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 (Ξ ). 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 (Ξ ) ⊂ 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