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?



Les deux théorèmes d'existence en théorie formelle des modules
You should look at A H M Levelt's notes:


[Sur la pro-representabilite de certains foncteurs en geometrie algebrique](http://www.math.ru.nl/~ahml/prorep.pdf)


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.


