derived categories of twisted sheaves on calabi-yau manifolds

Proposition 2.3.10. Let f : X → Y be a proper morphism <strong>of</strong> schemes or analytic
spaces, and let α ∈ ˇ H 2 (Y, O ∗ Y ). Let u : Y ′ → Y be a flat morphism, let X ′ =
X ×Y Y ′ , and let v, g be the projections, as shown:
X ′
v ✲ X
Y ′
g f
❄ u ❄
✲ Y.
Then there is a natural functorial isomorphism
for F · ∈ Dcoh(X, f ∗ α).
u ∗ Rf∗F · ∼
−→ Rg∗v ∗ F ·
Proposition 2.3.11. Let X be a scheme or analytic space, and let α, α ′ , α ′′ ∈
ˇH 2 (X, O∗ X ). Then there are natural functorial isomorphisms
and
L
·
F ⊗ G · ∼ ·
−→ G L
⊗ F ·
L L
· ·
F ⊗ (G ⊗ H · ) ∼ L
·
−→ (F ⊗ G · ) L
⊗ H ·
for F · ∈ D −
coh (X, α), G· ∈ D −
coh (X, α′ ), H · ∈ D −
coh (X, α′′ ).
Proposition 2.3.12. Let X be a scheme or analytic space, and let α, α ′ , α ′′ ∈
ˇH 2 (X, O∗ X ). Then there is a natural functorial isomorphism
RHom · (F · , G · ) L
⊗ H · ∼
−→ RHom · (F · , G · L
⊗ H · )
for F · ∈ D −
coh (X, α), G· ∈ D +
coh (X, α′ ) and H · ∈ Dcoh(X, α ′′ )fTd.
Proposition 2.3.13. Let X be a scheme or analytic space, let α, α ′ ∈ Br(X),
α ′′ ∈ ˇ H2 (X, O∗ X ), and assume that every coherent sheaf on X is a quotient <strong>of</strong> a
lffr. Then there is a natural functorial isomorphism
RHom · (F · , RHom · (G · , H · )) ∼
−→ RHom · L
·
(F ⊗ G · , H · )
for F · ∈ D −
coh (X, α), G· ∈ D −
coh (X, α′ ) and H · ∈ D +
coh (X, α′′ ).
Proposition 2.3.14. Let X be a scheme or analytic space, let α, α ′ , α ′′ ∈ ˇ H 2 (X,
O ∗ X ), and let L· be a bounded complex <strong>of</strong> α-lffr’s. Let L ·∨ = Hom · (L · , OX). Then
there are natural functorial isomorphisms
RHom · (F · , G · ) L
⊗ L · ∼
· · ·
−→ RHom (F , G L
⊗ L · ) ∼
−→ RHom · L
·
(F ⊗ L ·∨ , G · )
for F · ∈ D −
coh (X, α′ ), G · ∈ D +
coh (X, α′′ ).
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