I've seen the definition of a quasicoherent sheaf $\mathcal{F}$ on an arbitrary functor $$ X : CRing \to Sets $$ as a specification of an $R$-module $\mathcal{F}(x)$ for each $R$-point $x \in X(R)$ with a collection of isomorphisms $\alpha_{x,x'}: R' \otimes_R \mathcal{F}(x) \to \mathcal{F}(x') $ where $f : R\to R'$ is a morphism of commutative rings with $x' = X(f)x$.

There is a third condition, which is that if we have $f : R \to R' $ and $g: R' \to R$ and $x \in X(R)$ with $x' = X(f)x$ and $x'' = X(gf)x$ that then $\alpha_{x,x''}$ is given by composing the maps $$ R''\otimes_R \mathcal{F}(x) \to R'' \otimes_{R'} (R' \otimes_R \mathcal{F}(x)) \to R'' \otimes_{R'} \mathcal{F}(x') \to \mathcal{F}(x''), $$ where the second and third maps are induced by $\alpha_{x,x'}$ and $\alpha_{x',x''}$ respectively.

My question is how does one recover the notion of a quasicoherent sheaf ona scheme from the above notion applied to a schemes functor of points, that is if we have a quasicoherent sheaf on the functor of points $h_X$ of a scheme $X$ how does one naturally define a quasicoherent sheaf on $X$.

I know how to go in the opposite direction, you just pull back th sheaf along $R$-points to get a quasicoherent sheaf on $h_X$ from one on $X$, and this suggests to me that to get a quasicoherent sheaf on $X$ from one on $h_X$ one should just push forward all the $R$-modules $\mathcal{F}(x)$ along the $R$-points $x$, and glue these together somehow?

Any help on this would be really appreciated!

  • $\begingroup$ Yes, you can glue sheaves together: see Exercise 1.22 in [Hartshorne, Ch. II]. Quasicoherence is more or less automatic, since it is a local property. $\endgroup$ – Zhen Lin Feb 5 '16 at 13:57

In the general definition, you forgot to mention the condition that $\alpha_{x,x}$ is the canonical isomorphism.

Given a scheme $X$ and $R$-modules $M|_x$ for every $R$-point $x : \mathrm{Spec}(R) \to X$ equipped with coherence isomorphisms, choose an open affine covering $(u_i : \mathrm{Spec}(R_i) \to X)$ and consider the $R_i$-modules $ M|_{u_i}$. These induce quasi-coherent modules $M_i:=\widetilde{M|_{u_i}}$ on $\mathrm{Spec}(R_i)$. We have isomorphisms $M_i|_{\mathrm{Spec}(R_i) \cap \mathrm{Spec}(R_j)} \cong M_j|_{\mathrm{Spec}(R_i) \cap \mathrm{Spec}(R_j)}$ satisfying the cocycle condition. To see this, cover $\mathrm{Spec}(R_i) \cap \mathrm{Spec}(R_j)$ by open affines $\mathrm{Spec}(R_{ijk})$ and use the compatibility of the given modules for $R_i \to R_{ijk}$ and $R_j \to R_{ijk}$. Then we may glue the $M_i$ to some quasi-coherent module $M$ on $X$ with $u_i^* M \cong M_i = \widetilde{M|_{u_i}}$. We still need to check $x^* M \cong \widetilde{M|_x}$ holds for any $x : \mathrm{Spec}(R) \to X$. To see this, cover $\mathrm{Spec}(R)$ with open affines $\mathrm{Spec}(S_i)$ such that $x|_{\mathrm{Spec}(S_i)}$ factors through $\mathrm{Spec}(R_i)$, and use compatibility with respect to $R_i \to S_i$. Of course, there are many things to check here, but every step is rather straight forward.

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