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Let $K'/K$ be an extension of fields and set $X=\operatorname{Spec}(K)$ and $X'=\operatorname{Spec}(K')$. As the category of locally ringed spaces has fibre products (see arXiv:1103.2139 or here) we have a base change functor $F$ from the category of locally ringed spaces over $X$ to the category of locally ringed spaces over $X'$. As usual, a descent datum on a locally ringed space $Y'$ over $X'$ is an isomorphism $Y'\times_X X'\to X'\times_X Y'$ over $X'\times_X X'$ satisfying a cocycle condition. Then $F$ can be considered as a functor from the category of locally ringed spaces over $X$ to the category of locally ringed spaces over $X'$ equipped with a descent datum.

My question is, is this functor an equivalence of categories?

If we replace the category of locally ringed spaces with the category of schemes the answer is no. There the problem is that there might not be an open affine cover of $Y'$ that is stable under the descent datum. So if we get rid of the "cover with open affines" condition, i.e., replacing schemes with locally ringed spaces, there might still be hope for a positive answer.

If we replace the category of locally ringed spaces with ringed spaces the answer seems to be yes. In ringed spaces the underlying topological space of a fibre product is the fibre product of the underlying topological spaces. So base change over a field extension really does not change the topological space and it follows from the cocycle condition that the underlying map of the descent datum must be the identity. Then everything reduces to the standard descent result for algebras by looking at the sections.

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    $\begingroup$ Nice question! Can you perhaps give an example for schemes where it is not true? I was a bit surprised to read that. Maybe it works with qcqs schemes? $\endgroup$
    – Martin Brandenburg
    Commented May 30, 2021 at 9:08
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    $\begingroup$ After recalling how descent works for comm. algebras and dualizing it, we may want to define $Y$ as the coequalizer of $Y' \times_{X} Y' \to Y'$ and $Y' \times_{X} X' \to X' \times_{X} Y' \to Y'$. Remarks: Colimits of locally ringed spaces are just as in ringed spaces (so as in topological spaces, and limits of structure sheaves). Fiber products are more complicated. For example, points of $Y' \times_{X} X'$ consist of points $y' \in Y'$ with a prime ideal $\mathfrak{p}$ in $\mathcal{O}_{Y',y'} \otimes_K K'$, the stalk at that point is $(\mathcal{O}_{Y',y'} \otimes_K K')_{\mathfrak{p}}$. $\endgroup$
    – Martin Brandenburg
    Commented May 30, 2021 at 9:19
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    $\begingroup$ @MartinBrandenburg I think when working with quasi-coherent modules on schemes the (faithfully flat) descent works perfectly but descending schemes themselves does not always work. Counter examples are discussed in SGA 1, Chapter VIII, Remark 7.10. In particular, there is mentioned an example attributed to Hironaka of a complete three dimensional variety with a $\mathbb{Z}/\mathbb{Z}2$-action that does not preserve any affine open cover. $\endgroup$
    – Michael
    Commented May 30, 2021 at 10:30
  • $\begingroup$ @MartinBrandenburg Yes, I think considering this coequalizer is a good idea. In fact, I have been trying that. We get a morphism $Y'\to Y\times_X X'$ but I can't see how to use the cocylce condition to show that this is an isomorphism. $\endgroup$
    – Michael
    Commented May 30, 2021 at 10:38
  • $\begingroup$ I have a solution, but it will take some time to write it up. $\endgroup$
    – Martin Brandenburg
    Commented May 30, 2021 at 19:43

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