Simulating Algebraic Geometry with Algebra, I - University of Virginia

Proceedings of Symposia in Pure Mathematics
Volume 47 (1987)
Simulating Algebraic Geometry with Algebra, I:
The Algebraic Theory of Derived Categories
LEONARD
L. SCOTT
This article is the first of three in a series describing recent work by CPS
(Ed Cline, Brian Parshall, and myself). I will be concerned here primarily with
describing the context of our work and our results on the algebraic theory of de-
rived categories [CPSl]. I will also mention some interesting finite-dimensional
algebras of finite global dimension which have arisen in our work. Indeed this
work is partly inspired by the theory of finite-dimensional algebras, at least by
Happel’s use [Hal] of derived categories in that theory, and we are very grateful
to Klaus Roggenkamp for directing us to Happel’s work.
Brian Parshall in the second article will apply derived category theory to
stratify the derived category of representations of a semisimple algebraic group
G in characteristic p. This is done using an axiom system modeled by Beilinson-
Bernstein, and Deligne [BBD] on properties satisfied by sheaves on a stratified
topological space. While their point of view may be viewed as that of algebraic
geometry, ours is a little more in the spirit of classical algebra, and in particular,
there is no topological space in sight! It is this stratification which inspired the
title of this series, “Simulating algebraic geometry with algebra,” and has made
us feel very optimistic about our new methods.
The third article, by Ed Cline, is largely self-contained and touches on the
above themes only loosely, in that it involves derived categories (by way of in-
jective modules) and finite-dimensional algebras (but this time not at all of
finite global dimension). Ed’s focus is an approach to the Lusztig conjecture [L]
through understanding injective TGi-modules (modules for the restricted Lie al-
gebra of G together with a compatible action of a maximal torus). The ultimate
goal would be the construction of the injectives. One encouraging ingredient
of the theory to date is a CPS theorem [CPSS] asserting that injectivity of a
module can be determined just by looking at individual root groups. Another
ingredient is a conjecture [C2] of Ed’s, equivalent to the Lusztig conjecture,
1980 Mathematics Subject Classification (1985 Revision). Primary 18E30; Secondary 16A46,
18F20.
This research wsa supported in part by the National Science Foundation.
01967 American Mathematical Society
0082.0717/87 $1.00 + 8.25 per page
271

272 LEONARD L. SCOTT
which describes how injectives behave under Jantzen translation. There is some
suggestion of a Hecke algebra action on injectives here, which ultimately might
easily have a derived category formulation. Finally, Ed discusses some striking
empirical observations he has made [Cl], interpreting in detail certain coefficients
of the affine Kazhdan-Lusztig polynomials as dimensions of sections of
injective modules.
Context: The Lusztig conjecture and its characteristic 0 counterpart.
We begin by recalling the Kazhdan-Lusztig conjecture [KLl], which described
the characters of the irreducible constituents of Verma modules (for
semisimple Lie algebras over C):
chL, = c(-1) +‘)--l(y)Py,w (1) ch My
where the subscripts are an indexing with the Weyl group of the Verma and
irreducible modules in the “block” associated with the trivial module, and the
Py,zo’s are now called Kazhdan-Lusztig polynomials. The “Lusztig conjecture”
[L] is an analogue for the irreducible representations in characteristic p of an
algebraic group G. (The Lw’s are still irreducibles in the principal block, the
chM,‘s are essentially still characters of Verma modules, and the subscripts
range over values in the affine Weyl group restricted by the “lowest p2-alcove”;
cf. Cline’s article. This restriction has the effect of forcing p larger than the
Coxeter number, but there is some feeling that there will someday be versions
which avoid this.)
Kazhdan and Lusztig showed [KL2] that their polynomials Py,,,(q) were
PoincarC polynomials for the new “intersection cohomology theory” of Goreski-
McPherson, as adapted by Deligne to algebraic geometry. Deligne showed that
the theory thus obtained satisfied the Weil conjectures, an essential ingredient in
relating it to the Hecke algebra context of Kazhdan-Lusztig. (The Hecke algebra
multiplication constants are naturally rational points of varieties.) Meanwhile,
David Vogan had observed that, formally, one could view the “true” coefficients
describing the ch L,‘s in terms of the ch Mw’s as Poincare polynomials, but using
Ext groups involving the unknown irreducible modules L,. (The observation
also extends to characteristic p [S, An].)
Those of us thinking about such things at the time chuckled to ourselves,
“All we need now is something that behaves simultaneously like a topological
space and like the unknown irreducible modules.” Of course when the Kazhdan-
Lusztig conjecture was proved, at the end of 1980 by Brylinski-Kashiwara [BK],
and almost at the same time by Beilinson-Bernstein, that was precisely how it
happened!
The Brylinski-Kashiwara proof. The new-to us-ingredient was the
theory of D-modules. Of course, Bernstein and Kashiwara had been working
with them for years! These are sheaves of modules over a sheaf of differential

ALGEBRAIC THEORY OF DERIVED CATEGORIES 273
operators. The theory is indeed quite a blend of sophisticated algebraic and
topological-analytic
ideas.
The Brylinski-Kashiwara proof may be described schematically as follows:
equivalence
categories
of
D-modules
(holonomic, RS, . . . )
/ \
equivalence
of
derived cateoories
&iv constructible sheaves
(constant
on cells)
Here &iv is essentially the principal block for an abelian category generated
by the Verma modules. One ingredient in the proof is showing that this block is
equivalent to a category of D-modules satisfying reasonable conditions, including
the conditions that they be holonomic with regular singularities. We will not
elaborate on these, except to say that they are natural and make no reference to
the Schubert structure of G/B. There is a further condition, however, requiring
that the “support” of the D-modules involved be described suitably in terms
of Schubert cells (double cosets for B in G, with the right action of B factored
out).
Now we come to the remarkable part of the above diagram. Work of Mebkhout
[Me], approached independently by Kashiwara, implies that there is an equiva-
lence of derived categories between the (bounded) derived category of this cat-
egory of D-modules and the corrsponding derived category of sheaves, with co-
homology constant on Schubert cells, of finite-dimensional vector spaces over
C. (Strictly speaking, this is a “relative” derived categ0ry.l Some modern au-
thors prefer to think of the Mebkhout equivalence in terms of “perverse sheaves”
[BBD], though we do not require these here. I would like to take this opportunity
to thank Herb Clemens for first pointing out to us the implicit role of
Mebkhout’s work in the Brylinski-Kashiwara proof.)
This equivalence of derived categories is by no means induced by an equivalence
of categories! When composed with the first equivalence of categories, in fact,
we find that Verma modules in &riv do not correspond to sheaves at all, but
rather to translations of naturally defined sheaves in the derived category. (The
sheaves involved are direct images of extensions by zero of the constant sheaf C
on a cell, the codimension of the latter determining the extent of translation.)
The irreducible modules, it turns out, correspond precisely to sheaf complexes
defining the new intersection cohomology theory. The proof is then completed
by appealing to earlier work of Kazhdan-Lusztig [KL2], where they calculated
intersection cohomology in Schubert varieties in terms of their polynomials. As I
mentioned earlier, this Kazhdan-Lusztig part of the argument is quite an event in
itself, requiring results of Deligne which allow the Weil conjectures to be applied
to the new intersection cohomology theory.
‘Added in proof: We ignored this distinction at the conference, though it is clear now that
it is important. Also, the “perverse sheaf” formalism is increasingly attractive.

274 LEONARD L. SCOTT
What I wish to focus on, however, is the derived category aspect of the proof.
If one ignores the D-modules and Weil conjectures, the proof is still very remarkable:
I know of no previous major result established by a nontrivial equivalence
of derived categories. I think most mathematicians regarded derived categories
as a homological curiosity, or, at best, a technical convenience. Now they have
made a striking appearance in nature, and must be taken more seriously.
Perhaps it is time for
A quick kindergarten crash course in derived categories. Now every-
one knows that kindergarten children have a tendency to define words in terms
of actions. The result is often ungrammatical, but gets the point across. In this
spirit we have
DEFINITION. A derived category . . . is when you take complexes seriously!
More formally, let A be an abelian category. We shall also assume that A
has enough injectives. (Th’ is is not necessary for parts of the theory, while
other parts are best expressed assuming enough projectives.) The objects of
our derived categories will be complexes K’ of A, except that quasi-isomorphic
complexes are identified. A quasi-isomorphism of complexes is a map inducing
an isomorphism
on homology.
Derived categories come in various sizes and shapes, the most common called
D+(a D-(4, Db(4, and D(A). These arise from complexes of A whose cohomology
is bounded below, bounded above, bounded, and perhaps unbounded
in either direction, respectively. If K’ represents an object in Db(A), it is quasi-
isomorphic to a complex I’ of injectives which is bounded below. In this way
O+(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 de-
rived category level are the compositions 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 [Har].)
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-isomorphism 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 -+ B , where B is also
an abelian category. Namely, one sets RnF(A) = P(F(I’)), and argues that
this cohomology is well defined in 8. In fact, one shows that the homotopy class
of I’, and thus of the complex F(T), does not depend on the chosen resolution.
Prom the present
point of view, we think of this first of all as being a consequence
of the equivalence of the derived category Db(ff) 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

ALGEBRAIC THEORY OF DERIVED CATEGORIES 275
applies 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(I’) is well
defined in D+(B).
The derived category notion of a derived functor can now be described: This
functor is denoted RF, and assigns to K’ in D+(A) 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
RnF(A) = P(RF(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
RF(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 RF(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 RF and RG, if G: B -+ C is an additive functor on abelian categories
with enough injectives. (Often the result is the derived functor R(G o F) 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 further 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, Hom;l(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 invertible functor T on D+ (A), or Db( A),
etc., which sends K’ to the complex K’[l], where K[m]” = 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 sequence 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
be an 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 4 B as
its differential in degree -1. This complex is clearly isomorphic in the derived

276 LEONARD L. SCOTT
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’ -+ 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 ctegory 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 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(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-theoretic
way of thinking.
For the explicit axioms for triangulated categories, see Brian Parshall’s talk
in this series.
Programs for characteristic p. Of course, CPS would like to assimilate the
Brylinski-Kashiwara proof into characteristic p representation theory, in particular
with the goal of proving Lusztig’s analogous conjecture for the decomposition
of Weyl modules. There are various ways to approach this.
This first way is to try and carry over the techniques directly. This was our
first thought, and we were not too successful at it. There are really two steps
here, first replacing the analytic techniques with those of algebraic geometry in
characteristic 0, and then going from characteristic 0 to characteristic p. We
have learned that Bernstein’s version [Be] of the proof does indeed now avoid
the complex analysis. (We are grateful to Professor Hotta for this and many
other references on D-modules.)
The above approach is certainly due much further consideration, and we will
surely give it further thought ourselves. But our initial reaction (to not getting
anywhere) was to return to pure algebra [CPSl]. The program here is to understand
and eventually construct the injective modules, with the aid of clues

ALGEBRAIC THEORY OF DERIVED CATEGORIES 277
supplied by the Lusztig conjecture. Actually, one can get quite far with this; see
Ed Cline’s talk in this series.
The approach we now discuss here, however, as continued in Brian Parshall’s
talk, is an intermediate one. We wish to focus on the algebraic issues, derived
categories in particular, raised by the Brylinski-Kashiwara proof. Ideally, one
would only use the D-modules philosophically, and directly approach the all
important equivalence of derived categories:
&ii
, ’ b-module; 1
/ 0 ’ (holonomic, RS, . . . FL,
/
/
,
.
/
.
/ equivalence of
.
/
/ derived categories
constructible
(constant
.
.
.
sheaves
on cells)
Some good references for the algebraic theory of derived categories are Verdier,
in SGA 4; [De], Hartshorne’s Residues and Duality [Har], and the modern classic
Asterisque 100 [BBD]. The latter reference in particular gives an extremely
valuable discussion of the formalism involved in piecing together two derived
categories (recollement) as if one were the category of sheaves on an open set,
and the other on its closed complement, in some topological space.
These references are all geometrically oriented, however, and it was the theory
of finite-dimensional algebras (in the spirit of Auslander, who participated in this
conference) that supplied for us a key ingredient.
Tilting modules. Here is the original definition, due to Brenner-Butler [BrB]:
Let A be a finite-dimensional algebra, and T a finite-dimensional right A-
module. Then T is a tilting module for A provided the following three conditions
are satisfied:
(1) T has a short projective resolution
(2) Ext’(T, T) = 0,
(3) A has a resolution
O-A+To+Tl+O
by A-modules To, Tl in add T, the category of direct summands of a direct sum
of finitely many copies of T.
Happel proved [Hal] the following theorem concerning derived categories,
based on previous work on tilting modules of Brenner-Butler, Bongartz, and
Happel-Ringel [BrB, Bon, HaR]: Let T be a tilting module for A, [assume
the global dimension of A is finite], and put B = EndA[T]. Then the functor
RHomA(T, -) defines an equivalence of derived categories:
@(mod-A)
M @(mod-B).

278 LEONARDL.SCOTT
There are also consequences for global dimension estimates and Grothendieck
groups, but we will discuss them below along with a generalization of the theorem.
What we wish to make evident now is that this was just the kind of
concrete algebraic result we needed to get started on our algebraic study of
derived categories.
The notation mod-#A above for Happel refers to the category of finite-dimensional
right A-modules, though that is not essential, and it is convenient to use
the notation in the sequel, which discusses rings, for the category of all right
A-modules.
Derived categories and Morita theory. The work I will now describe is
from [CPSl], which gives a crude Morita theory for derived categories. The
inspiration for looking at such issues was, of course, the Brylinski-Kashiwara
proof. The numbering of results has been chosen to coincide with the cited
CPS paper. The notation R +vbF refers the derived of as
in (A) then to (A). “dim” below
homological
[Har].
THEOREM. A a and a A-module the
conditions:
T a resolution + + . + + + + where
Pi a generated in =
(ii) T) 0 all integers
(iii) has finite 0 A To . . Tg 0 each a
in T.
B the HomA(T,T), = let A B the
HomA(T, and B A functor T.
(a) functor + has in In
dim F f
(b) Simil&yi the functor + D(A) image in In fact,
5 g.
The functora +,bF and induce mutually equivalences be-
D”(A) and
(d) A Homr,(T, T), T satisfies left analogues (ii), and for
B. the projective pdim, T T is moat g, defined for
in (iii) and one choose g pdim, T as one of coume,
f = T).
In more recent [Ha2], Happel the formulation his finitealgebra
result include tilting with resolutions longer
length. so, and though our work was by his
preprint and work of we still not require finite global
mension assumption. difference lies our application a general
[Har, I, in Hartshorne’s attributed to [Har, p. which
extends functors on to all D(A) in circumstances.
This helps make possible to with general

ALGEBRAIC THEORY OF DERIVED CATEGORIES 279
A discussion of tilting modules for orders had been begun by Roggenkamp in
an unpublished manuscript [RI.
The next result, although formulated in our context, is indeed just a formal
consequence of the proof of an earlier result of Happel [Hal].
(2.5) COROLLARY (HAPPEL). Let A be a ring, T a tilting module in the
general aenae of (2.li, ii, iii), and put B = HomA(T,T). Let A = mod-A,
B = mod-B, and let J = R+9bF:Db(A) + Db(B) be the induced equivalence
of (2.1). Then there ia an induced iaomorphiam of Grothendieck groups
J: Grot(A) --t Grot( B). Furthermore, if A has finite global dimension then
(~xl7[yI)A = (JWL JW f
B or each [X],[Y] E Grot(A). Here ([X],[Y])4 =
x(RHorn~(X, Y)), the Euler characteristic of the Ext groups.
Theorem (2.1) says that (generalized) tilting modules give equivalences of de-
rived categories which are in fact derived functors. The theorem below provides
a converse. Again A and B are rings, A = mod-A, B = mod-B, and in addition
we let PA denote the category of finitely generated projective right A-modules.
The homotopy category of bounded complexes from PA is denoted Kb(P~); it
identifies naturally with a full subcategory of Db(A). Similar definitions and
remarks
apply of course to PB.
(3.2) THEOREM. Let A, PA, B, PB be as above. Let F: A + B be a left exact
additive functor. Assume that R+vbF: Db(ff) + D(B) maps Db(A) into Db(B),
inducing an equivalence of triangulated categories, as well as an equivalence of
Kb(P~) with Kb(P~). Then F S HomA(T, -), where T is a (generalized) tilting
. .. ..*
module in the aenae of (2.11, 11, 111) with B !Z HomA (T, T).
Conclusion. There are two further main consequences of our Morita theory
investigation, both partly inadvertent. Nevertheless, their discovery has made
us feel very optimistic about further progress.
The first is precisely the subject of Brian Parshall’s continuation of this lecture:
A natural question to ask, with the above Morita/Tilting theory in hand,
is “What happens if condition (iii) for a generalized tilting module in (2.1) is
dropped” One answer is that instead of an equivalence theory for derived
categories, one is led to a stratification theory using the recollement axioms of
Asterisque 100. In particular we give conditions for topological-style stratifications
at the derived category level for modules over a ring. But our theory
applies especially well to categories of rational representations for a semisimple
algebraic group, and we demonstrate stratifications of the derived categories here
which bear a remarkable resemblance to those for constructible sheaves in the
Brylinski-Kashiwara proof. As noted in the introduction, this is the source of
the title of this series, “Simulating algebraic geometry with algebra.”
In the context of these algebraic group investigations we have found a huge
new class A of finite-dimensional algebras of finite global dimension, containing
as their collective module categories all rational representations of semisimple

280 LEONARD L. SCOTT
algebraic groups. Morita equivalent versions of these algebras have been independently
discovered by Steve Donkin [Don]. These versions include the classical
Schur algebras studied by Schur, Green [G], and Aken-Buchsbaum [AkB]. The
latter pair established the finite global dimension of Schur algebras independently
of an earlier unpublished proof by Donkin.
We call the algebras in A “quasi-hereditary” because A shares the following
property with the class of hereditary algebras:
Any 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 = 0. (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 # 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. This is a
more refined list than I presented at the conference, but perhaps closer to an
axiomatization. Indeed, the above list does recursively axiomatize some class
of finite-dimensional algebras if, in (3), we replace “belongs to A” by “satisfies
these axioms (l), (2), and (3).” As of this writing, it seems reasonable to label
these the algebras of interest, and we give them the name “quasi-hereditary.”
Using Theorem (2.1) in Brian Parshall’s talk [CPSB, (3.1)] one can show these
algebras all have finite global dimension. The cited theorem gives a derived category
embedding Llb(A/ J-mod) + 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 Exti(A/J, S) = 0
for S any simple right A-module which is not an A/J module. But clearly
Ext;(S*, (A/J)*) is a summand of Exti(J/rad J, (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 of J.)
Notice the layering in the left ideal structure which is imposed by the above
axioms. This reflects at a ring-theoretic level the stratification of their module
derived categories. To illustrate the left ideal structure, and the difference
between hereditary and quasi-hereditary in simple cases, we give below the decomposition
of the left regular representation in schematic form of two algebras.
Each has exactly two irreducible modules, labeled a and b, of dimension one.
Hereditary
Quasi-hereditary
REFERENCES
[An] H. H. Andersen, An inversion formula for the Kazhdan-Luaztig polynomials for afine
Weyl groupq Aarhus Preprint 31, 1982; Adv. in Math. (to appear).
[AkB] L. Aken and D. Buchsbaum, Characteristic-free representation theory of the general
linear group, Preprint.
[Be] J. Bernstein, Algekaic theory of D-modules, Preprint.

ALGEBRAICTHEORYOFDERIVEDCATEGORIES 281
[Bon] K. Bongartz, Tilted algebras, Lecture Notes in Math., vol. 903, Springer-Verlag, 1982,
pp. 26-38.
[BrB] S. Brenner and M. C. R. Butler, Generalizations of the Bernstein-Gelfand-Ponomarov
rejection functors, Proc. Internat. Conf. on Representations of Algebras, Lecture Notes in
Math., vol. 832, Springer-Verlag, 1980, 103-169.
[BBD] A. Beilinson, J. Bernstein, and P. Deligne, Analyse et topologie sw les espaces sing&em,
Asterisque, no. 100, Sot. Math. France, Paris, 1982.
[BK] J. Brylinski and M. Kashiwara, Kazhdan-Lwztig conjecture and holonomic JYS~.WZJ,
Invent. Math. 64 (1981), 387-410.
[Cl] E. Cline, On injective modules for infinitesimal algebraic groups. II, Preprint.
[C2] -, Z+anslation fun&or and Hecke algebra actions, unpublished informal notes.
[CPSl] E. Cline, B. Parshall, and L. Scott, Derived categories and Morita theory, J. Algebra
104 (1986), 397-409.
[CPS2] -, Algebraic stratification in representation categories, J. Algebra (to appear).
[CPS3] -, On injective modules for infinitesimal algebraic groups I, J. London Math. Sot.
(2) 31 (1985), 277-291.
[De] P. Deligne, with an appendix by J. Verdier, Cat6gories dhivtes (SGA 4;), Lecture
Notes in Math., vol. 569, Springer-Verlag, 1977.
[Don] S. Donkin, On S&UT algebras and related algebras. I, II, J. Algebra 104 (1986), 310-
328; J. Algebra (to appear).
[G] J. A. Green, Polynomial representations of Gl,, Lecture Notes in Math., vol. 830,
Springer-Verlag, 1980.
(Har] R. Hartshorne, Residues and duality, Lecture Notes in Math., vol. 20, Springer-Verlag,
1966.
[Hal] D. Happel, Triangulated categories in representation theory of finite dimensional algebras,
Preprint, January 1985.
[Ha21 -, On the derived category of a finite-dimensional algebra, Preprint (apparently a
revision of [Hal]).
[HaR] D. Happel and C. Ringel, Tilted algebras, Trans. Amer. Math. Sot. 274 (1982),
33sb443.
[KLl] D. Kazhdan and G. Lusztig, Representations of Coseter groups and Hecke algebras,
Invent. Math. 63 (1979), 165-184.
[KL2] -, Schubert varieties and Poincart duality, Proc. Sympos. Pure Math., vol. 36,
Amer. Math. Sot., Providence, R.I., 1980, pp. 185-203.
[L] G. Lusztig, Some problems in the representation theory of finite Chevalley groups, Proc.
Sympos. Pure Math., vol. 37, Amer. Math. Sot., Providence, R.I., 1980, pp. 313-317.
[Me] 2. Mebkhout, Une equivalence de categories et une autre equivalence de categories, Compositio
Math. 51 (1984), 5142, 63-88.
(R] K. Roggenkamp, Tilted orders and order8 of finite global dimension, Preprint.
[S] L. Scott, RepreJen&OnJ in characteristic p, Proc. Sympos. Pure Math., vol. 37, Amer.
Math. Sot., Providence, R.I., 1980, pp. 319-331.
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