derived categories of twisted sheaves on calabi-yau manifolds

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

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

Lemma 1.3.4. Again, let R be a commutative ring, E a free R-module, and A = EndR(E). Then for any R modules F and G we have HomR(F, G) ∼ = HomA(F ⊗R E ∨ , G ⊗R E ∨ ), where HomA denotes the group <strong>of</strong> right A-module homomorphisms. Pro<strong>of</strong>. The map in one direction is · ⊗R idE ∨. In the other direction take · ⊗A idE, and use the previous lemma. Theorem 1.3.5. Let A be an Azumaya algebra over X, and let α ∈ Br ′ (X) be the element that A represents (we’ll <strong>of</strong>ten denote α by [A ]). Then there exists a locally free α-<strong>twisted</strong> sheaf E <strong>of</strong> finite rank (not necessarily unique) such that A ∼ = End(E ). Conversely, for any α ∈ Br ′ (X) such that there exists a locally free α-sheaf E <strong>of</strong> finite rank, End(E ) is an Azumaya algebra whose class in Br ′ (X) is α. Thus we have yet another characterization <strong>of</strong> the Brauer subgroup <strong>of</strong> the cohomological Brauer group: it is the subgroup <strong>of</strong> those twistings α for which there exist locally free α-<strong>twisted</strong> <strong>sheaves</strong> <strong>of</strong> finite rank (α-lffr’s, in short). Pro<strong>of</strong>. Find a cover {Ui}, OUi -lffr’s Ei and isomorphisms ϕi : End(Ei) → A |Ui as given by Theorem 1.1.6. By Lemma 1.3.1, the isomorphisms induce isomorphisms ϕ −1 j ◦ ϕi : End(Ei|Ui∩Uj ) → End(Ei|Ui∩Uj ) → Ej|Ui∩Uj . ¯ϕij : Ei|Ui∩Uj The threefold compositions ¯ϕijk are automorphisms <strong>of</strong> Ei such that the corresponding automorphisms on End(Ei) are the identity, hence by Lemma 1.3.2 they must be multiplications by sections αijk <strong>of</strong> O∗ X over Ui ∩ Uj ∩ Uk. Therefore we have found the data for an α-lffr E , where α is the element <strong>of</strong> ˇ H2 (X, O∗ X ) determined by {αijk}. It is easy to see that this is the same correspondence as described in Theorem 1.1.8. The fact that End(E ) is an Azumaya algebra follows immediately from Theorem 1.1.6. Proposition 1.3.6. Let A be an Azumaya algebra over X, let α = [A ], and let E be an α-lffr such that A ∼ = End(E ). Note that E is naturally a left A -module. Define a functor F between the category Mod(X, α) <strong>of</strong> α-<strong>twisted</strong> <strong>sheaves</strong> and the category Mod-A <strong>of</strong> <strong>sheaves</strong> <strong>of</strong> right A -modules on X by the formula ∨ F ( · ) = · ⊗OX E (the right A -module structure on F ( · ) is given by using the right A -module structure <strong>of</strong> E ∨ ). Then, for any pair <strong>of</strong> α-<strong>sheaves</strong> F and G , F induces a functorial isomorphism Hom Mod(X,α)(F , G ) ∼ = Hom Mod-A (F (F ), F (G )). 20
Pro<strong>of</strong>. Follows from Lemma 1.3.4 and the structure theorem for Azumaya algebras (1.1.6). Theorem 1.3.7. Let A be an Azumaya algebra over X. Then the functor F defined above is an equivalence <strong>of</strong> <strong>categories</strong> between Mod(X, α) and Mod-A . Pro<strong>of</strong>. Let E be as before, and define G : Mod-A → Mod(X, α) by the formula G( · ) = · ⊗A E . (The tensor product over A is defined locally, as usual.) From the formulas E ⊗OX E ∨ ∼ = End(E ) ∼ = A and E ∨ ⊗A E ∼ = OX (see lemma 1.3.3) it follows that F and G are inverse to one another. Taking global sections in Proposition 1.3.6 and using Proposition 1.2.12 we see that F is full and faithful, so it is an equivalence <strong>of</strong> <strong>categories</strong>. Remark 1.3.8. It is not hard to see that in fact all the functors we have defined in Proposition 1.2.10 (⊗, Hom, f ∗ , f∗, f!) are compatible with this equivalence. So from now on we’ll just freely switch between the viewpoint that <strong>twisted</strong> <strong>sheaves</strong> are “local <strong>sheaves</strong> that don’t quite match up” and the viewpoint that <strong>twisted</strong> <strong>sheaves</strong> are “modules over an Azumaya algebra”. As an application <strong>of</strong> this, we prove the following theorem, which shows that on a proper scheme over C, <strong>twisted</strong> coherent <strong>sheaves</strong> (in the étale topology) are the same as analytic <strong>twisted</strong> coherent <strong>sheaves</strong> (in the Euclidean topology) on the associated analytic space. Theorem 1.3.9. Let X be a proper scheme over C, let h : X h → X be the natural continuous map from the associated analytic space, and let α be an element <strong>of</strong> Br(X), represented by an Azumaya algebra A . Let A h = h ∗ A , and let α h = a(A h ). Fix an α-lffr E such that End(E ) ∼ = A and an α h -lffr E h such that End(E h ) ∼ = A h . Let F be the functor defined by Then F is exact, its restriction F : Mod(X, α) → Mod(X h , α h ) F ( · ) = h ∗ ( · ⊗OX E ∨ ) ⊗ A h E h . F |Coh : Coh(X, α) → Coh(X h , α h ) is an equivalence <strong>of</strong> <strong>categories</strong>, and we have, for any F , G ∈ Coh(X, α) h ∗ Hom Mod(X,α)(F , G ) = Hom Mod(X h ,α h )(F (F ), G(G )). 21
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: Proof. Since E and
Page 33 and 34: Definition 1.3.12. Let A be a ring.
Page 35 and 36: Definition 1.3.18. Two R-algebras A
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
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111 where ci = ci(i∗U · ). Obvio
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113 One space of i
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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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