Tags: #study-guides
Sheaves
Definitions
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What is a presheaf?
- What is a sheaf?
- What is the equalizer characterization of a sheaf?
- What is the sheafification of a presheaf?
- What is a morphism of presheaves? Of sheaves?
- What is a germ?
- What is the stalk of a sheaf?
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Some sheafy constructions:
- What is the kernel sheaf?
- What is the image sheaf?
- What is the cokernel sheaf?
- What is the pullback/pushforward of a sheaf?
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Some special sheaves:
- What is the structure sheaf of a scheme?
- What is a constant sheaf?
- What is the skyscraper sheaf?
- What is the canonical sheaf?
- What is the dualizing sheaf?
- What is the ideal sheaf?
- What is a local system?
- What is an invertible sheaf?
- What is a constructible sheaf?
- What is a locally free sheaf?
- What is a quasicoherent sheaf?
- What is a coherent sheaf?
- What is the sheaf of rational functions on a scheme?
- What is the tangent sheaf?
- What is the sheaf of relative differentials?
- What does it mean to twist a sheaf?
- What is the ideal sheaf?
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What is the six functor formalism?
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- How is the pushforward sheaf defined?
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How is the inverse image sheaf defined?
- What are its stalks?
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What is the extension by zero sheaf?
- What are its stalks?
- What is the exceptional inverse image sheaf?
- What are the twisting sheaves on \({\mathbb{P}}^n\)?
- How are the chains for sheaf cohomology defined?
- What is a flasque sheaf?
- What is a fine sheaf?
- What is the Leray map?
Results
- What’s special about a very ample invertible sheaf?
- What is the Euler sequence?
- What is the conormal sequence?
- For \(X\subseteq {\mathbb{P}}^2\) a plane curve of degree \(d\), show that the genus can be computed as \begin{align*} h^1(X; {\mathcal{O}}_X) = {(d-1)(d-2)\over 2} .\end{align*}
Problems
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Show that the following presheaves need not be sheaves:
- The tensor presheaf \(U\mapsto {\mathcal{F}}(U)\otimes_{{\mathcal{O}}_X(U)} {\mathcal{G}}(U)\).
- The image presheaf \(U \mapsto \operatorname{im}({\mathcal{F}}(U) \to {\mathcal{G}}(U))\)
- Show that the cokernel presheaf \(U\mapsto \operatorname{coker}({\mathcal{F}}(U) \to {\mathcal{G}}(U))\).
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Let \(X = {\mathbb{P}}^1\).
- Show that the skyscraper sequence is exact: \begin{align*} 0 \to {\mathcal{O}}_{{\mathbb{P}}^1}(-1) \to {\mathcal{O}}_{{\mathbb{P}}^1} \to K_{{\left[ {1: 0} \right]}} \to 0 \end{align*}
- Find \(\operatorname{coker}({\mathcal{O}}_X(-2) \to {\mathcal{O}}_X)\)
- Show that \({\mathcal{O}}_X(d_1) \otimes{\mathcal{O}}_X(d_2) { \, \xrightarrow{\sim}\, }{\mathcal{O}}_X(d_1+d_2)\).
- Find \(\operatorname{coker}({\mathcal{O}}_X \to {\mathcal{O}}_X(1){ {}^{ \scriptscriptstyle\oplus^{2} } })\).
- Show that \({\mathcal{O}}_X(d) {}^{ \vee } { \, \xrightarrow{\sim}\, }{\mathcal{O}}_X(-d)\).
- Produce a sequence that is exact on sheaves but not exact on the corresponding presheaves.
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Show that a pushforward of a locally constant sheaf need not be a locally constant sheaf, and conclude that the stalks do not entirely determine a sheaf.
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- Show that pushforward along \(f\) is left exact, and is fully exact if \(f: U\hookrightarrow X\) is injective with closed image.
- Show that if \(i: \left\{{x}\right\}\hookrightarrow X\) then \(i^{-1}{\mathcal{F}}\cong {\mathcal{F}}_x\)
- Show that if \(X\) is a Noetherian separated scheme with the Zariski topology, \({\mathcal{F}}\in{\mathsf{QCoh}}(X)\), and \({\mathcal{U}}\) is any open cover, then the Leray map is an isomorphism \({ {{\check{H}}}^{\scriptscriptstyle \bullet}} ({\mathcal{U}};{\mathcal{F}}) { \, \xrightarrow{\sim}\, } { {H}^{\scriptscriptstyle \bullet}} (X;{\mathcal{F}}) \coloneqq { {{\mathbb{R}}}^{\scriptscriptstyle \bullet}} \Gamma(X; {\mathcal{F}})\)
