Definition 87.9.9 (0AIF)—The Stacks project

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Part 5: Topics in Geometry
Chapter 87: Formal Algebraic Spaces
Section 87.9: Affine formal algebraic spaces
Definition 87.9.9 (cite)

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Definition 87.9.9. Let $S$ be a scheme. Let $A$ be a weakly admissible topological ring over $S$ , see Definition 87.4.8 ¹. The formal spectrum of $A$ is the affine formal algebraic space

$\text{Spf}(A) = \mathop{\mathrm{colim}}\nolimits \mathop{\mathrm{Spec}}(A/I)$

where the colimit is over the set of weak ideals of definition of $A$ and taken in the category $\mathop{\mathit{Sh}}\nolimits ((\mathit{Sch}/S)_{fppf})$ .

[1] See More on Algebra, Definition 15.36.1 for the classical case and see Remark 87.2.3 for a discussion of differences.

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