting A(S*) = S*, it follows immediately from the above remarks on

projective covers that %? is a highest weight category.

(c) Let $? be a complex semisimple Lie algebra and let % be the category 0 of Bernstein-Gelfand-Gelfand. Take for A the set of integral weights on a fixed Cartan subalgebra 2 of 3. The category 0 admits a duality functor * [4], and for each integral weight i, let A(l) = V(A)* be the dual of the Verma module v(1) of high weight 1. In this way, 0 g%” for a highest weight category %7. To verify this, it is sufficient using arguments analogous to [9], 0 4, to verify that Extk (I’(n), V(p)) + 0 implies A< p. However, in this case, 0 =l= Exth(,,(l/(il), V(p)) r Extb(,*,(& V(p)). If u I$ V(p) is a nonzero I-weight vector in a nonsplit extension of 1 by V(p), we must have that e, . v +O for some positive root vector e,. Hence, ;1+ a is a weight in I’(U) and so I< ,D.

(d) Let G be a semisimple algebraic group defined and split for FP. Put k = Fp, the algebraic closure of F,. Fix a maximal split torus T contained in a Bore1 subgroup B (corresponding to the negative roots). Let A =X(T) be the character group of T, partially ordered in the usual way. Fix an integer r >O, and let %? be the category of rational TG,-modules (cf. [S], [6] for a discussion of the infinitesimal thickenings TG, of T and BG, of B). The irreducible TG,.-modules are indexed by A. For each 1, set A(l) = II;;;. Then %? becomes a highest weight category. Similar remarks apply to the category of rational BG,-modules and the category of rational G-modules (cf. [16], $6, for more details). 0

Fix a field k. We recall the definition of the class Z! =2!(k) of (right) quasi- hereditary algebras, given in [18]. 2 is the class of finite dimensional algebras A over k defined recursively by assuming that A has a nonzero ideal J such that

(i) J is projective as a right A-module;

(ii) Horn, (JA, A/J) = 0 and J . rad (A) . J = 0;

(iii) if J + A, A/J belongs to 2.

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Cline, Parshall und Scott, Finite dimensional algebras and highest weight categories 93

Thus, if A is a quasi-hereditary algebra, there exists a strictly increasing sequence of ideals 0 = J,, c J, c J, c ... c J, = A such that each A/Ji is quasi-hereditary and Ji+l/Ji is a projective right ideal in A/Ji satisfying the quasi-hereditary condition above for J. We call such a sequence {Ji} of ideals a “defining system of ideals for the algebra A”.

Among the properties of a quasi-hereditary algebra A which we will use, we record the following (cf. [16], Theorem 4.3, [18]): (a) A has finite global dimension; (b) A is quasi-hereditary if and only if the opposite algebra AoP is quasi-hereditary; and (c) the inclusion i, : Db(mod-A/J) + Db(mod-A) is a full embedding of triangulated categories, and gives rise to a recollement

Db (mod-A/J) 3 Db (mod-A) 3 9”

of Db(mod-A), as discussed in 5 1 (with 9” as the derived category of a semisimple algebra).

The following key step is the first half of a strong connection between quasi- hereditary algebras and highest weight categories.

Lemma 3.4. Let A be a quasi-hereditary algebra over k. Then the category $?=mod-A of right A-modules is a highest weight category.

Proof. By property (b) of quasi-hereditary algebras mentioned above, the opposite algebra AoP is quasi-hereditary. Thus, let 0 = JO c J, c J, c ... c Jt = AoP be a defining sequence of ideals for AoP. We view Ji as an ideal in A so that the ideal Ji/Ji-1 in A/Jipl is a projective left ideal, etc. The process S H S* of taking linear duals establishes a bijection between the simple right A-modules S and the simple left A- modules S*. It is clear from the axioms for a quasi-hereditary algebra that, given a simple left A-module S *, there exists a unique integer i E [l, t] such that S* occurs in the radical quotient (Ji/Ji-l)/rad (Ji/Ji-l) of the left A-module Ji/Ji-1. Let Ai be a set indexing the irreducible left A-modules (2 H S(i)*) in the radical quotient of Ji/Ji-1 and put A = u Ai. We partially order A by setting 1”~ p iff 2 E Ai, ,U E Aj and i >j.

For A E Aj, let A(,?)* be the projective cover in A/A-i-mod of S(1)*. Note that A(Ji/Ji-l) is a direct sum of copies of A@)*, i E Ai. In fact, it is clearly sufficient to establish this for i = 1. However, since AJ1 is projective, it is a direct sum of projective indecomposable modules. Since Horn, (AJ1, A(A/J1)) = 0, these indecomposable sum- mands must be of the form A(I)*, i E A,.

Since clearly Horn, (,(Ji/Ji- 1), rad, (Ji/Ji-,)) = 0, the composition factors S(y)* of rad (A(I)*), 2 E Ai, satisfy p < 2. Hence, 3. 1 (b) is satisfied for the right A-module A(2).

Finally, let e be a primitive idempotent in A and consider the filtration {,Jie} of the projective indecomposable A-module Ae. Then A(Jie/Ji-1 e) is a direct summand of .(Ji/Ji-l) and hence a-direct sum of copies of A(I)*, 1 E Ai. Taking duals, we obtain an increasing filtration {Fi} of the injective indecomposable right A-module (Ae)* in which the section Fi/Fi-, is a direct sum of copies of A@), 2 E Ai. Clearly, this filtration can be refined to one as required in 3. l(b). This completes the proof that mod-A is a highest weight category. IJ

An ideal r in a poset A is called finitely generated if it is a finite union of sets of the form (- co, y] = {z E A 1 z 5 y}. The finitely many elements y are called generators of A. We can now prove the following local description of highest weight categories.

5, Journal fib Mathematik. Band 391

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94 Cline, Parshall and Scott, Finite dimensional algebras and highest weight categories

Theorem 3.5. Let % be a highest weight category with poset A of weights. Let I c A be a nonempty finitely generated ideal, and let Sz c I be a finite coideal.

(a) Let %?[I] be the full subcategory of %? consisting of objects with composition factors S(p), ,u E I. Then %?[I] is a highest weight category. The filtrations {4(n)} of 3. 1 (c) for qr] are finite.

(Remark : We will see in Theorem 3. 9 that the natural inclusion Db(W[Cr]) ---, Db(%) is a full embedding.)

(b) Let I be the direct sum of the injective envelopes in %‘[I] of the simple objects S(n), II E Q. Then A = A(I, s2) = End,(I) is a quasi-hereditary algebra, with highest weight category V(Q) =mod-A having poset s2 of weights. Furthermore, A is independent of the ideal I, in the sense that if 52 is a coideal in another ideal I’, then A(I’, 0) = A(I, 0).

The functor F = Horn, (-, I)* : Db(W [I]) + Db(V((a)) identifies the full subcategory of Db(V) represented by complexes of finite sums of injectives I (w)r in %‘[I] with o E s2 to Db(modf-A), where mods-A denotes the category %$(a) of finitely generated A- modules.

Proof. For i E I, note first that A(1) belongs to @?[I] since Z is an ideal in A. We use these modules A(i), 1 E I, to satisfy Axiom 3. 1 (b). Next, let I(&- be the largest subobject of I(1) lying in the subcategory %?[I]. Arguing as in [9], Lemma 4. 3, (except now I(& may be equal to Z(l)!), we see that I(&- is the injective envelope of S(1) which has a good filtration as required by Axiom 3. 1 (c) for a highest weight category. (Since A is interval-finite and I is finitely generated, there are only finitely many weights y in I above A. Using 3. 1 (b), we may arrange that all A(y)‘s, for such weights y, which appear as sections of a good filtration of I(n) in fact appear together at the bottom, as the sections of Fk(A) for some k>O. It is then easy to see that Fk(l)= I($.) The remaining assertion of (a) follows since A is interval-finite.

To see (b), first observe that if o E I, the good filtration {Fi(o)} of the injective envelope I(w) of S(o) in %?[I] has only finitely many terms, since I is finitely generated and A is interval-finite. Thus, Horn&I(w), I(p)) is finite dimensional for o, 1 E 0 by 3. 2(c), so that the algebra A= Hom,(I, I) is finite dimensional. Also, note that P(U) = Horn, (Z(a), I) is a projective indecomposable left A-module, and all such projective indecomposable modules arise this way, since clearly A= @ P(w) by

definition. Pick a maximal w E D and let A be the ideal in A generated by a\{~}. Put

4, = @ WA (in th e notation of the previous paragraph). Then I/I, is a direct sum of PER

copies of A(o) = I(U), and J = Horn, (Z/Z,, I) is a projective left ideal of A satisfying Hom,(J, A/J) g Horn, (IA, I/Z,) = 0. Thus, J is the largest left ideal J’ of A such that all the composition factors of A/J’ are distinct from the irreducible A-module S(o)’ = Horn, (S(m), I). It follows from the argument of [9], Cor. 3.8, that J is in fact a two-sided ideal of A. The kernel of the natural map P(U) + S(w)’ clearly does not have S(w)’ as a composition factor, so that J . rad(J) = 0. Finally, note that A/J g Horn, (Id, I) = Horn, (Id, Id) may be assumed quasi-hereditary by induction and (a). This completes the verification of the axioms. Next, note that Z is clearly independent of the containing ideal I, since I can be replaced by the ideal generated in A by Sz. The remaining assertion of (b) is easily obtained. (Note that A has finite global dimension since it is a quasi-hereditary algebra.) 0

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Cline, Parshall and Scott, Finite dimensional algebras and highest weight categories 95

As a corollary of the above results, we have the following fundamental theorem. It provides a bridge between representation theory in the sense of finite dimensional algebras and that in the sense of Lie theory.

Theorem 3.6. Let A be a finite dimensional algebra over a field k. The following are equivalent:

(a) A is a quasi-hereditary algebra;

(b) mod-A is a highest weight category;

(c) the category modf-A of finitely generated A-modules is a highest weight category.

Conversely, given a highest weight category W with finitely many simple objects, the category (8” is a highest weight category; in fact, %Yf g modf-A for some quasi-hereditary algebra A.

Proof. First, (b) e (c) is immediate. By 3. 4, we have that (a) * (b). Now suppose that (b) holds, and we show that A is a quasi-hereditary algebra. Let 1 be a maximal weight so that P(1) = A@)* is the projective cover of the irreducible left A-module S(n)*. Write P(I) = Ae, e an idempotent in A, and let J = AeA. Then J is projective as a left ideal in A (in fact, J is isomorphic to a direct sum of copies of P(1) since an nonzero homomorphism P(A) -+ J is an isomorphism onto its image by 3. 1). Also, Horn, (AJ, A/J) = 0 and J . rad (J) = 0. Finally, it is immediate that mod-A/J g V [Z]. It follows that A is a quasi-hereditary algebra, as desired.

Finally, if %? is a highest weight category with A finite, we apply 3. 5(b) to conclude that %” ~rnmod~-A for a quasi-hereditary algebra A. 0

In contrast to Example 2. 2, we have the following result.

Corollary 3.7. Let A be a quasi-hereditary algebra over k. Let

OcJ,c ... cJ,=A“P

be a defining sequence of ideals. Fix k >O, and let e E A be an idempotent such that the simple quotients of A(1 -e) and AJJ, coincide. Then the centralizer algebra

eAe E End, (Ae, Ae)

is a quasi-hereditary algebra. Moreover, the functor j*: mod-A + mod-e Ae (M H Me) induces a recollement of derived categories:

Dk,,-,(mod-A) 3 Db(mod-A) 3 Db(mod-eAe),

where B = A/AeA. Finally, the natural exact functor Dk,,-,(mod-A) -+ Db(mod-B) is an equivalence of categories.

Proof. Let ZcA be the ideal u Ai (in the notation of the proof of 3.4), and i>k

put fi = A\r, the complementary coideal. Then %‘(a) 2 mod-A’ where A’ = End(Z) with

I= @ Z(0). s ince eAe g End,(Ae, Ae) is certainly Morita equivalent to weR

End (I*) g End (I)“‘,

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96 Cline, Par-shall and Scott, Finite dimensional algebras and highest weight categories

it follows from 3. 5(b) that eAe is a quasi-hereditary algebra. In particular, eAe has finite global dimension and so Theorem 2. 3 applies to give a recollement above.

For the last assertion, note we can adjust e to assume that e = e, + ... + e,, where the e, are primitive orthogonal idempotents such that Aei is not isomorphic to Aej for i +j. If we take e, so that Ae, is the projective cover of S(o)* with o E Q maximal, then clearly J = Ae, A satisfies the conditions (a)-(b) in the definition of a quasi-hereditary algebra. In particular, the morphism Db(mod-A/Ae, A) + Db(mod-A) is a full embedd- ing. By induction on Y, we can assume that Db(mod-A/AeA) + Db(mod-A/Ae, A) is a full embedding. Composing these full embeddings, we can now apply Example 1. 3 to complete the proof. 0

For the next result we require the following lemma, which completes the discussion of Lemma 3. 2.

Lemma 3.8. Let Ce be a highest weight category with poset A of weights, let 1, ,u E A, and let n 2 0 be an integer.

(a) If M is an object of finite length in %I?, and V is any object in $7, then Ext: (M, V) is the union of the images of the various maps Ext”, (M, N) -+ Exts (M, V) for N ranging over the subobjects of finite length of V. Indeed, we have that

ind;Sim. Ext: (M, N) z Ext”, (M, V).

(b) If Ext; (S(u), A(i)) =t= 0, then necessarily p 2 1. Moreover, if n >O, we have

ExGW), 44*0 f or some o E A satisfying u 2 o >,I. In particular, u > 2. Also, n is bounded by the maximal length of any chain of weights between i and p.

(c) For M E Ob(%?) of finite length 1, the vector space Ext&(M, FJA)) is finite dimensional, and is equal to 0 for k sufficiently large, depending on the composition factors of M and on 1 (but not depending on n). In particular, Ext”, (M, A(),)) is finite dimensional and Ext”,(M, A(I)) = 0 for sufficiently large n, depending on the composition factors of M.

If all A(1)‘s have finite length, then for all integers II 2 0, Ext”,(M, N) is finite dimensional for all finite length M, N E Ob(%?).

Proof. First we prove (a). Let M E Ob(%?) have finite length. Observe, if E/V’g M for some E, V’ E Oh(V), then there is a finite length subobject F c E with F + V’ = E. (The proof of this is an easy induction on the length of M. For length 1, choose F to be any finite length subobject with F not contained in V’, etc..) Using M z F/F n V’, we have that the extension E is the image of the extension F under the natural map

Ext.& (M, F n V’) -+ Extk (M, V’).

Suppose, for some V E Oh(V), we have an element of Ext”, (M, V) which can therefore be represented as a Yoneda composite of an element of ExtG-’ (V’, V) and an element /I of Ext&(M, V’) for some V’ E Ob ($9’). Choose F as above for the element b of Extb(M, V’), and put F’ = F n V’, with the inclusion F’c V’ denoted by i. Write I= i,(/?‘) for some p’ E Ext$(M, F’). Thus, a/I = ai* is equivalent to i*(cr) p’. By induction on n, i*(a) E ExtG-l (F’, V) is the image, under j, for an inclusion j: N c V for a finite length submodule N of V, of an element a’ E ExtG-l (F’, N). Then j,(a’fY)=j*(cz’)f?‘=i*(a)f?‘Z:fi, p roving the first part of (a). The second part follows from the validity of the first for all n. (If y E ExtG(M, N) maps to zero in Ext!$(M, V), then y is in the image under the natural map Ext”,-’ (M, V/N) -+ Extb(M, N). Now

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Cline, Parshall and Scott, Finite dimensional algebras and highest weight categories 97

apply the first part above with V replaced by V/N and n replaced by n - 1, to conclude that there is a finite length subobject fi of V, containing N, with y mapping to zero under Ext”,(M, N) + ExtG(M, G).) This completes the proof of (a).

To prove (b), we may assume that n >O. Suppose y E Ext”,(S(p), A(1)) is nonzero. Regard A(A) c Z(A) and note that Ext”, (S(p), I(I)) = 0. Applying (a) (both parts, if necessary), we find that the image of y in Exti(S(p), Fk(IW)) 1s zero for k sufficiently large. Thus, y is the image of a nonzero element of ExtG-’ (S(p), Z(,I)/Fk(I)). Applying (a) again,

we find Ext”,-’ (S(y), Fktm(,4/Fk(4) =I 0 f or some m >O. Now (b) follows easily from the long exact sequence of cohomology, induction, and 3. 1 (c).

To prove (c), we note that there are only finitely many terms F,(Q/F,,-, (1”) 2 A(z,) with ,LL~ z,zJ. for some /J with S(p) a composition factor of M. Thus, for m sufficiently large, we have that Ext”, (M, F,(I)/F,,-, (1)) = 0 f or all integers n by (b). Choose k so that this occurs for all m> k. Then the long exact sequence of Ext, together with (a) applied to I(A)/Fk(A), shows that Exts (M, Fk(A)) E Ext: (M, L(1)) = 0.

Next observe that Ext”,(M, FkPl (2)) can be computed in terms of Ext”,-’ (M, A(T,)) and Extb (M, Fk(2)) = 0. Applying induction on n, we obtain that Extb (M, Fk-l (A)) is finite dimensional. Similarly, Extb (M, F,(1)) is finite dimensional for 1 = k - 2, . . . , 1. Thus, Extk (M, F,(A)) is finite dimensional for all integers 12 0. It is also clear (using the fact that /i is interval-finite) that ExtG(M, A(1)) = 0 for n B 0.

If the A(i)‘s are all of finite length, then so are the Fk(A)‘s. Obviously, any object N of finite length embeds in a direct sum of finitely many injectives 1(;1)‘s, and thus into a direct sum L of finitely many Fk(i)‘s. The statements regarding Extb(M, N) now follow easily by induction on n, applied to the quotient L/N. This completes the proof of (c), and hence of the lemma. 0

The following results show that highest weight categories admit stratifications at the derived category level resembling those for constructible sheaves on topological spaces.

Theorem 3.9. Let (8 be a highest weight category with poset A of weights.

(a) Given a finitely generated ideal r c A, the natural inclusion %‘[r] c %? (cf (3. 5)) induces a full embedding D”(??[Cr]) --* Db(W) of triangulated categories.

(b) Let rl c T, c T3 be ideals in A, such that T, is finitely generated and T3\T1 is a finite set. Note that r,\r, is a finite coideal in r,, so that %?((T,\T,) is defined (cf 3. 5(b)). Also, r,\r, is an ideal in r,\r,, so that %? [r,\r,] is defined (cf 3. 5(a)). Then we have a recollement

Proof, To prove that Db(Q?[Cr]) + Db(%?) is a full embedding, it is sufficient to prove the stronger result that i, : Df (W [r]) -+ Df (59) is a full embedding. We do this by exhibiting a right adjoint i! such that the adjunction morphism Id -+ i!i, is an isomorphism. We claim that such an adjoint is given by putting i! = R+ F, for F the left exact functor F: %? + Ce [r] which assigns to M E Ob (%) its largest subobject M,, all composition factors of which belong to %?[Cr]. Certainly, i! is right adjoint to i,, since i, is exact and left adjoint to F at the category level. Also, the composition of F with i, is the identity on %?[r], so it suffices to check that F(J) is acyclic for J an injective object

52 Journal fib Mathematik Band 391

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98 Cline, Parshall and Scott, Finite dimensional algebras and highest weight categories

in % rr]. We note that J has the form I, = F(Z) for some injective object I in %‘. If R” F(Z,) + 0 for some n > 0, then R” F(I/I,) + 0. Note that F may be viewed as a direct limit of all functors Hom,(M, -), for M an object in %?[r] of finite length. Thus, Ext”,(M, I/I,) + 0 for some such M. We claim to the contrary that Ext”,(M, I/Ir) = 0. Without loss, we can assume M = S(y) for some y E r, and I = I(i) for some ;1 E r. The proof of 3. 5(a) shows that we can choose a good filtration {$(A)} of 1(;l) so that Fk(i) = I, for some k. Thus, ExtZ, (S(y), 1(1)/F,(i)) = 0 by 3. 8(b). This completes the proof of (a). Finally, (b) follows easily from Theorem 3. 5 and Corollary 3. 7. 0

Remark 3. 10. As a practical matter, the theorem above holds if one replaces r,\r, with r,, noting that r, is the union of its finite coideals (by the interval-finiteness assumption): For any highest weight category %? with finitely generated poset A of weights, let Of”(%) denote the full strict triangulated subcategory of Db(%‘) consisting of objects represented by bounded complexes of injectives which are direct sums of only finitely many r(n)‘s. Using the identification of 3. 6(b), we have

with r,\r, ranging over the finite coideals of r,. A similar equation holds for L$(%[Cr,]), and it is easy to see that “f” subscripts can be added to the recollement diagram in Theorem 3. 9 above. The morphisms in these recollement diagrams are all compatible with each other, as r1 varies, and so effectively give a recollement diagram using r, itself.

In future work, the authors expect to elaborate on the above paragraph, and to investigate Dfb(%‘) in general defined as the inductive limit of the DJ”(%?[Cr]), for r ranging over the finitely generated ideals of A. (It appears that Of(%) contains the full subcategory of Db(%?) consisting of complexes with finite length cohomology and coincides with this in case all the A(I)‘s have finite length.) 0

Now let %? be a highest weight category with a finite poset A of weights. By Theorem 3. 6, the category %? contains enough projectives. For I E A, let V(2) be the largest quotient module of the projective cover of P(L) in g of S(L) all of whose composition factors S(p) satisfy p 5 1. By 3. 2(b), it follows that any composition factor S(p) of rad (V(A)) satisfies p ~1. We will make use of these modules in establishing the following result concerning the “multiplicity” of A(p) in I(i). The proof makes essential use of the recollement result of Theorem 3. 9.

Theorem 3.11 (Brauer-Humphreys reciprocity). Let %T be a highest weight category with a finite poset A of weights. For i, p E A, the number [I(p): A(i)] of times A(I) appears US a section Fj(~)/Fj~l(Y) in a good filtration {Fj(p)} of I(p) (cf. 3. 1) is independent of the good filtration chosen. In fact, we have

[I(P): A(41 = CV(4: S(PL)I.

Proof. We f t 1 us c aim that Exti(V(i), A(p)) = 0 for all A, p E A. (One can even replace Ext& by Ext”, for n >O here, using 3. 8(b). However, we do not require this stronger fact for the proof.) To see this, we may assume by 3. 2(b) that E,>p. Then V(1) and A(,a) belong to %‘[Cr], r = (- co, 21, and, in fact, V(2) is projective in %?[r]. Hence,

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C/ine, Parshall and Scott, Finite dimensional algebras and highest weight categories 99

by 3. 9(b), we obtain that Extb(V(A.), A(p))= Ext&,(V(A), A(p))=O, as desired. Note next that

Ho% (WJ, A(P)) = End(S(p)) if I= p, o otherwise

Therefore, the functor Horn, (V(n), -) may b e used to count the multiplicity of A(i) in any object with a good filtration. We conclude that

[Z(p): A(i)] =dimHom,(V(A), Z(p))/dimEnd(S(i))= [V(lv): S(p)]. 0

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Department of Mathematics, Clark University, Worcester, MA 01610

Department of Mathematics, University of Illinois, Urbana, IL 61801

Department of Mathematics, University of Virginia, Charlottesville, VA 22901

Eingegangen 3. November 1987