derived categories of twisted sheaves on calabi-yau manifolds

large enough to work for all Ui at once, so we conclude that p2∗(F ⊗ O(n)) is an
α-lffr, which implies that α ∈ Br(M).
The next theorem allows us to compare the absolute and the relative settings
for generating equivalences, when dealing with moduli problems:
Proposition 3.3.5. Let f : X → S be a morphism <strong>of</strong> schemes or analytic spaces,
with S <strong>of</strong> the form Spec R for a regular local ring R. If s is the closed point <strong>of</strong>
S, let i : Xs → X be the inclusion into X <strong>of</strong> the fiber Xs over s, and let F , G be
<strong>sheaves</strong> on Xs. Then
1. if Ext j
(F , G ) = 0 for all j then Extj
Xs X (i∗F , i∗G ) = 0 for all j;
2. if Ext j
Xs (F , G ) = 0 for all j > n0 then Ext j
X (i∗F , i∗G ) = 0 for all j >
n0 + dim S.
Pro<strong>of</strong>. (An adaptation <strong>of</strong> the pro<strong>of</strong> <strong>of</strong> [7, 3.1].) We have
R Hom ·
X(i∗F , i∗G ) = R Hom ·
Xs (Li∗ i∗F , G )
by the adjunction <strong>of</strong> Li ∗ and i∗. Furthermore,
Li ∗ i∗F = F L
⊗Xs Li ∗ i∗OXs
by the projection formula. Since S is smooth at s, writing down the Koszul
resolution for Os and pulling back via f we get a free resolution <strong>of</strong> OXs on X
which can be used to compute Li ∗ i∗F . This gives
H q (Li ∗ i∗F ) = F ⊗
q O ⊕m
where m = dim S. Now the hypercohomology spectral sequence
E p,q
2 = Ext p
Xs (Hq (Li ∗ i∗F ), G ) =⇒ H p+q (R Hom ·
Xs (Li∗ i∗F , G ))
= H p+q (R Hom ·
X(i∗F , i∗G ))
= Ext p+q
X (i∗F , i∗G )
gives the results.
Xs
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