## The local nature of formal smoothness and an apparently extraneous hypothesis of finite presentation

In showing that formal smoothness is a local property, EGA seems to use the following presentation (and in fact, seems to contain a mistake according to this question). The proposition is as follows:

Proposition (EGA$$\ IV_{4}, 16.5.18)$$: Suppose $$X,Y$$ are schemes over $$S$$ and $$Y_0$$ is a closed subscheme of $$Y$$ defined by the ideal sheaf $$\mathscr I$$ that is square $$0$$. Assume also that $$Y$$ is affine and $$\Omega^1_{X/S}$$ is finitely presented.

We are given a map $$u_0: Y_0 \to X$$. Then, if $$U_\alpha$$ is an open cover of $$X$$ and we can lift $$u_0$$ locally to maps $$u_\alpha$$ defined on the appropriate open subsets of $$Y$$, we can find a global lift $$u: Y\to X$$.

My understanding of this proof is as follows. They show that the sheaf $$\mathscr P$$ on $$Y$$ defined by $$\Gamma(V,\mathscr P) = \{f: V \to X: f_0 = u_0: V/\mathscr I \to X\}$$ is a torsor of the sheaf $$\mathscr G = \mathcal{Hom}_{Y_0}(u_0^{*}\Omega_{X/S}, \mathscr I)$$ and since $$Y_0$$ is affine, $$\mathscr P$$ is the trivial torsor. Then, the $$0$$ section of $$\mathscr G$$ gives the required lift.

Question: Assuming my outline of the argument above is correct, where does the finite presentation of $$\Omega_{X/S}$$ get used?

## References for formal powers of measures

In Information geometry, Ay et al. define the space of formal $$r$$th powers* of signed measures as the limit $$\mathcal S^r(\Omega) = \injlim L^{1/r}(\Omega, \mu)$$ of maps $$\phi\mapsto (\mu/\nu)^r\phi :$$ $$L^{1/r}(\Omega,\mu)\to L^{1/r}(\Omega,\nu)$$, $$\mu\leq\nu$$ (referencing only an exercise in Neveu’s Bases mathématiques du calcul de probabilités). It appears to me that this is essentially the same thing as Dmitri Pavlov’s definition (in arχiv:1309.7856 §5.10 or on MO) of a measure-independent space $$\mathrm L_r(\Omega)$$ as a limit of measure-dependent spaces $$\mathrm L_r(\Omega,\mu)\equiv L^{1/r}(\Omega,\mu)$$. It also (I think) reproduces the standard notion of $$r$$-densitites on manifolds.

Overall, this seems like a pretty elementary and even essential construction, but all the references above only mention it in passing and with scant references to the literature. I’d like to find an exposition that treats it with the attention it deserves, perhaps alongside or shortly after the usual classes of measures and measurable functions.

* The book also sometimes refers to them as formal $$r$$th roots (with the same $$r$$!)—mistakenly, I think.