# Friday, March 25 ## Flasque Sheaves :::{.remark} Important classes of sheaves: - Universal: flasque or flabby. - Classical topologies (Hausdorff, paracompact): fine $\implies$ soft. - AG: quasicoherent sheaves on affine sets and covers. ::: :::{.theorem title="Sufficient conditions for acyclicity"} Suppose $\cat A\in \Ab\Cat$ has enough injectives and $\mcf \in \Cat(\cat A,\cat B)$ is left exact. Suppose $\mcc \subseteq \Ob \cat A$ satisfies - Any $A\in \cat A$ admits an embedding $A\embeds C$ for some $C\in \mcc$. - If $A_1 \bigoplus A_2 \in \mcc$ then $A_1, A_2\in \mcc$. - Given a SES $0\to A\to B\to C\to 0$ with $A, B\in \mcc$, $C\in \mcc$ and $0\to FA\to FB\to FC\to 0$ is exact. Then every $C\in \mcc$ is $F\dash$acyclic. ::: :::{.exercise title="?"} Use this to show that flasque implies $F\dash$acyclic for $F(\wait) \da \globsec{\wait}$. ::: :::{.solution} Recall $U \subseteq X$ open $\implies F(X) \surjects F(U)$. - Take an embedding $0\to F\to \prod_{x\in X} (\iota_x)_* F_x$ where $\iota_x: \ts{x} \embeds X$. Use that for any group $A$, $\mcg \da (\iota_x)_* A$ satisfies $\mcg(X) \surjects \mcg(S)$ for any $S \subseteq X$ since $\mcg$ is flasque and soft and this is preserved under products. - Apply the lifting property to direct sums. - Use that restrictions of flasque sheaves to opens are again flasque to prove that there is a surjection: \begin{tikzcd} {B(X)} && {C(X)} \\ \\ {B(U)} && {C(U)} \arrow[two heads, from=1-1, to=1-3] \arrow[two heads, from=3-1, to=3-3] \arrow[two heads, from=1-1, to=3-1] \arrow["\therefore", dashed, two heads, from=1-3, to=3-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJCKFgpIl0sWzIsMCwiQyhYKSJdLFswLDIsIkIoVSkiXSxbMiwyLCJDKFUpIl0sWzAsMSwiIiwwLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzIsMywiIiwwLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzAsMiwiIiwxLHsic3R5bGUiOnsiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV0sWzEsMywiXFx0aGVyZWZvcmUiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifSwiaGVhZCI6eyJuYW1lIjoiZXBpIn19fV1d) ::: :::{.proof title="of theorem"} Any injective is in $\mcc$ by assumption: since $J\embeds C$ splits for any injective $J$, one has $C\cong J \oplus J'$, making $J$ a direct summand and thus in $\mcc$ by the 2nd property. Since there are enough injectives, form $0\to C\to I \to C''\to 0$. Take the LES, using that $\RR^{>0 } FI = 0$ to obtain \begin{tikzcd} 0 & FC & FI & {FC''} & 0 \\ 0 & {\RR^1 FC} & {\RR^1 FI = 0} & {\RR^1 FC''} \\ & {\RR^2 FC} & {\RR^2 FI = 0} & {\RR^2 FC''} \\ & \cdots \arrow[from=2-1, to=2-2] \arrow[from=2-2, to=2-3] \arrow[from=2-3, to=2-4] \arrow["\cong"{description}, from=2-4, to=3-2] \arrow[from=3-2, to=3-3] \arrow[from=3-3, to=3-4] \arrow["\cong"{description}, from=3-4, to=4-2] \arrow[from=1-1, to=1-2] \arrow[from=1-2, to=1-3] \arrow[from=1-3, to=1-4] \arrow[from=1-4, to=1-5] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsMTMsWzAsMCwiMCJdLFsxLDAsIkZDIl0sWzIsMCwiRkkiXSxbMywwLCJGQycnIl0sWzQsMCwiMCJdLFswLDEsIjAiXSxbMSwxLCJcXFJSXjEgRkMiXSxbMiwxLCJcXFJSXjEgRkkgPSAwIl0sWzMsMSwiXFxSUl4xIEZDJyciXSxbMiwyLCJcXFJSXjIgRkkgPSAwIl0sWzEsMiwiXFxSUl4yIEZDIl0sWzMsMiwiXFxSUl4yIEZDJyciXSxbMSwzLCJcXGNkb3RzIl0sWzUsNl0sWzYsN10sWzcsOF0sWzgsMTAsIlxcY29uZyIsMV0sWzEwLDldLFs5LDExXSxbMTEsMTIsIlxcY29uZyIsMV0sWzAsMV0sWzEsMl0sWzIsM10sWzMsNF1d) ::: :::{.remark} There is a canonical flasque resolution: \begin{tikzcd} 0 & \mcf & {S(F) \da \prod_{x\in X} (\iota_x)_* \mcf_X} \\ \\ 0 & \mcg & {S(\mcg)} \arrow[from=1-1, to=1-2] \arrow[from=1-2, to=1-3] \arrow[from=3-1, to=3-2] \arrow[from=3-2, to=3-3] \arrow[from=1-2, to=3-2] \arrow[dashed, from=1-3, to=3-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMCwwLCIwIl0sWzEsMCwiXFxtY2YiXSxbMiwwLCJTKEYpIFxcZGEgXFxwcm9kX3t4XFxpbiBYfSAoXFxpb3RhX3gpXyogXFxtY2ZfWCJdLFswLDIsIjAiXSxbMSwyLCJcXG1jZyJdLFsyLDIsIlMoXFxtY2cpIl0sWzAsMV0sWzEsMl0sWzMsNF0sWzQsNV0sWzEsNF0sWzIsNSwiIiwxLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d) This is useful e.g. for finite sets with the order topology, but less useful if $\abs{X}$ is infinite and there are non-closed points. ::: :::{.exercise title="?"} Show that if $X$ is Hausdorff paracompact, flasque implies soft. As a corollary, soft sheaves are acyclic for such spaces. ::: :::{.solution} See notes. ::: ## Fine Sheaves :::{.remark} Recall that a sheaf is fine iff it satisfies the POU property. - Classically: there is an open cover $\mcu \covers X$ and $\phi_i: U_i \to \RR$ with $\supp \phi_i \subseteq U_i$ where $\sum \phi_i = 1$ and locally there are only finitely many nonzero $\phi$. - For sheaves: there is an open cover $\mcu \covers X$ and $\phi_i: \mcf \to \mcf$ with $\supp \phi$ a closed set $Z_i$ where $\sum \phi_i = \id_\mcf$ and locally there are only finitely many $i$ with $\phi(\mcf)\neq 0$. ![](figures/2022-03-25_11-02-23.png) ::: :::{.example title="?"} Suppose $X$ is Hausdorff paracompact, set $\mcf \da \OO_X^\cts$. Thus $\OO_X$ has a POU property, as does any $\OO_X\dash$module. Take a usual POU $\ts{f_i}$ and define \[ \phi: \mcf &\to \mcf \\ s &\mapsto f_i s .\] So any $\mcf\in \mods{\OO_X^\cts}$ is soft. ::: :::{.remark} In this case, fine implies soft. ::: ## de Rham and Dolbeaut cohomology :::{.remark} Let $X$ be a smooth manifold over $\RR$. Note that $\constantsheaf{\RR}$ is not fine and not soft, and not even an $\OO_X\dash$module. However it admits a resolution $0 \from \constantsheaf{\RR} \cocovers \cocomplex{\Omega_{X}}$ where $\Omega^0_{X} \da \OO_X^{\smooth}$, and this resolution computes the sheaf cohomology $\cocomplex{H}(X; \constantsheaf{\RR})$. Similarly, $0 \to \constantsheaf{\CC} \cocovers_{\delbar} \Omega^{0, \bullet}$ where $\delbar = \sum \dd{}{\bar{z}_i } dz_i$. :::