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.

 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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