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 $1$.
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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?
