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Here is a proof which, I think, qualifies a ``independent''. As an $A$-module, $B$ is flat and finitely presented and therefore locally free (this is essentially Nakayama's lemma). Thus we may assume …

Fix $R$ (not necessarily local) and $M$. Let us call a map $u:M\to \overline{M}$ a free hull if $\overline{M}$ is finite free and for every finite free $F$ the induced map $\mathrm{Hom}(\overline{M},F …

Yes. Assume $f^n$ is flat for some $n>1$. Then since $f^n$ is local, it is faithfully flat. For any $A$-module $M$, put $M_1:=A\otimes_{f,A} M$ and recursively $M_i:=(M_{i-1})_1$. Let $u:E\to F$ be an …

So the only problem is flatness. Use the flatness criterion by fibers (EGA IV, 11.3.10). …

Probable counterexample: take $D=\mathbb{A}^2$ (or $\mathbb{P}^2$, if you prefer), $T=\mathbb{A}^1$, and let $X_t$ be the union of the four lines $x=0$, $y=0$, $x=y$, and $x=ty$. It is well known that …

I think what you are looking for is in the book, only later: see EGA IV, (11.2.6).

This is "flatness by fibers", EGA IV (11.3.10) (applied with $\mathcal{F}=\mathcal{O}_X$). …

On $X=\mathrm{Spec}\,\mathbb{C}[[t,h]]$, choose an irreducible curve of degree $\geq2$ (e.g. $t^2=h$) and let $\eta$ be its generic point, $j:\eta\to X$ the inclusion. Then $\mathcal{F}:=j_*\mathcal{O …