I have a question on the ''functor of points''-approach to schemes and $\mathcal{O}_X$-modules. Please let me first write up a defintion.
Let $Psh$ denote the category of presheaves on the opposite category of rings $Rng^{op}$. So $Psh$ is the category of functors from the category of rings $Rng$ to the category $Set$ of sets.
Fix an $X\in Psh$. Demazure and Gabriel define in their book ''Introduction to Algebraic Geometry and Algebraic Groups" (page 58, I.2.4.1) an $X$-module $M$ to be an object $M\in Psh$ and a morphism $f:M\to X$ in $Psh$ (a natural transformation $f:M\to X$, that's what they call an $X$-functor) such that for every ring $R$ and every map $p:*\to X(R)$ the set $M(R,p):=*\times_{X(R)}M(R)$ has an $R$-module structure with the property that for any ring map $\phi:R\to S$ the induced map $\psi:M(R,p)\to M(S,\phi(p))$ is additive and satisfies \begin{equation} \psi(\lambda m)= \phi(\lambda)\psi(m) \end{equation} for all $m\in M(R,p)$ and $\lambda\in R$.
They call $M$ quasicoherent if for any ring map $\phi:R\to S$, the induced map \begin{equation} M(R,p)\otimes_R S\cong M(S,\phi(p)) \end{equation} is an isomorphism.
I want to understand an $X$-module $M$ as a morphism $f:M\to X$ in $Psh$ for which some conditions are required to hold ''locally'', like a bundle, but let me more precise in what I mean: The map $p$ in the definition above corresponds by the Yoneda lemma to a map $p:R\to X$ (Here, I use the same notion for $R$ and its associated presheaf $\hom(R,-)$). Let the object $M_p'$ of $Psh$ be defined by the cartesian diagram \begin{eqnarray} M_p'&\to & M\\ \downarrow && \downarrow f\\ R&\xrightarrow{p} & X \end{eqnarray} in $Psh$. I want to formulate conditions on $M_p'$ (and not on $M(R,p)=*\times_{X(R)}M(R)$ as above) such that $M$ is an $X$-module. The set $M(R,p)$ is contained in the set $M_p'(R)$ but there are not equal, unfortunately. My question is thus: What are the conditions on the $M_p'$ such that $M$ (together with $f$) defines an $X$-module? How is quasicoherence defined in this situation?
- To be more precise, I would like the above definition of a quasicoherent $X$-module to be the same as something like this: An object $M\in Psh$ and a morphism $f:M\to X$ in $Psh$ such that for every ring $R$ and every map $p:R\to X$ the set $M_p'(R)=(R \times_X M)(R)$ has an $R$-module structure with the property that for any ring map $\phi:R\to S$ the induced map $\psi'(R):M_p'(R)\to M_{\phi(p)}'(R)$ is additive and satisfies $\psi'(R)(\lambda m)= \phi(\lambda)\psi'(R)(m)$ for all $m\in M_p'(R)$ and $\lambda\in R$ and $M_p'(R)\otimes_R S\cong M_{\phi(p)}'(S)$ is an isomorphism.
I hope that I was able to clarify my question. Thank you in advance for any hints.