My question is about a small detail on page 132 of the above-mentioned book.


Let $R'$ be a faithfully flat $R$ algebra and $M'$ a $R'$-module. Let $\varphi: p\_1^\* M' \cong p\_2^\* M'$ be a covering datum, where $p\_1$ and $p\_2$ are projections onto the first and second factor from $R'' = R'\otimes\_{R} R'$ to $R'$ (or rather, projection of the associated spectra). Using $\varphi$, the book derives two morphisms $$M' \rightrightarrows M'\otimes\_R R' $$ and say that this is co-cartesian over $$ R' \rightrightarrows R'\otimes\_R R'.$$ It's also said there that being co-cartesian over the second sequence is equivalent to giving a covering datum.


Next, they talk about descent datum (covering datum plus some co-cycle condition), relating descent datum with some diagram with triple arrows being co-cartesian over a similar sequence on the rings (involving $R'\otimes\_R R' \otimes\_R R'$). They then make a similar remark as above: descent datum = co-cartesian + some commutativity conditions such as eg. $p\_1\circ p\_{12} = p\_1 \circ p\_{13}$.


I don't quite understand what they mean by being co-cartesian. And so, I also don't see how it's related to giving a descent or descent datum. It would be helpful if someone can clarify.


Thanks!


EDIT: I started reading the relevant section in the Stacks project. They use the formalism of cosimplicial object, which makes everything much clearer. 



A small detail in Neron Models (Bosch-Lütkebohmert-Raynaud) on descent theory
Let $R,S$ be rings, $M$ a $R$-module and $N$ a $S$-module. Let $R \to S$ be a ring homomorphism. Then a $R$-module homomorphism $M \to N|\_R$ is said to be cocartesian over $R \to S$, when the corresponding $S$-module homomorphism $M \otimes\_R S \cong N$ is an isomorphism. If $M$ and $N$ are algebras, you recover the usual notion of a cocartesian square (a.k.a. pushout) in the category of rings. You can imagine it as follows: You pull the quasi-coherent sheaf $\tilde{M}$ on $\mathrm{Spec}(R)$ along $\mathrm{Spec}(S) \to \mathrm{Spec}(R)$ back to $\mathrm{Spec}(S)$ and arrive at $\tilde{N}$.


When there are two maps $R \rightrightarrows S$ and $M \rightrightarrows N$, then cocartesian means cocartesian for the two separated cases.


I found it quite hard to understand descent theory in this "affine language". Instead of BLR, I recommend the exposition by Angelo Vistoli on Descent Theory, or the original FGA by Grothendieck.


