Remark 74.22.2 (0ADH)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 74: Descent and Algebraic Spaces
Section 74.22: Descent data for spaces over spaces
Remark 74.22.2 (cite)

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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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