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Let $S$ be a connected scheme. Let $FEt_S$ be the Galois category of schemes $X$ finite etale over $S$. Let $I$ be a directed set, and $\{C_i\}_{i\in I}$ a projective system of objects in $FEt_S$. I'm interested in considering morphisms from $\{C_i\}_{i\in I}$ to a fixed scheme $X$ (especially in the case where $X\in FEt_S$).

By definition, such morphisms are given by \begin{equation}Hom_{Pro-FEt_S}(\{C_i\}_{i\in I},X) = \varinjlim_{i\in I} Hom_{FEt_S}(C_i,X)\end{equation}

Taking limits gives a functor from $Pro-FEt_S$ to $Sch/S$, so we may also consider the limit $C := \varprojlim_{i\in I}C_i$ in $Sch/S$.

If $S$ is Noetherian and $X\in FEt_S$, then by http://stacks.math.columbia.edu/tag/01ZC , we have that $$Hom_{Sch/S}(C,X) = \varinjlim_{i\in I} Hom_{Sch/S}(C_i,X)$$ so taking limits identifies $Pro-FEt_S$ with the full subcategory of $Sch/S$ consisting of inverse limits of objects in $FEt_S$.

My questions are:

  1. Is everything above correct?

  2. Unfortunately I have almost zero intuition for non-Noetherian rings, so if $S$ is not Noetherian (but $X$ is still in $FEt_S$), can someone explain what can go wrong in this case? Can someone provide an example where $Hom_{Sch/S}(C,X) \ne \varinjlim_{i\in I} Hom_{Sch/S}(C_i,X)$ ?

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  • $\begingroup$ You probably mean $Sch/S$ for the category you are asking about, it appears that's what they're using in the stacks project link. EDIT: to clarify, you used $Sch$ in the Hom sets $\endgroup$
    – Joe Berner
    Feb 22, 2016 at 0:41
  • $\begingroup$ I'm also a bit confused as the Stacks Project link doesn't mention being noetherian. It proves equivalence with $f$ being locally of finite presentation, but etale morphisms are all locally of finite presentation, and the structure map $X\rightarrow S$ is etale. $\endgroup$
    – Joe Berner
    Feb 22, 2016 at 0:54
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    $\begingroup$ @JoeB The stacks project link requires each $C_i$ to be quasi-separated and quasi-compact and $X$ to be locally of finite presentation. If $S$ is locally noetherian, then quasi-separatedness and LoFP is satisfied. If $S$ is noetherian, then quasi-compactness is also satisfied. $\endgroup$
    – Will Chen
    Feb 22, 2016 at 1:04
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    $\begingroup$ A finer result (EGA4 8.8.2) is that the canonical map is injective if $S$ is qc, and bijective if $S$ is qcqs. Here is an example of the failure of surjectivity, using an $S$ which is not connected (but it may give an idea of what can go wrong). Take $S=T\times I$, $I$ an infinite set, and let $V\to T$ be finite etale (and not the identity). Then $V\times I$ is finite etale over $S$, and it can be written as a limit of finite etale $S$-schemes of the form $(T\times (I-J))\amalg (V\times J)$ with $J\subset I$ finite. But the identity of $V\times I$ fails to factor through one of these schemes. $\endgroup$ Feb 22, 2016 at 2:08

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