---- > [!definition] Definition. ([[tensor product sheaf of modules]]) > Let $(X, \mathcal{O}_{X})$ be a [[ringed space]] and $\mathcal{F}$, $\mathcal{G}$ be ([[sheaf|sheaves of]]) $\mathcal{O}_{X}$-[[sheaf of modules|modules]]. We define the **tensor product sheaf of $\mathcal{O}_{X}$-modules** to be the [[sheaf]] [[sheafification|associated to]] the **tensor product presheaf** $U \mapsto \mathcal{F}(U) \otimes_{\mathcal{O}_{X}(U)} \mathcal{G}(U)$ and denote it by $\mathcal{F} \otimes_{\mathcal{O}(X)} \mathcal{G}$. ^definition > [!basicexample] Example. (Tensor product of line bundles)\ >Let $\mathcal{L}_{1}$, $\mathcal{L}_{2}$ be [[locally free sheaf|line bundles]] on $X$. We want to describe $\mathcal{L}_{1} \otimes_{\mathcal{O}_{X}}\mathcal{L}_{2}$. > Moral: don't think about the earnest tensor product, just think about the transition maps $g_{ij}h_{ij}$ defining $\mathcal{L}_{1} \otimes_{\mathcal{O}_{X}} \mathcal{L}_{2}$ on the common trivializing cover $\{ U_{i} \}$ of $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$. ^basic-example ---- #### [[lines bundles and transition functions]] ---- #### 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 > ```