I'm currently interested in the following result:

Let $f: X \to Y$ be a fpqc morphism of schemes. Then there is an equivalence of categories between quasi-coherent sheaves on $Y$ and "descent data" on $X$. Namely, the second category consists of quasi-coherent sheaves $\mathcal{F}$ on $X$ with an isomorphism $p_{1}^*(\mathcal{F}) \simeq p_2^*(\mathcal{F})$, where $p_1, p_2: X \times_Y X \to X$ are the two projections. Also, a diagram involving an iterated fibered product is required to commute as well (the cocycle condition).

In Demazure-Gabriel's *Introduction to Algebraic Geometry and Algebraic Groups,* it is proved (under the name ffqc (sic) descent theorem) that the sequence
$$ X \times_Y X \to^{p_1, p_2} X \to Y$$ is a coequalizer in the category of locally ringed spaces under the above hypotheses. If I am not mistaken, this is the same as (or very closely equivalent to) the theorem that says that representable functors are sheaves in the fpqc topology. On the other hand, D-G give a fairly explicit description of the quotient space.

**Question:** For a coequalizer diagram of ~~(locally) ringed spaces~~ schemes,
$$A \rightrightarrows^{f,g} B \to C,$$
is there a descent diagram for quasi-coherent sheaves on $A,B,C$? In particular, does the D-G form of the descent theorem directly, by itself, imply the more general one for quasi-coherent sheaves?

My guess is the answer is no. First, I've heard that the coequalizer condition above is actually very weak. So, suppose instead we have that $A \rightrightarrows^{f,g} B \to C,$ is a coequalizer *and* any base-change of it is a coequalizer. Does that imply that there is a descent diagram for quasi-coherent sheaves?

My guess is that the answer to the modified question is still no, for the meta-reason that Vistoli in *FGA Explained* spends much more time on proving descent for quasi-coherent sheaves than proving that the fpqc topology is subcanonical. On the other hand, I'd like to see a counterexample.

(N.B. I initially asked this question on Math.SE. I was advised to re-ask it here.)

usefulway) for abstract loc. ringed spaces. Can make def'n of quasi-coh. in cases with coh. str. sheaf (related to local injlim of coherent sheaves), such as for complex-analytic spaces (see "Coherent Analytic Sheaves"). But surprises happen, such as lost under direct limits; see Example 2.1.10 in my paper "Relative ampleness in rigid geometry", which applies to the unit disc in the complex plane. Likewise, base change for C-analytic spaces is specific to that category. Rewrite question to stick to schemes. $\endgroup$ – BCnrd Sep 7 '10 at 13:01exampleof a coequalizer diagram of schemes whose formation as such is compatible with arbitrary base change on $C$ and yet which is not arising from an fpqc cover (recall fpqc topology mixes in Zar. topology too, by the way)? The difficulty of making examples where that happens would be a good reason that nobody tries to prove theorems with that hypothesis away from the fpqc case (where more tools are available). There is a kind of "non-flat descent" (I think due to Oort) worked out in appendix to one of Kleiman's exposes in SGA6; very hard to use. $\endgroup$ – BCnrd Sep 7 '10 at 13:18