derived categories of twisted sheaves on calabi-yau manifolds

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

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

(Here (ϕ −1 i )∨ is the transpose <strong>of</strong> ϕ −1 i , and we have omitted the restrictions to Ui ∩ Uj.) For f ∗ F take ({f ∗ Fi}, {f ∗ ϕij}) on {f −1 (Ui)}. Now consider the case <strong>of</strong> f∗. Choose an open cover {Ui} <strong>of</strong> X such that α is trivial along Ui for all i, and thus f ∗ α is trivial on f −1 (Ui) for all i. Using Corollary 1.2.6 write F as ({Fi}, {ϕij}) on {f −1 (Ui)}. Take f∗F to be given by {f∗Fi}, {f∗ϕij}) on {Ui}. A similar construction works for f!. Remark 1.2.11. Note that if F and G are α-<strong>sheaves</strong>, then Hom(F , G ) is a sheaf without needing to choose a 1-cocycle g as in Lemma 1.2.8. Thus the following lemma makes sense (if we had to choose one, this choice <strong>of</strong> a line bundle would have influenced the space <strong>of</strong> global sections). Proposition 1.2.12. For F , G ∈ Mod(X, α) we have Γ(X, Hom(F , G )) ∼ = Hom(F , G ). (Since Hom(F , G ) is a regular sheaf, it makes sense to consider its global sections.) Pro<strong>of</strong>. Trivial chase through the definitions. Proposition 1.2.13. The functor f∗ is a right adjoint to f ∗ , as functors between Mod(X, α) and Mod(Y, f ∗ α). If f is an open immersion, then f! is a left adjoint to f ∗ . Pro<strong>of</strong>. On X consider the <strong>sheaves</strong> H1 = f∗Hom Mod(Y,f ∗ α)(f ∗ F , G ) and H2 = Hom Mod(X,α)(F , f∗G ). If U is a small enough open set to trivialize α then there are natural isomorphisms H1|U → H2|U which glue along intersections <strong>of</strong> such U’s, to give a natural isomorphism H1 ∼ = H2 and hence (taking global sections) a natural isomorphism HomMod(Y,f ∗ α)(f ∗ F , G ) ∼ = HomMod(X,α)(F , f∗G ). The same pro<strong>of</strong> works for f!. 1.3 Modules over an Azumaya Algebra When dealing with twisting classes α ∈ Br(X), there is a more natural description <strong>of</strong> Mod(X, α) in terms <strong>of</strong> <strong>sheaves</strong> <strong>of</strong> modules over an Azumaya algebra (thus avoiding any problems related to open covers, refinements, etc.) In this section we describe this correspondence. For a thorough treatment <strong>of</strong> Azumaya algebras over a scheme and <strong>of</strong> the Brauer group, the reader should consult [30, Chapter IV]. Lemma 1.3.1 (Skolem-Noether). Let R be a commutative ring, and let E and F be free R-modules <strong>of</strong> finite rank. Then every isomorphism End(E) → End(F ) (as R-algebras) is induced by an isomorphism E → F . 18
Pro<strong>of</strong>. Since E and F must have the same rank, we can choose an isomorphism q : F → E. Compose the given isomorphism End(E) → End(F ) with the isomorphism End(F ) → End(E) induced by q, to get an automorphism <strong>of</strong> End(E). By the classical Skolem-Noether theorem ([30, IV.1.4]), we conclude that there is an automorphism p : E → E that induces this automorphism <strong>of</strong> End(E). q −1 ◦ p is the desired isomorphism E → F . Lemma 1.3.2. Let R and E be as in the previous lemma, and let p be an automorphism <strong>of</strong> E. Then p induces the identity on End(E) if and only if p is multiplication by an unit in R. Pro<strong>of</strong>. Tracing through the definitions we see that the induced map on End(E) takes h ∈ End(E) to p◦h◦p −1 . Therefore p must be in the center <strong>of</strong> End(E), which is known to consist <strong>of</strong> multiplications by elements <strong>of</strong> R. Since p is also invertible, the result follows. Lemma 1.3.3. Let R and E be as before, and let A = EndR(E). It is a noncommutative R-algebra. Then E is naturally a left A-module, E ∨ = HomR(E, R) is naturally a right A-module (note that A = E ⊗R E ∨ ), and the evaluation map E ∨ ⊗A E → R is an isomorphism <strong>of</strong> R-modules. Pro<strong>of</strong>. The module structure on E is given by evaluation, and on E ∨ by composition. It is obvious that the evaluation map fi ⊗A ei ↦→ fi(ei) is well-defined (here fi ∈ E∨ and ei ∈ E). Let e1, . . . , en be a basis for E, and let e∨ 1, . . . , e∨ n be the dual basis. Then we have e ∨ i ⊗A ej = 0 for i = j e ∨ i ⊗A ei = e ∨ j ⊗A ej for all i, j. To see this, let a be ej ⊗R e ∨ j ∈ A. We have a · ej = ej and e ∨ i · a = 0 for i = j. Therefore we have e ∨ i ⊗A ej = e ∨ i ⊗A a · ej = e ∨ i · a ⊗A ej = 0. Also, let a be ej ⊗R e∨ i + ei ⊗R e∨ j ; it has the property that a · ej = ei and e∨ i · a = e∨ j . Therefore e ∨ i ⊗A ei = e ∨ i ⊗A (a · ej) = (e ∨ i · a) ⊗A ej = e ∨ j ⊗A ej. Every element <strong>of</strong> E∨ ⊗A E can be written as rije∨ i ⊗A ej. By the formulas just proved, this equals riie∨ 1 ⊗ e1, which maps under the evaluation map to rii. This map is clearly an R-module isomorphism. 19
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: Proof. Let U ′
Page 31 and 32: Proof. Follows fro
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
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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