derived categories of twisted sheaves on calabi-yau manifolds

Pro<strong>of</strong> <strong>of</strong> Lemma 1.2.3. We only give a pro<strong>of</strong> <strong>of</strong> this result in the analytic category,
for ease <strong>of</strong> notation. The corresponding pro<strong>of</strong> in the étale category is entirely
similar. Since U ′ = {U ′ j}j∈J is a refinement <strong>of</strong> U = {Ui}i∈I we are given a map
λ : J → I such that for each j ∈ J we have U ′ j ⊆ Uλ(j). If F is an α-<strong>twisted</strong> sheaf
given on U, then there is a natural notion <strong>of</strong> refinement <strong>of</strong> F to U ′ : this is the pair
({Fλ(j)|Uj }j∈J, {ϕλ(i)λ(j)|Ui∩Uj }j∈J),
and this construction clearly gives a refinement functor
Mod(X, α, U) → Mod(X, α, U ′ ).
In order to show that the refinement functor is an equivalence <strong>of</strong> <strong>categories</strong>, we
would need to show that it is fully faithful and that every α-sheaf G given on U ′ is
isomorphic to the refinement <strong>of</strong> an α-sheaf F on U (see [12, p.84] or [9, p.26]). The
first part (fully faithful) is an easy consequence <strong>of</strong> the definition <strong>of</strong> the morphisms,
and it only remains to construct F .
Let G = ({Gj}, {ψjk}) be an α-sheaf given along U ′ , and let Ui be any open set
in U (i ∈ I). Then define the sheaf Fi on Ui as follows: if V ⊆ Ui is an open set,
then
Fi(V ) = {(sj)j∈J ∈
G (U ′ j ∩ V ) | ψjk(sk) = αiλ(j)λ(k)sj for all j, k ∈ J}.
j∈J
It is not hard to see that this definition, along with the obvious restriction maps,
makes Fi into a sheaf on Ui.
(Note that what we are doing is in fact gluing the <strong>sheaves</strong> Gj|Ui∩Uj along the
isomorphisms
α −1
ψjk|Ui∩Uj∩Uk iλ(j)λ(k) .
Using Lemma 1.2.4 we can do the same construction in the étale setup.)
{ϕii
Define the isomorphisms ϕii ′ : Fi ′ → Fi that will make the collection ({Fi},
′}) into an α-sheaf as follows:
ϕii ′((sj)j∈J) = (tj)j∈J = (αii ′ λ(j)sj)j∈J
over any open set V ⊆ Ui ∩ Uj. One now easily verifies that (tj) is indeed a section
<strong>of</strong> Fi over V , and that ϕii ′ is indeed an isomorphism. Finally, one checks that
ϕii ′ ◦ ϕi ′ i ′′ ◦ ϕi ′′ i = αii ′ i ′′,
and thus that the collection ({Fi}, {ϕii ′}) is an α-sheaf whose refinement to U′ is
isomorphic to G .
Corollary 1.2.6. Let α ∈ Č2 (X, O ∗ X ) and let F be an α-sheaf given along an
open cover U. Let U ′ be any open cover over which α can be represented. Then F
can be represented by an α-sheaf on U ′ . In particular, let F be a sheaf (un<strong>twisted</strong>)
whose support is contained in an open set U over which α is trivial. Then F can
be also given the structure <strong>of</strong> an α-sheaf.
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