There seem to be two different definitions of the inverse image functor on sheaves of modules in the literature and I just wanted to make sure I am understanding properly. Suppose $f: X \rightarrow Y$ is a morphism of schemes and $\mathcal{F}$ is a quasicoherent sheaf on $Y$. How is the sheaf of $\mathcal{O}_{X}$-modules $f^{*}\mathcal{F}$ defined?

The most general appears to be, $$ f^{*} \mathcal{F} = \mathcal{O}_{X} \otimes_{f^{-1}\mathcal{O}_{Y}} f^{-1} \mathcal{F} \quad \text{call this sheaf $\mathcal{H}$}. $$ This should work (by "work" I mean provide an adjoint for the functor $f_{*}$ on the category of sheaves of modules) for any sheaf of modules, right?

However, I see another definition often,

$$ f^{*} \mathcal{F} = \mathcal{O}_{X} \otimes_{\mathcal{O}_{Y}} \mathcal{F} \quad \text{call this sheaf $\mathcal{J}$}. $$

This definition doesn't work for arbitrary sheaves of modules, right? But it should work if we restrict ourselves to quasicoherent modules, or possibly only quasicoherent modules locally of finite type?

My idea to prove they are the same in the case of quasicoherent sheaves: I would cover the base scheme $Y$ by affines $\{ U_{i} = \text{Spec}A_{i} \}$ and write $\mathcal{F}|_{U_{i}} \simeq \tilde{M_{i}}$ for $A_{i}$-modules $M_{i}$. Then we can cover $f^{-1}(U_{i})$ by affines $\{ V_{ij} = \text{Spec}B_{ij} \}$. Then over $V_{ij}$ we have, $$ \varphi_{ij}: (M_{i} \otimes_{A_{i}} B_{ij})^{\sim} \longrightarrow (\mathcal{O}_{X} \otimes_{f^{-1} \mathcal{O}_{Y}} f^{-1} \mathcal{F})|_{V_{ij}}. $$ The left hand side of this morphism is the sections of $\mathcal{J}$ and since $\mathcal{J}$ is clearly quasicoherent such a family is sufficient to provide a morphism which I claim will be an isomorphism on stalks.

Is this the correct approach? Am I correct that these are only equivalent on quasicoherent sheaves? I initially thought they were only equivalent for locally free sheaves, since you can use the projection formula, but I am quite certain this holds at least for coherent sheaves, if not quasicoherent.

  • $\begingroup$ I don't understand the second definition. $\mathcal{O}_X$ is a sheaf on $X$, while $\mathcal{O}_Y$ and $\mathcal{F}$ are sheaves on $Y$: how are you considering $\mathcal{O}_X$ as an $\mathcal{O}_Y$-module? The map $f: X \to Y$ gives you a map of sheaves $f^\#: \mathcal{O}_Y \to f_* \mathcal{O}_X$, but I still don't see how to make $\mathcal{O}_X$ an $\mathcal{O}_Y$-module. And how does this give a sheaf $f^* \mathcal{F}$ defined on $X$ in the end? $\endgroup$ – André 3000 Jun 15 at 4:50
  • $\begingroup$ @André3000 This was also something that confused me. For reference, I am getting this from the book by Mumford and Oda. See page 25 of this book dam.brown.edu/people/mumford/alg_geom/papers/AGII.pdf $\endgroup$ – Luke Jun 15 at 4:56

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