derived categories of twisted sheaves on calabi-yau manifolds
derived categories of twisted sheaves on calabi-yau manifolds
derived categories of twisted sheaves on calabi-yau manifolds
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Lemma 1.3.4. Again, let R be a commutative ring, E a free R-module, and<br />
A = EndR(E). Then for any R modules F and G we have<br />
HomR(F, G) ∼ = HomA(F ⊗R E ∨ , G ⊗R E ∨ ),<br />
where HomA denotes the group <str<strong>on</strong>g>of</str<strong>on</strong>g> right A-module homomorphisms.<br />
Pro<str<strong>on</strong>g>of</str<strong>on</strong>g>. The map in <strong>on</strong>e directi<strong>on</strong> is · ⊗R idE ∨. In the other directi<strong>on</strong> take · ⊗A idE,<br />
and use the previous lemma.<br />
Theorem 1.3.5. Let A be an Azumaya algebra over X, and let α ∈ Br ′ (X) be<br />
the element that A represents (we’ll <str<strong>on</strong>g>of</str<strong>on</strong>g>ten denote α by [A ]). Then there exists<br />
a locally free α-<str<strong>on</strong>g>twisted</str<strong>on</strong>g> sheaf E <str<strong>on</strong>g>of</str<strong>on</strong>g> finite rank (not necessarily unique) such that<br />
A ∼ = End(E ). C<strong>on</strong>versely, for any α ∈ Br ′ (X) such that there exists a locally free<br />
α-sheaf E <str<strong>on</strong>g>of</str<strong>on</strong>g> finite rank, End(E ) is an Azumaya algebra whose class in Br ′ (X) is<br />
α.<br />
Thus we have yet another characterizati<strong>on</strong> <str<strong>on</strong>g>of</str<strong>on</strong>g> the Brauer subgroup <str<strong>on</strong>g>of</str<strong>on</strong>g> the cohomological<br />
Brauer group: it is the subgroup <str<strong>on</strong>g>of</str<strong>on</strong>g> those twistings α for which there<br />
exist locally free α-<str<strong>on</strong>g>twisted</str<strong>on</strong>g> <str<strong>on</strong>g>sheaves</str<strong>on</strong>g> <str<strong>on</strong>g>of</str<strong>on</strong>g> finite rank (α-lffr’s, in short).<br />
Pro<str<strong>on</strong>g>of</str<strong>on</strong>g>. Find a cover {Ui}, OUi -lffr’s Ei and isomorphisms ϕi : End(Ei) → A |Ui as<br />
given by Theorem 1.1.6. By Lemma 1.3.1, the isomorphisms<br />
induce isomorphisms<br />
ϕ −1<br />
j ◦ ϕi : End(Ei|Ui∩Uj ) → End(Ei|Ui∩Uj )<br />
→ Ej|Ui∩Uj .<br />
¯ϕij : Ei|Ui∩Uj<br />
The threefold compositi<strong>on</strong>s ¯ϕijk are automorphisms <str<strong>on</strong>g>of</str<strong>on</strong>g> Ei such that the corresp<strong>on</strong>ding<br />
automorphisms <strong>on</strong> End(Ei) are the identity, hence by Lemma 1.3.2 they must<br />
be multiplicati<strong>on</strong>s by secti<strong>on</strong>s αijk <str<strong>on</strong>g>of</str<strong>on</strong>g> O∗ X over Ui ∩ Uj ∩ Uk. Therefore we have<br />
found the data for an α-lffr E , where α is the element <str<strong>on</strong>g>of</str<strong>on</strong>g> ˇ H2 (X, O∗ X ) determined<br />
by {αijk}. It is easy to see that this is the same corresp<strong>on</strong>dence as described in<br />
Theorem 1.1.8.<br />
The fact that End(E ) is an Azumaya algebra follows immediately from Theorem<br />
1.1.6.<br />
Propositi<strong>on</strong> 1.3.6. Let A be an Azumaya algebra over X, let α = [A ], and let<br />
E be an α-lffr such that A ∼ = End(E ). Note that E is naturally a left A -module.<br />
Define a functor F between the category Mod(X, α) <str<strong>on</strong>g>of</str<strong>on</strong>g> α-<str<strong>on</strong>g>twisted</str<strong>on</strong>g> <str<strong>on</strong>g>sheaves</str<strong>on</strong>g> and the<br />
category Mod-A <str<strong>on</strong>g>of</str<strong>on</strong>g> <str<strong>on</strong>g>sheaves</str<strong>on</strong>g> <str<strong>on</strong>g>of</str<strong>on</strong>g> right A -modules <strong>on</strong> X by the formula<br />
∨<br />
F ( · ) = · ⊗OX E<br />
(the right A -module structure <strong>on</strong> F ( · ) is given by using the right A -module structure<br />
<str<strong>on</strong>g>of</str<strong>on</strong>g> E ∨ ).<br />
Then, for any pair <str<strong>on</strong>g>of</str<strong>on</strong>g> α-<str<strong>on</strong>g>sheaves</str<strong>on</strong>g> F and G , F induces a functorial isomorphism<br />
Hom Mod(X,α)(F , G ) ∼ = Hom Mod-A (F (F ), F (G )).<br />
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