# Describing crystals of quasi-coherent sheaves using quasi-coherent sheaves on the de Rham prestack

Fix some field $$k$$ of characteristic $$0$$, a smooth $$k$$-scheme $$X$$, a commutative and unital ring $$R$$ with nilradical $$I$$, and two infinitesimally close $$R$$-points: $$x,y: \mathrm{Spec}R \rightrightarrows X$$ We define the de Rham prestack to be the following functor: $$X^{dR}: \mathsf{CRing} \to \mathsf{Sets}: R \mapsto X^{dR}(R) := X(R/I)$$ Then, is there is the following equivalence of categories: $$\mathsf{CrysQCoh}(X) \cong \mathsf{QCoh}(X^{dR})$$ between the categories of crystals of quasi-coherent sheaves over $$X$$ and of quasi-coherent sheaves over the de Rham prestack $$X^{dR}$$ ? Pointwise, the de Rham prestack is the coequaliser of the diagram: $$X(x), X(y): X \rightrightarrows X(R)$$ by the definition of infinitesimally close points. This suggests to me that if given some quasi-coherent sheaf $$\mathcal{F}$$ on $$X$$, it can then be pulled back along $$x, y$$, and along the canonical map $$i: \mathrm{Spec} R/I \to \mathrm{Spec} R$$ to yield an isomorphism of quasi-coherent $$R/I$$-modules: $$i^*x^*\mathcal{F} \cong i^*y^*\mathcal{F}$$ Thus, I suspect the equivalence to be true, but I'm having trouble articulating my thoughts.

• According to de Rham space in nLab the big site of $X_{dR}$ is the crystaline site of $X$. – Niels May 13 at 20:27
• @Niels Could you elaborate ? – Dat Minh Ha May 13 at 21:37
• Your coequalizer diagram looks rather ill-defined. How are $x$ and $y$ maps from the scheme $X$ to the set $X(R)$? I thought they were infinitesimally nearby elements of $X(R)$, meaning composition with the map from $\operatorname{Spec} R/I$ yields identical elements of $X(R/I)$. If $X$ is smooth, then you can use the coequalizer of the two projections from the completion of the diagonal in $X \times X$. – S. Carnahan May 15 at 13:02
• @S.Carnahan I applied the functor $h_X$ to the first diagram. – Dat Minh Ha May 15 at 13:04
• Presumably it goes something like the following: u have a universal pair of inf close points given by the two projections from the formal neighborhood of the diagonal, and this formal nhd is precisely the fiber product of X with itself, taken relative to the dR stack. A suitable version of descent should identify the two categories now. – EBz May 15 at 13:55