----- > [!proposition] Proposition. ([[lines bundles and transition functions]]) > Let $(X, \mathcal{O}_{X})$ be a [[ringed space]]. If $\mathcal{L}$ is a [[locally free sheaf|line bundle]] on $X$ with trivializing [[cover]] $\{ U_{i} \}$,[^1]so that we have ([[sheaf|sheaf of]]) $\mathcal{O}_{X}$-[[module]] [[linear map|isomorphisms]] $\mathcal{L} |_{U_{i}} \xrightarrow{\varphi_{i}} \mathcal{O}_{U_{i}}$ for each $i$. > > On a given overlap $U_{i} \cap U_{j}$, the composition > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12BbOnACwDGjYAHkAvgH1gAVQlYABJyFp5sgFZiQY0uky58hFACZyVWoxZtOPfkIajJMuYvbLVEjVp0gM2PASIARlNqemZWRA5uXkFhABkxeQAfKVkFJToVdU0xMxgoAHN4IlAAMwAnCC4kMhAcCCQTc3CrdnpytD4FOQA9YABaQMSUp3TXTPdPbTLK6sQm+qRg5stIznbOrA9k1OcMrI8c7wqqmupFxGWAIxgwKCR+gGZasNWkAF51ug6uqSxPagMOg3BgABT0-kMIHKWAKfBwWgoYiAA > \begin{tikzcd} > \mathcal{O}_{U_i \cap U_j} \arrow[r, "\varphi _i^{-1} |_{U_i \cap U_j}"] \arrow[rr, ":=\varphi_{ij}"', bend right] & \mathcal{L} |_{U_i \cap U_j} \arrow[r, "\varphi_j |_{U_i \cap U_j}"] & \mathcal{O}_{U_i \cap U_j} > \end{tikzcd} > \end{document} > ``` > > defines an [[automorphism]] $\varphi_{ij}:\mathcal{O}_{U_{i} \cap U_{j}} \to \mathcal{O}_{U_{i} \cap U_{j}}$[^2] of rank-$1$ [[free sheaf of modules|free sheaves]]. The collection of $\varphi_{ij}$s are called the **transition functions** of $\mathcal{L}$. > > In this [[locally free sheaf|line bundle]] case, specifying a transition function $\varphi_{ij}:\mathcal{O}_{U_{i} \cap U_{j}} \to \mathcal{O}_{U_{i} \cap U_{j}}$ [[lines bundles and transition functions#^justification|is equivalent to]] specifying [[unit|units]] $g_{ij} \in \mathcal{O}_{X}^{*}(U _{i } \cap U_{j})$,[^3] because any [[automorphism]] of a rank-$1$ [[free module]] is precisely multiplication by an invertible [[ring]] element.For this reason, the $g_{ij}$ are *also* called **transition functions** of $\mathcal{L}$. > > The $g_{ij}$s satisfy the [[cocycle conditions|cocycle conditions]] (on $U_{i} \cap U_{j} \cap U_{k}$ we have $g_{ik}=g_{ij}g_{jk}$ ). *In fact,* it follows from [[sheaf gluing]] that specifying the trivializing cover $\{ U_{i} \}$ and transition functions $\{ g_{ij} \}$ (equivalently, $\{ \varphi_{ij} \}$) *specifies the whole line bundle $\mathcal{L}$*. > > This is the content of the (short) proof below. > [!justification] > Indeed, once the $(U_{i} \cap U_{j})$-component (the 'global component') of the [[isomorphism]] $\varphi_{ij}$ is determined to be multiplication by $g_{ij} \in \mathcal{O}_{X}^{*}(U_{i} \cap U_{j})$, the other [[natural transformation|components]] (and hence the entire [[morphism of (pre)sheaves|sheaf morphism]]) are determined: if $(\varphi_{ij})_{W}$ denotes the $W$-[[natural transformation|component]] of $\varphi_{ij}:\mathcal{O}_{U_{i} \cap U_{j}} \to \mathcal{O}_{U_{i} \cap U_{j}}$, then the [[natural transformation|naturality square]] > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12BbOnACwDGjYAHkAvgH1gAVQlYABJyFp5sgFZiAFLIVK6K9QEoQY0uky58hFAEZyVWoxZtOPfkIajJMuXoMSNTQB1Y1NzbDwCIgAme2p6ZlZEEGDFdjgmACM4GBxVOTTlfLVQsxAMCKsiMhsHBOdk115BYXEpHUL9Yq0Ov2LS8Mso21Ja+Kckjm5mjy9233Yi9S0QkwcYKABzeCJQADMAJwguJDsQHAgkABZxxLZNTnoDtD4sKSwNQwkgkzLD46QZHOl0QAGZbg0QJt3hoQNQGHRMjAGAAFCyRawgA5YTZ8HC-fZHE6IIEXU4QyZKKAQPIAH2+BJA-2J4OB1wpLkW1LpDLEFDEQA > \begin{tikzcd} > \mathcal{O}_{U_i \cap U_j}(U_i \cap U_j) \arrow[d, "(-) \cdot g_{ij}"'] \arrow[r, "\cdot |_W"] & \mathcal{O}_{U_i\cap U_j}(W) \arrow[d, "(\varphi_{ij})_W"] & (W \subset U_i \cap U_j) \\ > \mathcal{O}_{U_i \cap U_j}(U_i \cap U_j) \arrow[r, "\cdot |_W"] & \mathcal{O}_{U_i\cap U_j}(W) & > \end{tikzcd} > \end{document} > ``` > > tells us $(\varphi_{ij})_{W}(s)=s \cdot (\varphi_{ij})_{W}(1)=s \cdot (\varphi_{ij})_{W}( 1 |_{W}) =s \cdot (g_{ij} |_{W})$ for any $s \in \mathcal{O}_{U _{i} \cap U_{j}}(W)$. ^justification > [!proof]+ Proof. ([[lines bundles and transition functions]]) > Let $(X, \mathcal{O}_{X})$ be a [[ringed space]], and view $\mathcal{O}_{X}$ as a rank-$1$ [[free sheaf of modules|free sheaf]] of $\mathcal{O}_{X}$-[[sheaf of modules|modules]]. > > Suppose we are given an [[cover|open cover]] $\{ U_{i} \}$ of $X$ together with elements $g_{ij} \in \mathcal{O}_{X}^{*}(U_{i} \cap U_{j})$ for all overlaps $U_{i} \cap U_{j}$ satisfying the [[cocycle conditions]] $g_{ii}=\id_{U_{i}}$ and $g_{ik}=g_{ij}g_{jk}$ on $U_{i}\cap U_{j} \cap U_{k}$. > > > > The open cover $\{ U_{i} \}$ gives rise to [[sheaf|sheaves]] $\{ \mathcal{F}_{i}:=\mathcal{O}_{X} |_{U_{i}} =\mathcal{O}_{U_{i}}\}$. The $g_{ij}$ determine [[morphism of sheaves of modules|isomorphisms]] $\varphi_{ij}: \mathcal{F}_{i} |_{U_{i} \cap U_{j}}=\mathcal{O}_{U_{i} \cap U_{j}} \xrightarrow{\sim}\mathcal{F}_{j} |_{U_{i} \cap U_{j}}= \mathcal{O}_{U_{i} \cap U_{j}}$ satisfying the [[cocycle conditions]]. Thus the [[sheaf gluing|sheaf gluing procedure]] produces a unique [[sheaf]] of $\mathcal{O}_{X}$-modules $\mathcal{L}$ on $X$ for which: > - There are isomorphisms $\varphi_{i}: \mathcal{L} |_{U_{i}} \xrightarrow{\sim} \mathcal{F}_{i}=\mathcal{O}_{X} |_{U_{i}}$ ; > - This means $\mathcal{L}$ is a [[locally free sheaf|line bundle]]; > - We have $\varphi_{j} \circ \varphi_{i} ^{-1}=\varphi_{ij}$ on $U_{i} \cap U_{j}$. > - This means that transition functions of $\mathcal{L}$ are precisely the $\varphi_{ij}$, or equivalently the $g_{ij}$ . > > ----- #### [^1]: In other words, $\mathcal{L}$ is a rank-$1$ [[locally free sheaf|locally free]] [[sheaf]] of $\mathcal{O}_{X}$-[[sheaf of modules|modules]], as witnessed by the fact that $\mathcal{L} |_{U_{i}} \cong \mathcal{O}_{X} |_{U_{i}} =\mathcal{O}_{X}^{\oplus 1}|_{U_{i}}$ for each $i$. [^2]: Maybe a warning re: conventions is in order here. [^3]: $\mathcal{O}_{X}^{*}$ is the [[sheaf]] of [[abelian group|abelian groups]] given by $\mathcal{O}_{X}^{*}(U):=\mathcal{O}_{X}(U)^{*}$. ---- #### 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 > ```