# A small detail in Neron Models (Bosch-Lütkebohmert-Raynaud) on descent theory

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.

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.
• a) Yes, separatedly. b) When $M$ is some quasi-coherent sheaf on $X$ and $Y \to X$ is a morphism, a descent datum is just an isomorphism between $p_1^* M \cong p_2^* M$ on $Y \times_X Y$ which satisfies the cocycle condition on $Y \times_X Y \times_X Y$. This is the usual commutative diagram. – Martin Brandenburg Jul 1 '12 at 8:35
• For notation, it is quite useful to work more generally with a set of morphisms $\{Y_i \to X\}$ instead. Then $p_1^* M \cong p_2^* M$ lives on each $Y_i \times_X Y_j$, and the cocycle condition takes place on $Y_i \times_X Y_j \times_X Y_k$. This way you see exactly where you are ... – Martin Brandenburg Jul 1 '12 at 9:59