---- > [!definition] Definition. ([[sheaf image]]) > Let $\mathcal{F}$ and $\mathcal{G}$ be [[sheaf|sheaves]] over a [[topological space]] $X$, and let $f:\mathcal{F} \to \mathcal{G}$ be a [[morphism of (pre)sheaves|morphism of sheaves]]. The **sheaf image** of $f$ is the [[subpresheaf|subsheaf]] of $\mathcal{G}$ given by [[sheafification]] of its [[presheaf image]]. The notation $\im f$ is (still) used. ^definition > [!basicproperties] > - Taking images and [[(pre)sheaf stalk|stalks]] are compatible: $(\im f)_{p}=\im(f_{p}:\mathcal{F}_{p} \to \mathcal{G}_{p})$.[^1] > - If $f$ is an [[injective sheaf morphism]], then the [[presheaf image]] is a [[sheaf]] and [[sheafification|sheafifying]] therefore adds no new information. ^properties [^1]: It may seem like there is something tricky going on here, since the definition of $\im f$ involves a [[sheafification]]. But recall that sheafification preserves stalks, and this is a result about stalks. So the sheafification does not affect the discussion. > [!justification]+ > We need to justify that the sheaf image can be naturally identified with a subsheaf of $\mathcal{G}$ — that 'the sheafification can be done inside $\mathcal{G}. For the discussion that follows, that [[presheaf image]] will be denoted $\im f$ and the sheaf image (its sheafification) will be denoted $\im ^{+}f$. > > Let $\iota$ denote the inclusion [[morphism of (pre)sheaves|morphism]] of $\im f$ into $\mathcal{G}$. it induces, via the [[sheafification|universal property of sheafification]], a unique [[morphism of (pre)sheaves|sheaf morphism]] $\iota^{+}:\im ^{+}f \to \mathcal{G} \cong \mathcal{G}^{+}$. > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12cYAPHYLAFsABAF8AZiFGl0mXPkIoAjOSq1GLNp0F0cACwDGjYAHFRwzgYIBzC+x36jDU6IB6AaikyQGbHgJEZEpq9MysiBxcvPxCbu6c3HzAYsKSomowUNbwRKDiAE4QgkhkIDgQSABM1KGaEQl6MDh0INQMdABGMAwACnL+iiD5WNZ6OF55hcWIpeVIKmV0WAxsehAQANat6mFa7PjNEyAFRVXUc4gLteGRvFhwOHDCAIR2B3Qe2+1dvf0KbMNRuN0qIgA > \begin{tikzcd} > \text{im }f \arrow[d, "\theta"'] \arrow[r, "\iota", hook] & \mathcal{G} \cong \mathcal{G}^+ \\ > \text{im}^+\text{ } f \arrow[ru, "\exists ! \iota^+"'] & > \end{tikzcd} > \end{document} > ``` > > $\iota^{+}$ is an [[injective sheaf morphism]], since it is [[sheaf morphism injectivity and surjectivity can be tested on stalks|since it is evidently]] [[injection|injective]] on the stalks $\iota^{+}_{p}=\iota_{p}$. [[sheaf morphism injectivity and surjectivity can be tested on stalks#^corollary|It is therefore an isomorphism onto its image.]] This is the natural identification sought. > [!proof] Proof of Basic Properties. > Suppose $f:\mathcal{F} \to \mathcal{G}$ is an [[injective sheaf morphism]] (a sheaf embedding). We want to show that its [[presheaf image]] $\mathcal{I}$ is actually a [[sheaf]]. > > > **Locality.** Suppose we are given an [[cover|open cover]] $\{ U_{i} \}_{i \in I}=U$ and $f_{U}(s) \in \mathcal{I}(U)$ satisfying $f_{U}(s) |_{U_{i}}=0$ for all $i \in I$. Since $f_{U}(s)$ in particular belongs to $\mathcal{G}(U)$, we can use the locality condition satisfied by $\mathcal{G}$ to conclude that $f_{U}(s)=0$. (Injectivity was not required here.) > > **Gluing.** Suppose we are given an [[cover|open cover]] $\{ U_{i} \}_{i \in I}=U$ and $t_{i} \in \mathcal{I}(U_{i})$ satisfying $t_{i} |_{U_{i} \cap U_{j}}=t_{j} |_{U_{i} \cap U_{j}}$ for all $i,j \in I$. The gluing condition satisfied by $\mathcal{G}$ gives rise to unique $t \in \mathcal{G}(U)$ satisfying $t |_{U_{i}}=t_{i}$ for all $i \in I$. In general, it is not expected that $t$ should belong to the $\mathcal{I}(U_{})$. Our claim is that in this case, because $f$ is [[injective sheaf morphism|injective]], we do in fact have $t \in \mathcal{I}(U)$. > > Indeed, since $f$ is an isomorphism onto its image, each $t_{i} \in \mathcal{I}(U_{i})$ lifts uniquely to some $s_{i} \in \mathcal{F}(U_{i})$, such that the vertical maps in the [[natural transformation|naturality square]] are [[isomorphism|isomorphisms]]: > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12BbOnACwDGjYADEAvgAoAqgH0sAShBjS6TLnyEUARnJVajFm049+QhqMmysAAk5C012QCtFy1djwEiZLXvrNWRA5uXkFhAElLOVcVEAwPDSIdX2p-QyDjULNgSOk5W3Z7RxkXJT0YKABzeCJQADMAJwguJDIQHAgkACZUg0CQOplgKzECsc5sFuoGOgAjGAYABTVPTRAGrEq+HCVYxuakHXbOxABmXoCjdknx9mtB4fy7OgdnMRBpuYXlhK8ghhgdR20ywYH6cAgDCwUA+ID4MDoMMQYCYDAY1BwdCwDDYkDBu3qTRaiDaHUOF3SwQEUAgOAKNBgDRwQysBSKbwJAyJ3QxJ3O+kuGUKNLpnAZTJZT0KL2KTnen3mSxWiSCGy2OzEFDEQA > \begin{tikzcd} > \mathcal{F}(U_i) \arrow[d, "f_{U_i} \ \ \sim"'] \arrow[r, "\cdot \vert_{U_i \cap U_j}"] & \mathcal{F}(U_i \cap U_j) \arrow[d, "\sim \ \ f_{U_i \cap U_j}"] \\ > \mathcal{I}(U_i) \arrow[r, "\cdot \vert_{U_i \cap U_j}"'] & \mathcal{I}(U_i \cap U_j) > \end{tikzcd} > \end{document} > ``` > Call their inverses $\varphi_{U_{i}}$ and $\varphi_{U_{i} \cap U_{j}}$. By the diagram: $s_{i} |_{U_{i} \cap U_{j}}= \varphi_{U_{i} \cap U_{j}}\big( f_{U_{i}}(s_{i}) |_{U_{i} \cap U_{j}} \big)=\varphi_{U_{i} \cap U_{j}}(t_{i} |U_{i} \cap U_{j})$ > Since $t_{i} |_{U_{i} \cap U_{j}}=t_{j}|_{U_{i} \cap U_{j}}$, we get that this equals $s_{j} |_{U_{i} \cap U_{j}}$. Thus, the $s_{i}$ agree on their overlaps. Since $\mathcal{F}$ is a [[sheaf]], they glue into a section $s \in \mathcal{F}(U)$, and we can now write $f_{U}(s) |_{U_{i}}\overbrace{=}^{\text{naturality}}f_{U_{i}}(s |_{U_{i}})\overbrace{=}^{\text{gluing consequence}}f_{U_{i}}(s_{i})=t_{i}=t |_{U_{i}}\begin{align} . > \end{align}$ > Since $f_{U}(s)$ and $t$ restrict identically for all $U_{i}$, the locality axiom of $\mathcal{G}$ implies that they are in fact equal. Thus, $t \in \mathcal{I}(U)$. ---- #### ---- #### 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 > ```