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.

1. We say $a$ is formally smooth if given any solid diagram as above there exists a dotted arrow making the diagram commute1.

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

3. 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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