Finite étale covers of a projective flat family over a complete DVR lift uniquely

For any finite locally free sheaf $E$ on $Z$, multiplication by $t^n$ is injective because $Z$ is flat over $R$. Its exact quotient sequence gives $0⟶H^0(Z,E)\mathrm{/}{t}^n⟶H^0(Z_n,E\mathrm{/}{t}^{n}E)⟶H^1(Z,E)[t^n]⟶0.$ Here the groups are the Čech groups in [F1], and the sequence follows from its exact-sequence calculation. The transition map on the last group is multiplication by $t$, as follows by comparing the two quotient exact sequences for $t^{n+1}$ and $t^n$. The torsion subgroup of the finite $R$-module $H^1(Z,E)$ is killed by one power of $t$, so a compatible system in these last groups is zero. Thus every compatible system of sections comes uniquely from the limit of $H^0(Z,E)\mathrm{/}{t}^n$; by [F3] this is $H^0(Z,E)$. Applying the result to $\mathcal{H}om(E,G)$ proves full faithfulness of completion for finite locally free sheaves.

By [F2] lift the cover on $Z_1$ successively to compatible finite étale algebras $D_n$ on $Z_n$. Each is locally free. Flatness of $Z\mathrm{/}R$ gives $\ker (D_{n+1}\to D_n)\cong D_1$ as a sheaf on $Z_1$, by multiplication by $t^n$. Choose $d$ so that $D_1(d)$ is globally generated and $H^1(Z_1,D_1(d))=0$ using [F1]. Its finite generating list of global sections lifts compatibly to every $D_n(d)$ because the obstruction group at every successive stage is that same zero group. Nakayama makes these lifts generate each $D_n(d)$. Thus there is a compatible system of surjections ${\mathcal{O}}_{Z_n}(-d{)}^r\to D_n$, with locally free kernels $K_n$ compatible under reduction: each sequence splits locally because $D_n$ is locally free. The same argument for $K_1$, with another twist $e$, gives a system of presentations ${\mathcal{O}}_{Z_n}(-e{)}^s⟶{\mathcal{O}}_{Z_n}(-d{)}^r⟶D_n⟶0.$ By step 1.1 the compatible first maps algebraize to a map on $Z$. Let $D$ be its coherent cokernel; right exactness gives $D\mathrm{/}{t}^{n}D\cong D_n$ for all $n$.

The sheaf $D$ is locally free near the closed fibre. At such a local ring $B$, let $M$ be its finite module. If $tm=0$, the fact that $M\mathrm{/}{t}^{n}M$ is free over $B\mathrm{/}{t}^{n}B$ and that $t$ is a nonzerodivisor in $B$ implies $m\in t^{n-1}M$ for every $n$. Krull intersection in [F3] gives $m=0$, so $t$ acts injectively on $M$. Lift a basis of $M\mathrm{/}tM$ to a map $B^r\to M$; Nakayama makes it surjective. For its finite kernel $K$, reduction modulo $t$ remains left exact because $M$ has no $t$-torsion (apply the two-term resolution of $B\mathrm{/}tB$). Hence $K\mathrm{/}tK=0$, and Nakayama gives $K=0$. The chosen basis spreads to an open neighbourhood by finite presentations. The failure-of-local-freeness locus of a coherent module is closed, as seen from minors in finite presentations. If nonempty it would have a nonempty closed image in $\mathrm{Spec}R$, hence meet the closed fibre by properness, contradicting what was just proved. Thus $D$ is locally free everywhere.

The products $D_n\otimes D_n\to D_n$ and units algebraize by step 1.1 because $D$ and its tensor powers are locally free. Associativity, commutativity and unit identities hold by the injectivity in that step, since they hold on every $Z_n$. The resulting algebra is finite locally free. Its differentials vanish on the closed fibre by [F2]–[F3] and then near that fibre by Nakayama; the support of this coherent differential module is closed and proper over $R$, so it is empty by the same argument as step 3.1. Thus $D$ is finite étale by [F3], and its relative spectrum is the required lift. Maps between two lifted covers lift uniquely through all $Z_n$ by [F2] and algebraize by step 1.1; multiplication identities are again detected by that injectivity. This proves the equivalence.