The Stacks project

Remark 74.22.2. Let $S$ be a scheme. Let $Y \to X$ be a morphism of algebraic spaces over $S$. Let $(V/Y, \varphi )$ be a descent datum relative to $Y \to X$. We may think of the isomorphism $\varphi $ as an isomorphism

\[ (Y \times _ X Y) \times _{\text{pr}_0, Y} V \longrightarrow (Y \times _ X Y) \times _{\text{pr}_1, Y} V \]

of algebraic spaces over $Y \times _ X Y$. 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 74.22.1 and 74.22.3 we should have the opposite direction to what was done in Definition 74.3.1 by the general principle that “functions” and “spaces” are dual.

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