Definition 76.13.1 (049S)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 76: More on Morphisms of Spaces
Section 76.13: Formally smooth, étale, unramified transformations
Definition 76.13.1 (cite)

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Definition 76.13.1. Let $S$ be a scheme. Let $a : F \to G$ be a transformation of functors $F, G : (\mathit{Sch}/S)_{fppf}^{opp} \to \textit{Sets}$ . Consider commutative solid diagrams of the form

$\xymatrix{ F \ar[d]_ a & T \ar[d]^ i \ar[l] \\ G & T' \ar[l] \ar@{-->}[lu] }$

where $T$ and $T'$ are affine schemes and $i$ is a closed immersion defined by an ideal of square zero.

We say $a$ is formally smooth if given any solid diagram as above there exists a dotted arrow making the diagram commute ¹.
We say $a$ is formally étale if given any solid diagram as above there exists exactly one dotted arrow making the diagram commute.
We say $a$ is formally unramified if given any solid diagram as above there exists at most one dotted arrow making the diagram commute.

[1] This is just one possible definition that one can make here. Another slightly weaker condition would be to require that the dotted arrow exists fppf locally on

$T'$ . This weaker notion has in some sense better formal properties.

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