4 PRAMOD N. ACHAR
Proof. (1) Suppose V = 0, and let m be the smallest integer such that
Hm(V
) = 0. Then, for any object X V n1 · · · V nk , it follows that the
map
Hm(V
n1 )
Hm(X)
is injective. But if X = 0, this contradicts the as-
sumption that
Hm(V
) = 0.
(2) The argument given for part (1) shows that
Hi(V
) = 0 for i 0, and that
the map H0(V
)
H0(k)

= k is injective. Let Y V n1∗···∗ V nk be such that
there is a distinguished triangle V k Y →. If H0(V
) = 0, it would follow that
H0(Y ) = 0, leading to a contradiction with the fact that H0(k) = 0, so it must be
that H0(V )

=
k. We then see from that distinguished triangle that
Hi(V
)

=
Hi−1(Y
) for all i 1.
Because all the ni are strictly positive, it follows from the fact that
H0(V
)

= k
that
H0(Y
) is concentrated in strictly positive degrees, and hence so is
H1(V
).
Thereafter, we proceed by induction on i: if
Hi(V
) is concentrated in strictly
positive degrees, so is
Hi(Y
), and therefore so is
Hi+1(V
).
2.3. Quasi-hereditary categories. Let S be a set equipped with a partial
order ≤. Assume that every principal lower set is finite, i.e., that
(2.1) For all s S, the set {t S | t s} is finite.
Let A be a finite-length abelian category, and assume that one of the following
holds:
“Ungraded case”: There is a fixed bijection Irr(A)

= S.
“Graded case”: A is equipped with a Tate twist, and there is a fixed
bijection Irr(A)

=
S ×Z with the property that for a simple object L A,
L corresponds to (s, n) if and only if L 1 corresponds to (s, n + 1).
In the ungraded case, choose a representative simple object Σs for each s S,
and let
(≤s)A
(resp.
(s)A)
be the Serre subcategory of A generated by all simple
objects Σt
with t s (resp. t s).
In the graded case, let Σs
denote a representative simple object corresponding
to (s, 0) S × Z. In this case, (≤s)A (resp. (s)A) denotes the Serre subcategory of
A generated by all simple objects Σt n with t s (resp. t s) and n Z. More
generally, for any subset Ξ S × Z, we let ΞA denote the Serre subcategory of A
generated by the Σt n with (t, n) Ξ.
In the sequel, we will focus mostly on the graded case. With the above notation
in place, the corresponding definitions and statements for the ungraded cases can
usually be obtained simply by omitting Tate twists and by changing “Hom” and
“Ext” to “Hom” and “Ext,” respectively. For instance, it is left to the reader to
formulate the ungraded version of the following definition.
Definition 2.3. A category A as above is said to be graded quasi-hereditary
if for each s S, there is:
(1) an object Δs and a surjective map φs : Δs Σs such that
ker φs
(s)A
and Hom(Δs, Σt) =
Ext1(Δs,
Σt) = 0 if t s.
(2) an object ∇s and an injective map ψs : Σs ∇s such that
cok
ψs

(s)A
and Hom(Σt,
∇s)
=
Ext1(Σt, ∇s)
= 0 if t s.
4
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