derived categories of twisted sheaves on calabi-yau manifolds

Theorem 2.2.8. Let A ′ and B ′ be thick sub<strong>categories</strong> <strong>of</strong> abelian <strong>categories</strong> A and
B, respectively, and let F : A → B be an exact functor that takes A ′ to B ′ . Assume
furthermore that the following properties are satisfied:
1. A and B have enough injectives;
2. F is an equivalence <strong>of</strong> <strong>categories</strong> when restricted to A ′ → B ′ ;
3. F induces a natural isomorphism
for any X, Y ∈ A ′ and any i.
Ext i
A(X, Y ) ∼ = Ext i
B(F (X), F (Y ))
Then the natural functor ˜ F : Db A ′(A) → DbB ′(B) induced by F is an equivalence
<strong>of</strong> <strong>categories</strong>.
Pro<strong>of</strong>. As a first step we want to prove that ˜ F is full and faithful, i.e. that for any
X · , Y · ∈ D b A ′(A), ˜ F induces an isomorphism
Hom D b
A ′(A)(X · , Y · ) ∼ = Hom D b
B ′(B)( ˜ F (X · ), ˜ F (Y · )).
We shall prove this statement by induction on n = n(X · ) + n(Y · ). If n = −∞,
then one <strong>of</strong> X · or Y · is the zero complex, so there is nothing to prove. If n = 0,
then both X · and Y · consist <strong>of</strong> a single object <strong>of</strong> A ′ , hence the statement follows
from [23, I.6.4] and property 3 above. This is the basis for the induction process.
Assume that ˜ F induces an isomorphism
Hom D b
A ′(A)(X · , Y · ) ∼ = Hom D b
B ′(B)( ˜ F (X · ), ˜ F (Y · ))
for all X · , Y · ∈ D b A ′(A) with n(X· ) + n(Y · ) < n, and let X · , Y · be objects <strong>of</strong>
D b A ′(A) with n(X· )+n(Y · ) = n > 0. Without loss <strong>of</strong> generality assume n(Y · ) > 0,
and that H i (Y · ) = 0 for i < 0, and H 0 (Y · ) = 0. Using Remark 2.2.3 we can
assume that Y i = 0 for i < 0, and H 0 (Y · ) = 0.
Just as before there is a map <strong>of</strong> complexes between Y ′· = H 0 (Y · ) (the complex
whose single nonzero object is H 0 (Y · ) in degree 0) and Y · , that induces an
isomorphism in cohomology in all degrees ≤ 0. Let Y ′′· be the third vertex <strong>of</strong> a
triangle built on this morphism. Again, we have n(Y ′′· ) < n(Y · ); also, from the
assumption, n(Y ′· ) = 0 < n(Y · ). From the long exact sequence <strong>of</strong> Hom’s ([23,
I.1.1b]), the five-lemma and the induction hypothesis we conclude that
Hom i
D b
A ′(A)(X· , Y · ) ∼ = Hom i
D b
B ′(B)( ˜ F (X · ), ˜ F (Y · )),
which is what we needed to prove that ˜ F is full and faithful. (The case when
n(Y · ) = 0 but n(X · ) > 0 follows in a similar way.)
We are left with proving that any object Y · <strong>of</strong> Db B ′(B) is isomorphic (in Db B ′(B))
to an object <strong>of</strong> the form ˜ F (X · ) for some X · ∈ Db A ′(A). We prove this by induction
on n = n(Y · ): the case n = −∞ is trivial, and n = 0 follows from property 2.
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