---
sort: 022
title: Sheaves
flashcard: "Orals::Sheaves"
---

Tags: #study-guides

# Sheaves

- [x] What is a **presheaf**? #syllabus ✅ 2022-11-28
	- [ ] A functor $F: \Open(X)\op\to \CRing$ (say) preserving terminal obejcts, i.e. for each open $U \subseteq X$ a ring $F(U)$ where $F(\emptyset) = 0$, for every $U\injects V$ a restriction $\Res_{V< U}: F(V)\to F(U)$, where $\Res_{U, U} = \id_U$ and $U\injects V\injects W \leadsto \Res_{VU} \circ \Res_{WV} = \Res_{WU}$.
- [x] What is a **sheaf**? #syllabus ✅ 2022-12-05
	- [ ] A presheaf satisfying unique gluing: given $\ts{\phi_i \in F(U_i)}$ with $\ro{\phi_i}{U_{ij}} = \ro{\phi_j}{U_{ij}}$ there exists a unique $\Phi\in F(\Union_i U_i)$ such that $\ro{\Phi}{U_i} = \phi_i$ for all $i$. 
- [x] Give an example of a presheaf that is not a sheaf. ✅ 2022-11-28
	- [ ] Any presheaf of constant functions, since functions can be constant on disjoint open sets (thus agreeing on overlaps trivially) without being globally constant (jump discontinuities)
- [x] What is the **equalizer characterization** of a sheaf? ✅ 2022-12-04
	- [ ] $$F(U) \rightarrow \prod_{i} F(U_i) {{{} \atop \longrightarrow}\atop{\longrightarrow \atop {}}} \prod_{i, j} F(U_i \cap U_j)$$
      where the maps are the restrictions $U_i \to U_{ij}$ and $U_j\to U_{ij}$.
- [x] What is the **sheafification** of a presheaf? #syllabus ⏫ ✅ 2022-12-05
	- [ ] For $\mcf$ a presheaf, define $\mcf^+(U) \da \Top(U, \Disjoint_{x\in X} \mcf_x)$ as the functions $s$ where $s(p) \in \mcf_p$ and for each $p\in U$ there is a neighborhood $V$ and $t\in \mcf(V)$ such that $\forall q\in V,\,\, t\mid_q = s(q)$. ![](attachments/Pasted%20image%2020221205143016.png)
- [x] What is a **morphism of presheaves**? Of sheaves? #syllabus ✅ 2022-11-28
	- [ ] $\phi: \mcf \to \mcg$ is a collection $\phi_U: \mcf(U) \to \mcg(U)$ fitting into commuting squares with restrictions.
- [x] What is a **germ**? ✅ 2022-11-28
	- [ ] Elements in the stalk $\mcf_p$ are germs of sections of $\mcf$ over $p$.
- [x] What is the **stalk** of a sheaf? ✅ 2022-11-28
	- [ ] $\mcf_p \da \colim_{U\ni p} \mcf(U)$.
- [x] What is the **kernel sheaf**? #syllabus ✅ 2022-11-28
	- [ ] $U\mapsto \ker \phi(U)$, which is already a sheaf.
- [x] What is the **image sheaf**? #syllabus ✅ 2022-11-28
	- [ ] Sheafify $U\mapsto \im \phi(U)$. 
- [x] Give an example to show that the image presheaf is not necessarily a sheaf. ✅ 2022-12-05
	- [ ] Problem: may fail existence.
	      Take $0\to \ul \ZZ \mapsvia{2\pi i} \OO_X \mapsvia{\exp} \OO_X\units \to 0$; this is exact as a sequence of sheaves since it's exact on stalks, but not exact as a sequence of presheaves, showing that $\im_{\pre} \exp \neq (\im_\pre \exp)^+$ (where $\mcf = \mcf^+$ for any sheaf).
	      Cover $\CC\units$ by contractible $U_i$, then take sections $f_i \da \id_{U_i} \in \OO_X\units$; then $f_i\in (\im \exp)^-$ since $\log(z)$ exists on each $U_i$.
	      The sectors glue to $\CC\units$, but there is no global $\log(z)$. So $\id_{\CC\units} \not\in (\im \exp)^-$, so it fails the gluing axiom.
- [x] What is the **cokernel sheaf**? #syllabus ✅ 2022-11-28
	- [ ] For $\phi: \mcf\to \mcg$, sheafify $U\mapsto \coker \phi(U) \da \mcg/ \im \phi(U)$.
- [x] Give an example to show that the cokernel presheaf is not a sheaf. ✅ 2022-11-28
	- [ ] Problem: may fail uniqueness.
	      Let $\mcg \leq \Hol(\CC, \CC)$ be the sheaf of nonvanishing holomorphic functions and take $\exp: \Hol(\CC, \CC)\to \mcg$ and $U = \CC\smz$.
	      Set $f(z) = z$, then $f\in \mcg(U)$ since it's nonvanishing. Cover $U$ by sectors $U_i$, then $f\in \im \ro{\exp}{U_i}$ for all $i$ since $\log f$ exists and $\exp(\log f) = f$, so $f = 0$ in $\coker \ro{\exp}{U_i}$ for all $i$. But $f\not\in \im \exp$ since there is no global logarithm, so $f\neq 0\in \coker \exp$ as a morphism of presheaves.
- [x] What is the **pullback/pushforward** (inverse/direct images) of a sheaf? ⏫ #syllabus ✅ 2022-12-05
	- [ ] For $f:X\to Y$, $\mcf\in \Sh(X), \mcg\in \Sh(Y)$.
	      Pushforward/direct image: already a sheaf
	      $$f_*\mcf(U) =\mcf(f\inv (U)),\qquad f_*\mcf\in \Sh(Y)$$
	      Pullback/inverse image: **sheafify**
	      $$f\inv \mcg(U) \da \colim_{V\contains f(U)} \mcg(V)$$
	      For $f: X\to Y$ an inclusion, $f\inv \mcg(U) = \ro{\mcg}{\im(f)}(U) = \mcg(U)$, so denote $f\inv \mcg \da \ro \mcf X$
- [x] What is the **structure sheaf** of a scheme? ✅ 2022-11-28
	- [ ] Part of the definition of a scheme as a locally ringed space $(X, \OO_X)$; $\OO_X$ is the structure sheaf.
- [x] What is a **constant sheaf**? #syllabus ✅ 2022-11-28
	- [ ] For $A\in \Ab\Grp$, $\ul{A} \da C^0(\wait, A^{\disc})$. On connected sets $U$, $\ul{A}(U) = A$. This is the sheaf of locally constant functions, i.e. constant on every connected component. Stalks are all $A$.
- [x] What is the **skyscraper sheaf**? Its stalks? Its adjoint characterization? #syllabus ✅ 2022-12-04
	- [ ] For the inclusion $\iota: \ts{p} \injects X$, and $\ul{A}$ the constant sheaf on $\ts{p}$, the direct image $\iota_*\ul{A}$.
	      Assigns $U\mapsto \ul{A}(U) \chi_{p\in U}$, stalks are $A$ at each point in $\cl_X(p)$ and 0 elsewhere. Adjoint to taking stalks: $\Set(\mcf_x, A) \iso \Sh(X)(\mcf, (\iota_x)_* A)$.
- [x] What is the **canonical sheaf**? ✅ 2022-11-28
	- [ ] $\omega_{X/S} \da \det \Omega_{X/k} \da \Extpower^n \Omega_{X/S}$ where $n\da \dim X$.
- [x] What is the **dualizing sheaf**? ✅ 2022-12-04 #unknown
	- [ ] For $X$ proper over a field, $\omega_X^\circ\in \Coh(X)$ together with a trace $\trace: H^n(X; \omega_X^\circ)\to k$ such that $\forall \mcf\in \Coh(X)$, the composition $$H^n(X; \mcf) \times \Hom(\mcf, \omega_X^\circ) \to H^n(X, \omega_X^\circ) \mapsvia{\tr}k$$
	      yields an isomorphism $\Hom(\mcf, \omega_X^\circ) \iso H^n(X; \mcf)\dual$ and more generally $\Ext^i(\mcf, \omega_X^\circ) \cong H^{n-i}(X; \mcf)\dual$.
	      Exists for any projective scheme, not always equal to $\det \Omega_{X/k}$.
- [x] What is an **ideal sheaf**? ✅ 2022-12-04 #unknown
	- [ ] For $\iota:Y\to X$ the inclusion of a closed subscheme, $\mci_Y \da \ker(\OO_X \mapsvia{\iota^\sharp} \OO_{Y})$.
- [x] What is a **local system**? ✅ 2022-11-28
	- [ ] A locally constant sheaf, i.e. a sheaf such that every $x\in X$ admits a neighborhood $U\ni x$ such that $\ro\mcf U$ is a constant sheaf.
- [x] What is an **invertible sheaf**? #syllabus ✅ 2022-11-28
	- [ ] $\mcl \in \Sh(X)$ such that $\exists \mcl\inv \in \Sh(X)$ with $\mcl\tensor_{\OO_X} \mcl\inv \cong \OO_X$. 
	- [ ] Equivalently, every $x\in X$ admits a neighborhood $U\ni x$ such that $\ro \mcf U \cong \OO_U$.
	- [ ] Equivalently, a locally free $\OO_X\dash$module of rank 1.
- [x] Discuss invertible sheaves on affine schemes. ✅ 2022-12-04 #unknown 
	- [ ] $\mcf\in \Pic(\spec A)\implies \mcf$ corresponds to $M\in\amod$ which is finite projective locally free of rank 1 and thus flat and finitely presented. Every invertible sheaf on $\AA^2\slice k$ is trivial.
- [x] What is a **vector bundle** over a scheme? #syllabus ✅ 2022-11-28
	- [ ] For $\mce$ a locally free sheaf on $Y$, the associated geometric vector bundle is $\mathbf{V}(\mce)\da \spec \Sym(\mce)$.
	- [ ] More plainly: for $Y\in\Sch$, a geometric vector bundle of rank $n$ over $Y$ is a scheme $X$ with a morphism $f:X\to Y$ along with the data of $\mcu\covers Y$ and isomorphism $\psi_i: f\inv(U_i)\to\AA^n\slice{U_i}$ such that for any $\spec A \subseteq U_{ij}$ each $\psi_{ij} \da \psi_j\circ \psi_i\inv \in \Aut(\AA^n\slice{\spec A})$ is given by a linear automorphism $\theta \in \Aut_{\amod}(A[x_1,\cdots ,x_n])$, so $\theta(a) = a$ for $a\in A$ and $\theta(x_i) = \sum a_{ij} x_j$.
- [x] What is a **constructible sheaf**? ✅ 2022-11-28
	- [ ] $\mcf\in \Sh(X)$ is constructible iff $\exists \ts{i_{Y_i}: Y_i\injects X}_{i\leq N}$ with $X = \Union_{i\leq N}Y_i$, i.e. $Y$ is a finite union of locally closed subschemes, such that $\ro\mcf {Y_i} \da \iota_{Y_i}^* \mcf$ is a finite locally constant sheaf.
- [x] What is a **free sheaf**? ✅ 2022-12-04
	- [ ] $\mcf \cong \OO_X\sumpower{n}$ for some $n$.
- [x] What is a **locally free sheaf**? ✅ 2022-12-04
	- [ ] Each stalk $\mcf_x$ is a free $\OO_{X, x}\dash$module.
      Equivalently, $\exists \mcu\covers X$ such that each $\ro\mcf {U_i}$ is a free $\ro{\OO_X}{U}\dash$module.
- [x] What is the **sheaf of rational functions** on a scheme? ✅ 2022-12-04
	- [ ] Define the function field of $X$ to be $K\da \OO_{X, \eta_X}$, the stalk at the generic point, and define $\mck_X \da \ul{K}$, the constant sheaf associated to this field.
	      Equivalently, sheafify $U\to \ff(\OO_X(U))$.
- [x] What is the fundamental exact sequence involving the tangent sheaf for $\PP^n$? ✅ 2022-12-04
	- [ ] $0\to \OO_X \to \OO_X(1)\sumpower{n+1}\to \mct_X\to 0$.
- [x] Define the **twisting sheaf** on $\PP^n\slice Y$. ✅ 2022-12-04
	- [ ] $\OO_{\PP^n\slice Y}(1) \da g^* \OO_{\PP^n\slice \ZZ}(1)$ where $g: \PP^n\slice Y\to \PP^n\slice \ZZ$.
- [x] Define the **twisting sheaf** on $\Proj S$ for $S$ a graded ring. ✅ 2022-12-04
	- [ ] $\OO_X(n) \da S[n]\tilde{}$, and the twisting sheaf is $\OO_X(1)$.
- [x] What does it mean to **twist a sheaf**? ✅ 2022-11-28
	- [ ] $\mcf(n) \da \mcf\tensor_{\OO_X} \OO_X(n)$.
- [x] Characterize coherent sheaves over $\spec A$ for $A$ Noetherian. ✅ 2022-12-04
	- [ ] $\mcf\in \Coh(\spec A)\implies \exists \bigoplus_{i\in I} \OO(n_i) \surjects \mcf$, i.e. always a quotient of a free sheaf.
	      Easy proof: $\mcf(n)$ is globally generated for some $n$, so $\exists \bigoplus_{i\leq N} \OO_X \surjects \mcf(n)$, now twist this exact sequence by $-n$.
- [x] What is the **six functor formalism**? ✅ 2022-11-25
	- [ ] Direct image (pushforward) and inverse image (pullback), extension by zero (lower shriek) and exceptional image (upper shriek), internal tensor and internal hom.
	- [ ] ![](attachments/Pasted%20image%2020220315152915.png)
- [x] How is the **inverse image sheaf** defined? What are its stalks? ✅ 2022-11-25
	- [ ] $f\inv \mcg(U) \da \colim_{V\contains f(U)} \mcg(V)$. Stalks: $\mcg_p \iso (f\inv \mcg)_p$.
- [x] What is the **extension by zero** sheaf? What are its stalks? ✅ 2022-11-25
	- [ ] Sheafify $U\mapsto \chi_{V\subseteq U}\mcf (V)$ to get $\iota_! \mcf$ for $\iota: U \subseteq X$. Stalks are $\mcf_p \chi_{p\in U}$.
	- [ ] Alternatively take $\iota_* \mcf$.
- [x] How is the pushforward sheaf defined? What are its stalks? ✅ 2022-11-25
	- [ ] $f_*\mcf(V) \da \mcf(f\inv(V))$. Can't say much in general about its stalks.
- [x] What is the restriction of a sheaf? ✅ 2022-11-28
	- [ ] For $\iota: X\injects Y$, $\ro \mcg X \da \iota\inv \mcf$, same stalks as $\mcg$.
- [x] What is a **flasque sheaf**? ✅ 2022-11-28
	- [ ] $U\injects V \implies \mcf(V) \surjects \mcf(U)$.
- [x] Discuss flasque sheaves. ✅ 2022-11-28
	- [ ] $H^{\geq 1}(X; \mcf) = 0$ for any flasque sheaf on any topological space, so are $\Gamma\dash$acyclic.
- [x] What is **Grothendieck vanishing**? ✅ 2022-11-28
	- [ ] For $X$ a noetherian topological space, for all $\mcf \in \Sh(X, \Ab\Grp)$, one has 
	      $$H^{i\geq \dim X+1}(X; \mcf) = 0.$$
	- [ ] How to remember: $\mathrm{cd}(X) \leq \dim X$.
- [x] Is a morphism of sheaves injective if injective on sections? Surjective? ✅ 2022-12-07
	- [ ] Injective: yes, surjective: no.