derived categories of twisted sheaves on calabi-yau manifolds

property “free on stalks” for the purpose <strong>of</strong> this chapter. Note that these <strong>sheaves</strong>
are not locally free, as they are not quasi-coherent!
Lemma 2.1.4. If α ∈ Br(X), and if on X every coherent sheaf is the quotient <strong>of</strong>
a locally free sheaf <strong>of</strong> finite rank (lffr), then the same holds for coherent α-<strong>sheaves</strong>.
Pro<strong>of</strong>. Let G be an α −1 -lffr, and let F be any coherent α-sheaf; then F ⊗ G is a
coherent sheaf on X, hence we can find a lffr L that surjects onto F ⊗ G . Tensor
this surjection with G ∨ , to get a surjective map L ⊗G ∨ → F ⊗G ⊗G ∨ . But there
is a surjective trace map G ⊗ G ∨ → OX given by the evaluation, and therefore a
surjective composite map
which is what we wanted.
L ⊗ G ∨ → F ⊗ G ⊗ G ∨ → F
Lemma 2.1.5. For any injective α-sheaf I and for any open set U ⊆ X we have
I |U is an injective α-sheaf on U. In particular, if I = ({Ii}, {ϕij}) along an
open cover U = {Ui}, then the sheaf Ii is injective on Ui for all i.
Pro<strong>of</strong>. For the first part, use a pro<strong>of</strong> identical to [22, III, 6.1] (note that, in fact,
that pro<strong>of</strong> can be written entirely in functorial terms, using the adjointness properties
<strong>of</strong> f! and f ∗ , and using only the fact that f! takes injective maps to injective
maps). For the last statement use Lemma 1.2.3 and the fact that the injectivity
<strong>of</strong> an object is a categorical property.
Proposition 2.1.6. Assume that X is smooth, <strong>of</strong> dimension n. Then for any
coherent α-sheaf F and any coherent α ′ -sheaf G we have
Ext i
X(F , G ) = 0 for any i > n.
(We define Ext i (F , · ) as the right <strong>derived</strong> functors <strong>of</strong> Hom(F , · ).)
Remark 2.1.7. Note that we are, unfortunately, unable to conclude from this proposition
that any coherent α-sheaf has injective dimension at most n on X. Indeed,
to prove this, we would need to prove that the property
“Ext i X(F , I ) = 0 for any coherent F and any i > 0”
for an α-sheaf I implies that I is injective. Of course, if I were coherent,
this property would say that I is indeed injective in Coh(X, α). But we can not
hope, in general, to find resolutions <strong>of</strong> coherent α-<strong>sheaves</strong> by coherent injective
α-<strong>sheaves</strong>, so we are forced to go to the larger category <strong>of</strong> non-coherent α-<strong>sheaves</strong>.
On a scheme, a possible alternative would be to use the results in [23, II.7], but
we do not know <strong>of</strong> similar results for analytic spaces. Fortunately, we can get by
with the slightly weaker form <strong>of</strong> Proposition 2.1.6.
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