Remark 35.31.2. Let $X \to S$ be a morphism of schemes. Let $(V/X, \varphi )$ be a descent datum relative to $X \to S$. We may think of the isomorphism $\varphi$ as an isomorphism

$(X \times _ S X) \times _{\text{pr}_0, X} V \longrightarrow (X \times _ S X) \times _{\text{pr}_1, X} V$

of schemes over $X \times _ S X$. So loosely speaking one may think of $\varphi$ as a map $\varphi : \text{pr}_0^*V \to \text{pr}_1^*V$1. The cocycle condition then says that $\text{pr}_{02}^*\varphi = \text{pr}_{12}^*\varphi \circ \text{pr}_{01}^*\varphi$. In this way it is very similar to the case of a descent datum on quasi-coherent sheaves.

[1] Unfortunately, we have chosen the “wrong” direction for our arrow here. In Definitions 35.31.1 and 35.31.3 we should have the opposite direction to what was done in Definition 35.2.1 by the general principle that “functions” and “spaces” are dual.

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