PERVERSE COHERENT SHEAVES IN GOOD CHARACTERISTIC 13
Finally, let
Ξ = Ξ (t, m)
Σt m occurs as a composition factor in
some Δs r or ∇s r with a0 r b0
.
It is clear that s / F ). Note that all the new pairs (t, m) Ξ Ξ have t s. It
follows that every element s F0(Ξ ) either belongs to F0(Ξ) or is s. Therefore,
F ) F (Ξ) {s}, so by induction, Ξ , and therefore Ξ, is contained in a finite
convex set.
Lemma 3.13. If Ξ S × Z is a finite convex set, and (s, n) Ξ, then Δs
Ξ
n
belongs to
ΞA
and has Σs n as its unique simple quotient. Moreover, for all d 1,
we have
(3.5)
Homd(Δs Ξ
n , Σt m ) = 0 if (t, m) Ξ and t s,
Homd(Δs Ξ
n ,
∇t
m ) = 0 for all (t, m) Ξ.
In particular, if (s, n) is a costandard-maximal element of Ξ, then Δs Ξ n is a
projective cover of Σs n .
Proof. Let as be as in Definition 3.11. Because Ξ is convex, the standard
objects Δs as , Δs as + 1 , . . . , Δs n all belong to
ΞA,
so it follows by Proposi-
tion 3.4 that Δs
Ξ
n =
˜
Δ
(n−as)
s
n does as well. We also already know that Σs n
is the unique simple quotient of Δs
Ξ
n . Finally, note that if (s, m) Ξ, then
as m, so n m (n as) + d. Therefore, (3.5) is an immediate consequence of
Proposition 3.8 and Corollary 3.9.
Proposition 3.14. Let Ξ S × Z be a finite convex set. Every simple object
Σs n
ΞA
admits a projective cover P with a Ξ-standard filtration. Moreover, we
have
(3.6)
Homd(P,
X) = 0 for all d 1 and all X
ΞA.
Proof. We proceed by induction on the size of Ξ. If Ξ = ∅, there is nothing
to prove. Otherwise, let (s, n) be a costandard-maximal element of Ξ, and let
Ψ = Ξ {(s, n)}. Lemma 3.13 tells us that Δs
Ξ
n is a projective cover of Σs n
satisfying (3.6).
Next, let R denote the kernel of the map Δs
Ξ
n Σs n . R has a composition
series consisting of Rs n and various Δs m with r m n. We see thus that R
is contained in
ΨA.
Consider a pair (t, m) Ψ. We assume inductively that Σt m admits a projec-
tive cover P in
ΨA
with a Ψ-standard filtration and satisfying (3.6). In particular,
we have
Hom2(P
, R) = 0. We have already seen above that
Hom2(Δs Ξ
n , R) = 0,
so we may invoke Proposition 2.11 to obtain a projective cover P of Σt m in
ΞA.
The key observation now is that every Ψ-standard object is also Ξ-standard.
(This would not have been the case if we had instead defined Ψ by deleting a
standard-maximal element of Ξ.) Thus, we now see from the exact sequence (2.3)
that P has a Ξ-standard filtration.
We must now establish (3.6). If (u, p) Ψ, we already know that
Homd(P
, Σu p ) =
Homd(Δs
Ξ
n , Σu p ) = 0
13
Previous Page Next Page