derived categories of twisted sheaves on calabi-yau manifolds

Definition 1.3.12. Let A be a ring. A right A-module is said to be an Aprogenerator
if it satisfies the following two conditions:
1. F is finitely generated projective;
2. F is a generator, i.e. the functor HomR(F, · ) : Mod-A → Ab is faithful.
Theorem 1.3.13 (Fundamental Theorem <strong>of</strong> Morita Theory). Let A, B be
rings. Then A ∼M B if and only if there exists an A-progenerator F such that
B ∼ = EndA(F ).
In this case, the functors
Mod-A → Mod-B M ↦→ M ⊗A F ∨
Mod-B → Mod-A N ↦→ N ⊗B F,
are mutually inverse, where F ∨ = HomA(F, A), (note that F ∨ is naturally a left
A-module and F is naturally a right B-module)
Pro<strong>of</strong>. [27, Chap. 18, especially 18.24].
Lemma 1.3.14. Let A, B and C be R-algebras over a commutative ring R, with
C being flat as an R-module. If A ∼M B then A ⊗R C ∼M B ⊗R C.
Pro<strong>of</strong>. Write B ∼ = EndA(F ) for an A-progenerator F . Then we have isomorphisms
<strong>of</strong> R-algebras:
B ⊗R C ∼ = EndA(F ) ⊗R C ∼ = EndA⊗RC(F ⊗R C).
To prove the last isomorphism, use [14, 2.10], noting that A ⊗R C is indeed a
flat A-module (by the assumption that C is a flat R-module), and F is A-finitely
presented being a progenerator.
On the other hand, it is easy to see that F ⊗R C is a progenerator for A ⊗R C,
using the fact that an A-module F is a progenerator if and only if F is a direct
summand <strong>of</strong> a finite direct sum <strong>of</strong> copies <strong>of</strong> A and A is a direct summand <strong>of</strong> a
finite direct sum <strong>of</strong> copies <strong>of</strong> F ([27, 18.9]).
Theorem 1.3.15. Two Azumaya algebras A and B over a commutative ring R are
Morita equivalent if and only if there exist finitely generated projective R-modules F
and F ′ such that A⊗R End(F ) ∼ = B ⊗R End(F ′ ). (We assume that A, B, F and F ′
have positive rank on each component <strong>of</strong> Spec R, to avoid trivial counterexamples.)
Pro<strong>of</strong>. Using [27, 18.11 and Ex. 2.24], we conclude that any finitely generated
projective R-module (with positive rank on each component <strong>of</strong> Spec R) is an
R-progenerator. Thus, one implication is easy using Lemma 1.3.14 and Theorem
1.3.13.
Now let’s assume A ∼M B. Then, by Lemma 1.3.14, we have A ⊗R A ∨ ∼M
B ⊗R A ∨ . Since A ⊗R A ∨ ∼ = EndR(A) we conclude using Theorem 1.3.13 that
B ⊗R A ∨ ∼M R. Now using again Theorem 1.3.13, we conclude that
B ⊗R A ∨ ∼ = EndR(F )
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