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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. Edit: There is a further condition; a diagram involving an iterated fibered product is required to commute as well.

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 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, $A \to^{f,g} B \to C,$ 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?

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    Ok, it's posted (http://mathoverflow.net/questions/37970/do-coequalizers-in-ringspc-automatically-lead-to-descent) on MO now; thank you for the kind words.2010-09-07

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