derived categories of twisted sheaves on calabi-yau manifolds

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$Lecture notes for Math 522 Spring 2012 (Rudin chapter 7)$

$Math 320 - Fall 2010 HW #7 Selected Solutions Problem 5.1.32 Let ...$

for some R-progenerator F . Tensoring this isomorphism over R with A, we get B ⊗R EndR(A) ∼ = A ⊗R EndR(F ), and thus taking F ′ = A we get the result. This shows that in the local situation (over an affine scheme) the Brauer group <strong>of</strong> a commutative ring R is precisely the group <strong>of</strong> isomorphism classes <strong>of</strong> Azumaya algebras under the tensor product operation, modulo Morita equivalence. Unfortunately this does not generalize well to Azumaya algebras over a scheme, as the following example shows: Example 1.3.16. Work over the ground field C. Let X be a double cover <strong>of</strong> P 2 branched over a smooth sextic curve. Then it is known that X is a smooth K3 surface (see Chapter 5 for more information on K3 surfaces and their Brauer groups), and in this case we have Br(X) = T ∨ X ⊗ Q/Z (Lemma 5.4.1), where TX is the transcendental lattice <strong>of</strong> X, and T ∨ X is the dual lattice to TX. There is a natural involution ι on X which interchanges the two sheets. The +1-eigenspace <strong>of</strong> the induced action <strong>of</strong> ι on H2 (X, Z) is precisely the algebraic part <strong>of</strong> H2 (X, Z), and therefore ι acts by −1 on TX, and from Lemma 5.4.1 it also acts by −1 on Br(X). Let A be a sheaf <strong>of</strong> Azumaya algebras on X whose image [A ] in Br(X) is nonzero and <strong>of</strong> order not equal to 2. Obviously ι induces an equivalence <strong>of</strong> <strong>categories</strong> ι ∗ : Coh(X, A ) → Coh(X, ι ∗ A ). But we assumed that [ι ∗ A ] = −[A ] = [A ], so we have found Azumaya algebras A and ι ∗ A that are Morita equivalent, but not equal in the Brauer group. In view <strong>of</strong> this example it makes sense to state the following conjecture: Conjecture 1.3.17. On a projective scheme X we have A ∼M B for <strong>sheaves</strong> <strong>of</strong> Azumaya algebras A and B if and only if there exists an automorphism ϕ : X → X such that [A ] = [ϕ ∗ B] in Br(X). We’ll be interested in studying a coarser equivalence relation on the group <strong>of</strong> isomorphism classes <strong>of</strong> Azumaya algebras, and that is the notion <strong>of</strong> <strong>derived</strong> Morita equivalence (see the definition below). In the course <strong>of</strong> this work we’ll not go into the details <strong>of</strong> the theory <strong>of</strong> <strong>derived</strong> equivalences for non-commutative rings (and tilting modules, tilting complexes, etc.) which is a whole subject in itself (see [38] for an introduction to the problem and main results), but just point out a number <strong>of</strong> surprising results that are consequences <strong>of</strong> the geometric theory we study here. 24
Definition 1.3.18. Two R-algebras A and B (or <strong>sheaves</strong> <strong>of</strong> algebras on a scheme X, A and B) are said to be <strong>derived</strong> Morita equivalent if and only if the bounded <strong>derived</strong> <strong>categories</strong> Db (Mod-A) and Db (Mod-B) (or Db coh (Mod(X, A )) and Db coh (Mod(X, B)) in the case <strong>of</strong> a scheme) are equivalent as triangulated <strong>categories</strong>. For interesting examples <strong>of</strong> <strong>derived</strong> Morita equivalence <strong>of</strong> <strong>sheaves</strong> <strong>of</strong> Azumaya algebras, see Section 5.5 and Section 6.6. (These examples do not come from Morita equivalences <strong>of</strong> the algebras involved, at least if one believes Conjecture 1.3.17.) These examples should be contrasted with the following result for Azumaya algebras over local rings: Theorem 1.3.19. Let R be a commutative local ring, A and B Azumaya algebras over R. Then the following are equivalent: 1. [A] = [B] in Br(R). 2. A is Morita equivalent to B. 3. A is <strong>derived</strong> Morita equivalent to B. Pro<strong>of</strong>. [45]. 25
Page 1 and 2: DERIVED CATEGORIES OF TWISTED SHEAV
Page 3 and 4: DERIVED CATEGORIES OF TWISTED SHEAV
Page 5 and 6: Tuturor celor din care am fost făc
Page 7 and 8: Table of Contents
Page 9 and 10: List of Figures 6.
Page 11 and 12: Introduction and Overview Twisted <
Page 13 and 14: twisted by α ∈
Page 15 and 16: fibers introduces interesting featu
Page 17 and 18: Chapter 1 Twisted Sheaves In this c
Page 19 and 20: to be H2 an(X, O∗ X ). If we do n
Page 21 and 22: is bijective. Then A is called an A
Page 23 and 24: such that α = δa and β = δb. Si
Page 25 and 26: Lemma 1.2.3. Let α ∈ Č2 (X, O
Page 27 and 28: Proof. Let U ′
Page 29 and 30: Proof. Since E and
Page 31 and 32: Proof. Follows fro
Page 33: Definition 1.3.12. Let A be a ring.
Page 37 and 38: property “free on stalks” for t
Page 39 and 40: 2.2 The Derived Category and Derive
Page 41 and 42: to reduce to the case when F · als
Page 43 and 44: So assume n > 0, and as before cons
Page 45 and 46: Proposition 2.3.4. Let f : X → Y
Page 47 and 48: 2.4 Duality for Proper Smooth Morph
Page 49 and 50: complex manifolds, and let X × Y
Page 51 and 52: if v consists only of</stro
Page 53 and 54: The reason for the notation is that
Page 55 and 56: Proof. The first s
Page 57 and 58: which by definition associates to a
Page 59 and 60: Part II Applications 49
Page 61 and 62: X which would intersect a general f
Page 63 and 64: to C ′ , independent of</
Page 65 and 66: Jij → S. This means that if Fij w
Page 67 and 68: Li = ρ∗ i L −1 |Xi . Then we h
Page 69 and 70: It is easy to see that ϕi(ti) = [O
Page 71 and 72: Remark 4.5.3. Note that we have fix
Page 73 and 74: allow us to get X; however, finding
Page 75 and 76: endowed with the product ((r, l, s)
Page 77 and 78: Theorem 5.1.10. Under the hypothese
Page 79 and 80: groups of differen
Page 81 and 82: Let c be the image of</stro
Page 83 and 84: Proof. Trivial cha
Page 85 and 86:
and therefore rk(V ) = (v(V ), (0,
Page 87 and 88:
for some k ≫ 0. By the same argum
Page 89 and 90:
and therefore Now consider the map
Page 91 and 92:
In order to identify the kernel, no
Page 93 and 94:
for all F , G stable sheave
Page 95 and 96:
Chapter 6 Elliptic Calabi-Yau Three
Page 97 and 98:
Definition 6.1.2. A projective morp
Page 99 and 100:
Example 6.2.1. We’ve already seen
Page 101 and 102:
contract 45 curves 2H+D-E contract
Page 103 and 104:
Remark 6.3.2. This theorem basicall
Page 105 and 106:
But this is impossible, because the
Page 107 and 108:
We have H 0 (C, IQ) = 0 and χ(IQ)
Page 109 and 110:
Having fixed a smooth point P ∈ C
Page 111 and 112:
101 Proof. Since X
Page 113 and 114:
103 Therefore, α can be represente
Page 115 and 116:
Corollary 6.5.6. Under the hypothes
Page 117 and 118:
107 Remark 6.6.4. We could also pro
Page 119 and 120:
109 where the intersection form on
Page 121 and 122:
111 where ci = ci(i∗U · ). Obvio
Page 123 and 124:
113 One space of i
Page 125 and 126:
Open Questions and Further Directio
Page 127 and 128:
Bibliography [1] Aspinwall, P., Mor
Page 129 and 130:
119 [28] D. Bayer and M. Stillman,
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derived categories of twisted sheaves on calabi-yau manifolds

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