TITLE: Pullback bundle and inverse image sheaf
QUESTION [6 upvotes]: Let $\varphi \colon X \rightarrow Y$ be a morphism of varietys, $\mathcal{F}$ a locally free sheaf of $\mathcal{O}_{Y}$-modules and $L$ the associated vector bundle. Then we can construct the inverse image sheaf $\varphi^* \mathcal{F} = \varphi^{-1} \mathcal{F} \otimes_{\varphi^{-1} \mathcal{O}_{Y}} \mathcal{O}_{X}$.
Furthermore there is the so called pullback bundle $\varphi^*L = L \times_Y X$.
According to http://en.wikipedia.org/wiki/Pullback_bundle,"the pullback of bundles corresponds to the inverse image of sheaves" but how can I see this?
Thanks in advance!

REPLY [3 votes]: There are lots of nice answers to this question already, and this may not add much that isn't already present, but just to emphasize, there's no reason to analyze things affine locally, as this is a consequence of largely formal properties of the construction of the equivalence (anti-equivalence if you choose to omit the dual) between geometric vector bundles and finite locally free $\mathscr{O}_X$-modules via the relative spectrum (on any scheme $X$ whatsoever). Namely once one constructs the relative spectrum of a quasi-coherent algebra (which at some point does require one to work with affine opens), the compatibility with pullback follows from whatever characterizing property one chooses to use for the relative spectrum (my personal preference is for using the functor it represents).
If $\mathscr{F}$ is any finite locally free $\mathscr{O}_X$-module, the corresponding geometric vector bundle is $\mathbf{A}(\mathscr{F})=\underline{\mathrm{Spec}}_X(\mathrm{Sym}_{\mathscr{O}_X}(\mathscr{F}^\vee))$ (I'm not totally sure if this is the standard convention or if omitting the dual is the standard convention). In general, if $\mathscr{A}$ is a quasi-coherent $\mathscr{O}_X$-algebra (such as $\mathrm{Sym}_{\mathscr{O}_X}(\mathscr{F}^\vee))$, and $g:X^\prime\to X$ is a morphism, then there is a canonical isomorphism $\underline{\mathrm{Spec}}_{X^\prime}(g^*(\mathscr{A}))\simeq X\times_X\underline{\mathrm{Spec}}_X(\mathscr{A})$ (one can verify this very easily using the description of the functor of points of the relative spectrum which is sometimes taken as its defining property). In the vector bundle case, one also uses that pullback commutes with formation of the symmetric algebra (and duality, if you include that dual) for finite locally free modules (formation of the symmetric algebra commutes with pullback for general quasi-coherent modules, not just finite locally free ones). For details about these things, I recommend http://stacks.math.columbia.edu/tag/01LQ and http://stacks.math.columbia.edu/tag/01M1, as well as Chapter 11 of the wonderful book by Görtz-Wedhorn.