4.6. (拟) 凝聚层

Now we jump to quasi-coherent/coherent sheaves on stack.

定义 4.6.0.1. We define Lisse-etale site $\mathrm{Lis}-\mathrm{et}(\mathfrak{X})$ on $\mathfrak{X}$ as follows (the topos is denoted by ${\mathfrak{X}}_{\mathrm{Lis}-\mathrm{et}}$). The site $\mathrm{Lis}-\mathrm{et}(\mathfrak{X})$ has objectswith $T$ being scheme, and morphismsThe coverings are given by family of diagrams of the formsuch that we have etale surjection $\coprod T_i\twoheadrightarrow T$.

Then, we define ${\mathcal{O}}_{\mathfrak{X}}\in {\mathfrak{X}}_{\mathrm{Lis}-\mathrm{et}}$ as ${\mathcal{O}}_{\mathfrak{X}}(T\stackrel{t}{\to }\mathfrak{X}):=\Gamma ({\mathcal{O}}_T)$.

Now let $\mathcal{C}$ be the following category. The objects are: for all $(T,t)\in \mathrm{Lis}-\mathrm{et}(\mathfrak{X})$ a choice of etale sheaf of sets ${\mathcal{F}}_{(T,t)}\in T_{et}$ and for all $(f,f^{♭}):(T^{\prime },t^{\prime })\to (T,t)$ a choice$$f^{-1}{\mathcal{F}}_{(T,t)}\stackrel{{\rho }_{(f,f^{♭})}}{\to }{\mathcal{F}}_{T^{\prime },t^{\prime }}$$such that:

1.	if $f:T^{\prime }\to T$ is etale then ${\rho }_{(f,f^{♭})}$ is isomorphism.
2.	For diagramwe have

A morphism between $(\{{\mathcal{F}}_{(T,t)}\},\{{\rho }_{(f,f^{♭})}\})\to (\{{\mathcal{G}}_{(T,t)}\},\{{\lambda }_{(f,f^{♭})}\})$ in $\mathcal{C}$ is a collection of morphisms ${\gamma }_{(T,t)}:{\mathcal{F}}_{(T,t)}\to {\mathcal{G}}_{(T,t)}$ so the following diagram commutes

We have $\mathcal{C}\to {\mathfrak{X}}_{\mathrm{Lis}-\mathrm{et}}$ given by $(\{{\mathcal{F}}_{(T,t)}\},\{{\rho }_{(f,f^{♭})}\})$ maps to the sheaf $\mathcal{F}$ given by $\mathcal{F}(T\stackrel{t}{\to }\mathfrak{X})={\mathcal{F}}_{(T,t)}(T)$.

This is a presheaf where transition maps of $\mathcal{F}$ come from ${\rho }_{(f,f^{♭})}$.

It is a sheaf because etale covers in $\mathrm{Lis}-\mathrm{et}(\mathfrak{X})$ are already coverings in $T_{et}$.

Conversely, ${\mathfrak{X}}_{\mathrm{Lis}-\mathrm{et}}\to \mathcal{C}$ given by $\mathcal{F}$ maps to the object ${\mathcal{F}}_{(T,t)}:=\mathcal{F}{|}_{T_{et}}$.

Therefore, we get $\mathcal{C}\cong {\mathfrak{X}}_{\mathrm{Lis}-\mathrm{et}}$.

定义 4.6.0.2. Let $\Lambda$ be a sheaf of rings on $\mathrm{Lis}-\mathrm{et}(\mathfrak{X})$. Let $\mathcal{F}$ be a sheaf of $\Lambda$-module. Then:

1.	We say $\mathcal{F}$ is Cartesian if for all $(T^{\prime },t^{\prime })\to (T,t)$, we get$$f^{\ast }{\mathcal{F}}_{(T,t)}:=f^{-1}{\mathcal{F}}_{(T,t)}{\otimes }_{f^{-1}{\Lambda }_{(T,t)}}{\Lambda }_{(T^{\prime },t^{\prime })}\stackrel{\sim }{\to }{\mathcal{F}}_{(T^{\prime },t^{\prime })}$$
2.	We say $\mathcal{F}$ is quasi-coherent if it is Cartesian ${\mathcal{O}}_{\mathfrak{X}}$-module and for all $(T,t)$, we have ${\mathcal{F}}_{(T,t)}\in \mathsf{Qcoh}(T_{et})$ is quasi-coherent.
3.	We say $\mathcal{F}$ is coherent if $\mathcal{X}$ is locally Noetherian (note this implies for anyu $(T,t)\in \mathrm{Lis}-\mathrm{et}(\mathfrak{X})$ we get $T$ locally Noetherian), $\mathcal{F}$ is quasi-coherent and for all $(T,t)$, we have ${\mathcal{F}}_{(T,t)}\in \mathsf{Coh}(T_{et})$.