PERVERSE COHERENT SHEAVES IN GOOD CHARACTERISTIC 3

use of the term “admissible” is slightly more restrictive than the deﬁnition used

in [BBD].) If A ⊂ T is admissible, then it is automatically an abelian category,

and every short exact sequence in A gives rise to a distinguished triangle in T.

The heart of any t-structure on T is admissible. For the following fact, see [BBD,

Remarque 3.1.17] or [BGS, Lemma 3.2.4].

Lemma 2.1. Let A be an admissible abelian subcategory of a triangulated cate-

gory T. The natural map

ExtA(X,

i

Y ) → HomT(X,

i

Y )

is an isomorphism for i = 0, 1. If it is an isomorphism for i = 0, 1,...,k, then it

is injective for i = k + 1.

Next, we recall the “∗” operation for objects of a triangulated category D. If

X and Y are classes of objects in D, then we deﬁne

X ∗ Y = A ∈ D

there is a distinguished triangle

X → A → Y → with X ∈ X , Y ∈ Y

.

By [BBD, Lemme 1.3.10], this operation is associative. In an abuse of notation,

when X is a singleton {X}, we will often write X ∗ Y rather than {X} ∗ Y. Note

that the zero object is a sort of “unit” for this operation. For instance, we have

X ∗ Y ∗ 0 = X ∗ Y. Given a class X , X ∗ 0 is the class of all objects isomorphic to

some object of X .

2.2. Tate twist. Many of our categories will be equipped with an automor-

phism known as a Tate twist, and denoted X → X 1 . We will always assume that

Tate twists are “faithful,” meaning that for any nonzero object X, we have

X

∼

=

X n if and only if n = 0.

A key example is the category Vectk of graded k-vector spaces, where the Tate

twist is the “shift of grading” functor. For X ∈ Vectk, let Xn denote its nth graded

component. Then X m is the graded vector space given by

(X m )n = Xn−m.

We regard k itself as an object of Vectk by placing it in degree 0.

If X and Y are objects of an additive category equipped with a Tate twist, we

let Hom(X, Y ) denote the graded vector space deﬁned by

Hom(X, Y )n = Hom(X, Y −n ).

Notations like

Homi(−,

−),

Exti(−,

−), and RHom(−, −) are deﬁned similarly.

The following lemma is a graded analogue of [B2, Lemma 5].

Lemma 2.2. Let V be an object in

D+Vectk.

(1) If there are integers n1,...,nk such that 0 ∈ V n1 ∗ · · · ∗ V nk , then

V = 0.

(2) If there are integers n1,...,nk 0 such that k ∈ V ∗ V n1 ∗ · · · ∗ V nk ,

then

Hi(V

) = 0 for i 0, and

H0(V

)

∼

= k. For i 0,

Hi(V

) is concen-

trated in strictly positive degrees.

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