In Exposé 195 of the Séminaire Bourbaki, Grothendieck states the following two theorems of non-flat descent.

Theorem 1. Let $\Lambda$ be a noetherian ring and $C$ the category of $\Lambda$-algebras which are finite type artinian $\Lambda$-modules. Let $F:C\to (Set)$ be a covariant functor. Then $F$ is pro-representable if and only if the following two conditions are satisfied:

i) $F$ commutes with finite products,
ii) for each $A\in C$ and each homomorphism $A\to A'$ in $C$ such that the diagram $A\to A'\rightrightarrows A'\otimes_A A'$ is exact, the diagram $F(A)\to F(A')\rightrightarrows F(A'\otimes_A A')$ is exact.

Moreover, it is enough to check ii) when $A$ is local and when moreover we are in one of the following two cases:

a) $A'$ is a free $A$-module,
b) the quotient $A'/A$ is an $A$-module of length $1$.

$\ $

Theorem 2. Let $A$ be a local artinian ring with maximal ideal $m$, $A'$ an $A$-algebra that contains $A$ such that $mA'\subset A$ and $A\to A'\rightrightarrows A'\otimes_A A'$ is exact (note that this is the case in particular when $A'/A$ is an $A$-module of length $1$). Let $\mathcal{F}$ be the fibered category of flat quasi-coherent sheaves on variable schemes. Then the morphism ${\rm Spec}(A')\to {\rm Spec}(A)$ is a morphism of strict $\mathcal{F}$-descent.

(See Sém. Bourbaki, Exp. 195 for undefined terms.)

Concerning Theorem 1, Grothendieck writes: "The proof of this theorem is quite delicate and can not be sketched here". The proof of Theorem 2 is not given either.

I don't know any place where these theorems are proved. Does anyone know?

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    $\begingroup$ Do you know Schlessinger's article "Functors on Artin rings"? $\endgroup$ – Martin Brandenburg May 16 '12 at 6:21
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    $\begingroup$ Yes, I do, but whereas Thm 1 seems to have an analogue in Schlessinger's paper, Thm 2 does not. Moreover Schlessinger has finiteness assumptions which Grothendieck hasn't. That being said, it is probably possible to prove the two theorems above along the lines of the arguments given in Schlessinger. But my question really is about the existence of a place where the statements of Grothendieck in Exposé 195 are proved: I'm curious to know! $\endgroup$ – Matthieu Romagny May 16 '12 at 7:01

You should look at A H M Levelt's notes:

Sur la pro-representabilite de certains foncteurs en geometrie algebrique

where, I think, proofs of both Theorem 1 and Theorem 2 are written out. Also, he has written a paper "Foncteurs exacts à gauche" (Inventiones + errata) which discusses the relation between Schlessinger's conditions and these. I tend to view these particular instances of non-flat descent as a pre-Schlessinger/Artin theory that has been obsoleted. On the other hand, I do find non-flat descent interesting.

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  • $\begingroup$ Great, thanks David! In fact I have been looking for this paper by Levelt for some time, without success. $\endgroup$ – Matthieu Romagny Feb 5 '13 at 20:12
  • $\begingroup$ Had I simply gone to Levelt's webpage... $\endgroup$ – Matthieu Romagny Feb 5 '13 at 20:14

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