14 PRAMOD N. ACHAR
for all d 1, so it follows that
Homd(P,
Σu p ) = 0 as well. It remains to show
that
Homd(P,
Σs n ) = 0.
Consider the exact sequence
Homd−1(P,
Qs n )
Homd(P,
Σs n )
Homd(P, ∇s
n ).
The first term is already known to vanish because Qs n
ΨA,
and the last term
vanishes by Corollary 3.9 because P has Ξ-standard filtration. Thus, we have
Homd(P,
Σs n ) = 0, as desired.
3.4. Main result. Given a set Ξ S × Z, let ΞD D denote the full tri-
angulated subcategory generated by
ΞA.
In other words,
ΞD
is full subcategory
consisting of objects X all of whose cohomology objects
Hi(X)
lie in
ΞA.
Note
that
ΞA
is the heart of a bounded t-structure on
ΞD,
so we have a realization func-
tor real :
Db(ΞA)

ΞD.
Composition with the inclusion functor gives us a natural
functor
Db(ΞA)
D.
Theorem 3.15. For any convex set Ξ S × Z, the natural functor Db(ΞA)
D is fully faithful. In particular, the realization functor
real :
Db(A)
D
is an equivalence of categories.
Proof. We first treat the case where Ξ is finite. For a fixed object X ΞA,
{Extd(·,X)}d≥0
is a universal δ-functor, and Proposition 3.14 tells us that the
δ-functor
{Homd(·,X)}d≥0
is effaceable. Since their 0th parts agree, there is a
canonical isomorphism of functors
Extd(·,X)


Homd(·,X).
Therefore, the natural
functor Db(ΞA)
D is full and faithful.
Now, suppose Ξ is infinite. For any finite convex subset Ψ Ξ, consider the
chain of maps
ExtΨ(X,
i
Y )
g
ExtΞ(X,
i
Y )
h

Homi(X,
Y ).
We claim that g and h are both isomorphisms for all i. This is clearly the case
for i = 0 and i = 1. Suppose, in fact, that it is known for i = 0, 1,...,d 1.
By Lemma 2.1, both g and h are injective for i = d. We know by the previous
paragraph that the composition h g is an isomorphism for all i, so it follows that
g and h are isomorphisms as well.
We have shown that ExtΞ(X,
d
Y )
Homd(X,
Y ) when X, Y ΨA. But given
any two objects X, Y ΞA, we know from Lemma 3.12 that there exists a finite con-
vex subset Ψ Ξ such that X, Y ΨA. It follows that ExtΞ(X,
d
Y )
Homd(X,
Y )
for all X, Y ΞA, so Db(ΞA) D is full and faithful.
4. Notation for semisimple groups
4.1. Representations and varieties. As noted in Section 2, we will work
over a fixed algebraically closed field k. For an algebraic group H over k, let Rep(H)
denote the category of rational representations of H, and Repf (H) Rep(H) the
subcategory of finite-dimensional representations. If H K, we have the usual
induction and restriction functors indH
K
: Rep(H) Rep(K) and resH
K
: Rep(K)
Rep(H). We also use the derived functor RindH
K
:
DbRep(H)

DbRep(K).
(Of
course, resH
K
is exact.)
14
Previous Page Next Page