---- > [!definition] Definition. ([[sheaf cohomology]]) > If $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0$ is a [[short exact sequence]] of [[sheaf|sheaves]], [[global sections functor is left-exact|then]] [[exact functor|left-exactness]] of the global sections functor $\Gamma(X, -)$ implies $0 \to \Gamma(X, \mathcal{F}') \to \Gamma(X, \mathcal{F}) \to \Gamma(X, \mathcal{F}'')$ is [[exact sequence|exact]]. But we don't know anything about surjectivity on the right. In some sense, the goal of sheaf cohomology is to extend this sequence to better understand $\Gamma(X, \mathcal{F}'')$ > > No time to develop the language of right [[derived functor|derived functors]] in this course, so he will tell us the answer. > > **Fact.** There exist [[covariant functor|covariant functors]] ($i \geq 0$) $H^{i}(X, \bullet):\mathsf{Shv}_{\mathsf{Ab}}(X) \to \mathsf{Ab}$ with the following properties. > > **1.** $H^{0}(X, \mathcal{F})=\Gamma(X, \mathcal{F})$ > > **2.** (Zig-Zag) ([[long exact sequence on homology induced by short exact sequence of chain complexes|relation here?]]) Whenever $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0$ [[short exact sequence|short]] [[exact sequence|exact]], there are **connecting maps** $\delta:H^{i}(X, \mathcal{F}'') \to H^{i+1}(X, \mathcal{F}')$ fitting into a long [[exact sequence]] > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \tikzset{ Curved/.style={ rounded corners,to path={ -- ([xshift=2ex]\tikztostart.east) |- (#1) [near end]\tikztonodes -| ([xshift=-2ex]\tikztotarget.west) -- (\tikztotarget) } } } \begin{document} \begin{tikzcd} 0 \arrow[r] & H^0(X, \mathcal{F}') \arrow[r] & H^0(X, \mathcal{F}) \arrow[r] \arrow[d, phantom, ""{coordinate, name=A}] & H^0(X, \mathcal{F}'') \arrow[dll, Curved=A, "\delta"] \\ & H^1(X, \mathcal{F}') \arrow[r] & H^1(X, \mathcal{F}) \arrow[r] \arrow[d, phantom, ""{coordinate, name=B}] & H^1(X, \mathcal{F}'') \arrow[dll, Curved=B, "\delta"] \\ & H^2(X, \mathcal{F}') \arrow[r] & \dots & \ \end{tikzcd} \end{document} > ``` > > > **3.** (Naturality) Given a morphism of [[short exact sequence|short exact sequences]] > > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRGJAF9T1Nd9CKAIzkqtRizYAdKQFs6OABYBjRsABinAORceIDNjwEiAJlHV6zVohAz5S1Qw2ddvQwKIBmc+KvS5Cipqmlo63G78xigALD6WkjYc4fp8RoLIsUJi8dbsrinuUSSkWRYSuUl6BpHpIqW+CbYBDmoA4tr51WmmJdnl-vZBTu2dqR4o3vU5A4GOwO2hXGIwUADm8ESgAGYAThCySGQgOBBIQsm7+2fUJ0gmF3sHiGbHp4ieD1fvN2-Rn08ANh+SAA7P9QcDEAAOcHQyEATlh8MhAFZYSJXqDYS9btDYd5MYhERROEA > \begin{tikzcd} > 0 \arrow[r] & \mathcal{F}' \arrow[r] \arrow[d] & \mathcal{F} \arrow[r] \arrow[d] & \mathcal{F}'' \arrow[r] \arrow[d] & 0 \\ > 0 \arrow[r] & \mathcal{G}' \arrow[r] & \mathcal{G} \arrow[r] & \mathcal{G}'' \arrow[r] & 0 > \end{tikzcd} > \end{document} > ``` > > the following sequence commutes for all $i$: > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAAkA9LACgA1SAAgA6wgLZ0cACwDGjYADEAvgHIVAShBLS6TLnyEUARnJVajFmy7AsAaiNL+Q0ROlyGi1Zu27seAkRkRmb0zKyIHNxOIuKSsvIA4qoaWjogGH4GRCbB1KGWEdZ2DtEuce7ASSlKZjBQAObwRKAAZgBOEGJIZCA4EEgATD4g7Z1IJr39iADMw6NdiD1943kW4SCisAw4dKmtHQsD1Mszq2FsmzDbuzVKQA > \begin{tikzcd} > {H^i(X, \mathcal{F}'')} \arrow[d] \arrow[r, "\delta"] & {H^{i+1}(X, \mathcal{F}')} \arrow[d] \\ > {H^i(X, \mathcal{G}'')} \arrow[r, "\delta"] & {H^{i+1}(X, \mathcal{G}')} > \end{tikzcd} > \end{document} > ``` > > **4.** When $\mathcal{F}$ is [[flasque sheaf|flasque]], we have $H^{i}(X, \mathcal{F})=0$ for all $i>0$. > > Moreover, these four properties completely determine $H^{i}$. > [!basicproperties] > - [[sheaf cohomology is well-behaved with respect to the dimension of Noetherian spaces]] > - [[when do Čech and sheaf cohomology agree?]] ^properties ![[CleanShot 2025-05-31 at [email protected]]] **(a).** Put $U=X-\text{supp }s=\{p \in X: s_{p}=0 \in \mathcal{F}_{p}\}=\{ p \in X:s |_{V_{p}}=0 \text{ for some open } V_{p} \ni p \}$. Evidently, $p \in U$ implies $V_{p} \subset U$. Since the $V_{p}$ [[cover]] $U$ (one per $p \in U$), have that $U=\bigcup_{}V_{p}$ is open. **(b).** The first map ([[inclusion map|inclusion]]) is of course [[injection|injective]]. We want to show the kernel of the second map (restriction) can be identified with $\Gamma_{Z}(X, \mathcal{F})$. So let $s \in \Gamma(X, \mathcal{F})$ be such that $s |_{X-Z}=0$. We want to show $\text{supp }s \subset Z$, i.e., $s_{p} \neq 0 \implies p \in Z$. It is clear that $p \in X-Z \implies s_{p}=0$, which is the contrapositive of this, so good. Hence the sequence $0 \to \Gamma_{Z}(X, \mathcal{F}) \to \Gamma(X, \mathcal{F}) \to \Gamma(X-Z, \mathcal{F})$ is exact. Let $0 \to \mathcal{F}_{1} \xrightarrow{f} \mathcal{F}_{2} \xrightarrow{g} \mathcal{F}_{3} \to 0$ be exact. We know this implies $f$ is an [[injective sheaf morphism]] and $g$ is a [[surjective sheaf morphism]]. By definition of [[injective sheaf morphism]] this means **(c)** The hypotheses in [[when do Čech and sheaf cohomology agree?]] apply, so we can use [[Čech sheaf cohomology|Čech cohomology]]? To begin, $H^{0}_{Z}(X, \mathcal{O}_{X})=\Gamma_{Z}(X, \mathcal{O}_{X})$. Since $\langle x,y \rangle$ is maximal, $V(\langle x,y \rangle)=\langle x,y \rangle$. So a section $s \in \Gamma(X, \mathcal{O}_{X})=k[x,y]$ has support in $Z$ if and only if $s_{\mathfrak{p}}=0$ for all [[prime ideal|prime ideals]] $\mathfrak{p}$ of $k[x,y]$ other than $\langle x,y \rangle$. So $\begin{align} H^{0}_{Z}(X, \mathcal{O}_{X})=\Gamma_{Z}(X, \mathcal{O}_{X})&=\{ f \in k[x,y]: f |_{X-Z}=0 \} \\ &= \{ f \in k[x,y]: f \in \langle x,y \rangle \} \\ &= \langle x,y \rangle . \end{align}$ ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```