Projective Čech finiteness and Serre vanishing for the étale lifting construction

On a finite affine cover with affine intersections, quasi-coherent sections on each intersection are module localizations. A short exact sequence therefore gives a termwise exact sequence of Čech complexes, hence a long exact sequence of their cohomology by the elementary kernel/image diagram chase. The localization Čech complex for a principal cover of an affine scheme is exact in positive degrees: localizing it at any prime, one covering function is a unit and insertion of its index is a contracting homotopy. Its cohomology modules thus vanish at every prime, and vanish globally. Refining finite affine covers by principal covers and using this contraction in the rows or columns of the double Čech complex preserves their cohomology. In particular the standard cover may be used consistently for sheaves, exact sequences and closed-immersion pushforwards. Its complex has length $N$, so its cohomology vanishes above $N$.

Every coherent sheaf on ${\mathbb{P}}_A^N$ is a quotient of a finite direct sum of twists. Here is the section-extension argument. Choose finitely many module generators on each $D_{+}(T_i)$. A generator on that chart, restricted to $D_{+}(T_i)\cap D_{+}(T_j)$, is a module element with denominators powers of $T_i\mathrm{/}{T}_j$. Multiplying by a sufficiently large power permits extension to $D_{+}(T_j)$ as a section of a suitable twist. On a pair of charts, two such extensions agree after further localization at $T_i$; their difference is killed by some power of that element. There are finitely many charts, pairs and generators, so increase the twisting exponent to kill all these finitely many differences. The extensions now glue. Their restrictions to the original chart are the original generators multiplied by an invertible power, so the finitely many maps $\mathcal{O}(-a_i)\to F$ constructed this way are jointly surjective. Their kernel is coherent by [F1]. Since $\mathcal{O}(d-a_i)$ is generated by degree $d-a_i$ monomials for $d\ge a_i$, this also proves global generation of $F(d)$ for all sufficiently large $d$. The same construction applies to the coherent pushforward of $F$ from $Z$, whose affine modules are the same finite modules regarded over the ambient quotient ring.

For $N>0$, the Čech complex of $\mathcal{O}(d)$ on ${\mathbb{P}}_A^N$ decomposes into complexes indexed by Laurent monomials $T_0^{e_0}\cdots T_N^{e_N}$ with $\sum e_i=d$. Such a monomial appears on precisely the intersections whose index set contains $J=\{i:e_i<0\}$. If $J$ is empty, the simplex complex has one copy of $A$ in degree-zero cohomology. If $J$ is the full index set, it has one copy of $A$ in degree $N$. In every other case choose an index outside $J$; insertion of that index with the usual alternating sign is a contracting homotopy, so the complex is acyclic. Thus $H^0(\mathcal{O}(d))$ has the finite basis of monomials with all $e_i\ge 0$, $H^N(\mathcal{O}(d))$ has the finite basis with all $e_i<0$, and all intermediate groups vanish. The top group is zero for $d\ge -N$. For $N=0$ the space is affine, its sole group is $A$ in degree zero for every twist, and the assertions follow directly.

Descend on $q$ from $q>N$. For any coherent $F$, take the finite-twist surjection $E\to F$ from step 1.2, with coherent kernel $K$. The exact segment $H^q(E)\to H^q(F)\to H^{q+1}(K)$ shows finiteness of $H^q(F)$ because the end groups are finite by step 2.1 and the induction hypothesis, and $A$ is Noetherian. For $q>0$, apply the same segment after twisting by $d$. Its left group is zero for large $d$ by step 2.1, and its right group is zero for large $d$ by induction applied to $K$. This gives the required vanishing. Finally the Čech complex on $Z$ equals that of its coherent closed-immersion pushforward on projective space; twists commute with that pushforward on every standard chart. The assertions therefore hold on $Z$ as well.