---- (This is a necessarily specialized treatment in light of time constraints.) > [!definition] Definition. ([[Čech sheaf cohomology]]) > Let $X$ be a [[topological space]] and $\mathcal{F}$ a [[sheaf]] (say) of [[abelian group|abelian groups]] on $X$. Let $\mathscr{U}:=\{ U_{i} \}_{i \in I}$ be an [[cover|open cover]] of $X$; assuming $I$ is [[well-ordered set|well-ordered]]. Write $U_{i_{0}\dots i_{p}}:=U_{i_{0}} \cap \dots \cap U_{i_{p}} \text{ for }i_{0},\dots,i_{p} \in I.$ > The **group of Čech $p$-cochains** is $\check{C}^{p}(\mathscr{U}, \mathcal{F}):=\prod_{i_{0}<\dots<i_{p} }^{}\mathcal{F}(U_{i_{0}\dots i_{p}}).$ > We write $\alpha \in \check{C}^{p}(\mathscr{U}, \mathcal{F})$ as a tuple $\alpha=(\alpha_{i_{0}\dots i_{p}})_{i_{0}<\dots<i_{p}}$. The **Čech codifferential** is $\begin{align} > d:\check{C}^{p}(\mathscr{U}, \mathcal{F}) &\to \check{C}^{p+1}(\mathscr{U}, \mathcal{F}) \\ > (d\alpha)_{i_{0}<\dots<i_{p+1}} &:= \sum_{k=0}^{p+1} (-1)^{k} \alpha_{i_{0}\dots \widehat{i_{k}} \dots i_{p+1}} |_{U_{i_{0}\dots i_{p+1}}}. > \end{align}$ > The alternating sign ensures $d^{2}=0$, giving us a **Čech [[chain complex of modules|cochain complex]]** $\check C^{\bullet}\check (\mathscr{U}, \mathcal{F}): C^{0}(\mathscr{U}, \mathcal{F}) \xrightarrow{d^{0}} \check{C}^{1} (\mathscr{U}, \mathcal{F})\xrightarrow{d^{1}} \cdots$ > whose $i$th [[(co)homology of a complex|cohomology]] is called the **$i$th Čech cohomology of $\mathcal{F}$ with respect to the cover $\mathscr{U}$** and denoted $\check H^{i}(\mathscr{U}, \mathcal{F})$. > > > [!note] When does $H^{*}(X,\mathcal{F})=\check H^{*}(\mathscr{U}, \mathcal{F})$? > See [[when do Čech and sheaf cohomology agree?]] ^note > [!basicexample] A non-algebro-geometric example. > Put $X:=\mathbb{S}^{1}$, $\mathcal{F}=\underline{\mathbb{Z}}$ the [[constant sheaf]] with coefficients $\mathbb{Z}$. Cover $\mathbb{S}^{1}$ by $\mathscr{U}=\{ U_{0},U_{1} \}$ like so:[^1] > ![[Pasted image 20250518180640.png|200]] > We have >- $\check C^{0}(\mathscr{U}, \mathcal{F})=\mathcal{F}(U_{0}) \times \mathcal{F}(U_{1}) \cong \mathbb{Z} \times \mathbb{Z}$[^2] >- $\check{C}^{1}(\mathscr{U}, \mathcal{F})=\mathcal{F}(\underbrace{ U_{0} \cap U_{1} }_{ \text{disconnected} }) \cong \mathbb{Z} \times \mathbb{Z}$ >- $\check{C}^{n}(\mathscr{U}, \mathcal{F})=0$ for $n \geq 2$ >- If $\alpha=(\alpha_{0},\alpha_{1}) \in \check{C}^{0}(\mathscr{U}, \mathcal{F})$, then $d^{0}\alpha=\alpha_{1} |_{U_{0} \cap U_{1}}-\alpha_{0} |_{U_{0} \cap U_{1}}$. All the other codifferentials are zero. > From these observations, obtain >- $\check H^{0}(\mathscr{U}, \mathcal{F}) \cong \operatorname{ker }d^{0}=\mathbb{Z} \langle 1,1 \rangle$. This is isomorphic to $\Gamma(\mathbb{S}^{1}, \mathcal{F})$, [[sheaf cohomology|as one would hope]]. >- $\check H^{1}(\mathscr{U}, \mathcal{F}) = \operatorname{coker }d^{0} \cong \frac{\mathbb{Z} \times \mathbb{Z}}{\mathbb{Z} \langle 1,-1 \rangle} \cong \mathbb{Z}$. > Thus, $\check {H}^{*}(\mathbb{S}^{1}, \underline{\mathbb{Z}})$ is precisely the [[singular cohomology]] of $\mathbb{S}^{1}$ with $\mathbb{Z}$-[[(co)homology with coefficients|coefficients]]. This is not a coincidence. ^basic-example [[projective space|Consider]] $X:=\mathbb{P}^{1}_{k}=\text{Proj }k[X_{0},X_{1}]$ for $k$ a [[field]]. $X$ is covered by the [[affine scheme|principal open affines]] $U_{0}=D_{+}(X_{0})$, $U_{1}=D_{+}(X_{1})$; write $\mathscr{U}=\{ U_{0},U_{1} \}$. [[universal line bundle on projective space|Take]] $\mathcal{F}:=\mathcal{O}_{\mathbb{P}^{1}_{k}}(-2)=\big(\mathcal{O}_{\mathbb{P}^{1}_{k}}(1) ^{\otimes 2} \big)^{\vee}$. *What is $\check{H}^{i}(X, \mathcal{F})$?* Recall that $\mathcal{F}$ [[lines bundles and transition functions|is obtained]] by starting with the cover $\mathscr{U}$ and a single transition function $g_{01}$ as that of $\mathcal{F}$ (=$\frac{X_{0}}{X_{1}}$) [[tensor product sheaf of modules|squared]] and [[dual sheaf of modules|inverted]] . In other words, $\mathcal{F}$ has trivializing cover $\mathscr{U}$ and transition functions $\frac{X_{1}^{2}}{X_{0}^{2}} \in \mathcal{O}^{*}_{\mathbb{P}^{1}_{}}(-2)(U_{0} \cap U_{1})$. The Čech cochain groups are $\begin{align} \check C^{0}(\mathscr{U}, \mathcal{F})&=\mathcal{O}_{\mathbb{P}^{1}}(-2)(U_{0}) \times \mathcal{O}_{\mathbb{P}^{1}}(-2)(U_{1}) \\ &=\mathcal{O}_{\mathbb{P}^{1}}(D_{_{+}}(X_{0})) \times \mathcal{O}_{\mathbb{P}^{1}}(D_{_{+}}(X_{1})) \\ &= k[X_{0},X_{1}]_{(X_{0})} \times k[X_{0},X_{1}]_{(X_{1})} \\ &= k\left[ \frac{X_{1}}{X_{0}} \right] \times k\left[ \frac{X_{0}}{X_{1}} \right] \end{align}$ and $\begin{align} \check C^{1} (\mathscr{U}, \mathcal{F}) &= \mathcal{F}(U_{01}) \\ &= \mathcal{O}_{\mathbb{P}^{1}} (-2) (U_{0} \cap U_{1}) \\ &= \underbrace{ \mathcal{O}_{\mathbb{P}^{1}} (-2) |_{U_{0}} }_{ \cong \mathcal{O}_{\mathbb{P}^{1}} |_{U_{0}} } (U_{0} \cap U_{1}) \\ &\cong \mathcal{O}_{U_{0}} \big(\underbrace{ D_{+}(X_{0}) \cap D_{+}(X_{1}) }_{ =D_{+}(X_{0}X_{1}) } \big) \\ &= k\left[ \frac{X_{1}}{X_{0}} \right] _{(X_{0}X_{1})} \\ &= k\left[ \frac{X_{1}}{X_{0}} , \frac{X_{0}}{X_{1}}\right]. \end{align}$ As for the codifferental $\begin{align}d: \check{C}^{0}(\mathscr{U}, \mathcal{F}) &\to \check{C}^{1}(\mathscr{U}, \mathcal{F}) \\\end{align}$, let $f\left( \frac{X_{1}}{X_{0}} \right)$, $g\left( \frac{X_{0}}{X_{1}} \right)$ be an arbitrary pair of elements in $\mathcal{F}(U_{0})=k\left[ \frac{X_{1}}{X_{0}} \right]$, $\mathcal{F}(U_{1})=\frac{X_{0}}{X_{1}}$ respectively. We have $\begin{align} d \ \left( f\left( \frac{X_{1}}{X_{0}} \right), g\left( \frac{X_{0}}{X_{1}} \right) \right) &= g\left( \frac{X_{0}}{X_{1}} \right) |_{U_{0} \cap U_{1}} - f\left( \frac{X_{1}}{X_{0}} \right) |_{U_{0} \cap U_{1}} \\ &= g\left( \frac{X_{0}}{X_{1}} \right) \frac{X_{0}^{2}}{X_{1}^{2}} - f\left( \frac{X_{1}}{X_{0}} \right) \in k\left[ \frac{X_{1}}{X_{0}} , \frac{X_{0}}{X_{1}}\right]. \end{align}$ (Recall that viewing $\check{C}^{1}(\mathscr{U}, \mathcal{F})$ as $k\left[ \frac{X_{1}}{X_{0}}, \frac{X_{0}}{X_{1}} \right]$ involved choosing a trivialization to restrict to; we chose $U_{0}$. $g$ comes from $U_{1}$, however, so in order to compare it to $f$ (which comes from $U_{0}$) in this [[ring]], we need to multiply by $\frac{X_{0}^{2}}{X_{1}^{2}}$. ) The kernel of $d$ is zero: fixing notation $T=\frac{X_{1}}{X_{0}}$, $T^{-1}=\frac{X_{0}}{X_{1}}$, it is clear that one can never have $g(T ^{-1})T^{-2}=f(T)$. For the cokernel of $d$: the image of $d$ has any polynomial in $T$, and any polynomial $T^{-1}$ starting in degree $2$. The one thing that we can't get are the degree 1 polynomials in $T ^{-1}$... so $\operatorname{coker }d=k \cdot \frac{X_{0}}{X_{1}}$, a one-dimensional [[vector space]]. ---- #### [^1]: Usually in a algebro-topological setting it will usually not be helpful to choose a coarse cover like this when doing Čech cohomology. More common is to work with all open covers at once and take some [[categorical limit|limit]]. Fortunately, in algebraic geometry the covers are frequently nice.o [^2]: Using that $U_{0},U_{1}$ are [[connected]] — locally constant implies globally constant for connected spaces, so we get one function per element of $\mathbb{Z}$. [[sheaf cohomology]] ---- #### 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 > ```