Polynomial lifting on punctured affine three-space

Set $u_1=x,u_2=y,u_3=z$. The opens $D(u_i)$ cover $U$, because the only prime containing all three variables is $\mathfrak{m}$; all intersections are $D({\prod }_{i\in I}{u}_i)$ with their localization rings by [F1]. Thus $U$ is quasi-compact. Apply the overlap criterion of [F1] over $\mathrm{Spec}\mathbb{Z}$: each map $A_{u_i}{\otimes }_{\mathbb{Z}}{A}_{u_j}\to A_{u_i{u}_j}$ is surjective, since its image contains $A$, $1\mathrm{/}{u}_i$ and $1\mathrm{/}{u}_j$. Hence $U$ is separated in the absolute sense required by [F2]. On each chart a coordinate of $v$ is a unit; eliminating that coordinate proves injectivity of $v$, local splitting of its image, and a free rank-two quotient $F$ using the sheaf-cokernel construction in [F4]. It also splits the row map with kernel $Q$. Dualizing that split finite free sequence chartwise gives $F\cong Q^{\vee }$, with the identifications agreeing on overlaps by evaluation. Each localization embeds in $A_{xyz}$: multiplication by a monomial just shifts the exponents in the coefficient description of [F1], so it is injective. By locality and gluing in [F5], $\mathrm{\Gamma }(U,{\mathcal{O}}_U)$ is $A_x\cap A_y\cap A_z$ inside $A_{xyz}$. Write this common localization as $k[x^{\pm 1},y^{\pm 1},z^{\pm 1}]$, meaning finite sums of monomials with integer exponents. Multiplication by a sufficiently large power of $xyz$ gives a polynomial, and uniqueness of polynomial coefficients proves uniqueness of this expression. Membership in all three subrings permits no negative exponent of any variable, so this intersection is $A$. The augmented ordered localization complex is $L^0=A$, $L^1=A_x\oplus A_y\oplus A_z$, $L^2=A_{xy}\oplus A_{xz}\oplus A_{yz}$, $L^3=A_{xyz}$, with alternating inclusions as differentials. Define $K_n^p={⨁}_{\mathrm{\mid }I\mathrm{\mid }=p}A$ for $0\le p\le 3$, with differential adding $j\mathrm{\notin }I$ with coefficient $u_j^n$ and the usual alternating sign. This is the cochain Koszul complex on $x^n,y^n,z^n$. The complement-of-subsets pairing in $\bigwedge A^3$ identifies it, with degree signs, with the homological Koszul complex of [F3] in reverse degrees: the coefficient of a subset and its complement is their wedge coefficient of $e_1\wedge e_2\wedge e_3$, and deleting an index on one side corresponds to adding it on the other. The sequence is regular: the quotients successively have monomial bases with the preceding variable exponents bounded by $n-1$, so multiplication by the next variable's $n$-th power is injective; the final quotient has basis $x^a{y}^b{z}^c$ with $0\le a,b,c<n$. Thus [F3] makes $H^p(K_n^{\bullet })=0$ for $p<3$. The transition from $n$ to $n+1$ multiplies the summand indexed by $I$ by ${\prod }_{i\in I}{u}_i$ (the empty product is 1). It commutes with the differentials, and $a\mapsto a\mathrm{/}({\prod }_{i\in I}{u}_i{)}^n$ identifies its colimit summand with the corresponding localization, hence identifies the colimit complex with $L^{\bullet }$. Exactness of this filtered colimit gives $H^2(L^{\bullet })=0$. Removing $L^0$ shifts the degrees by one and gives precisely the ordered Čech complex for $D(x),D(y),D(z)$ and ${\mathcal{O}}_U$. Therefore its first cohomology vanishes, and [F2] applies to this quasi-coherent sheaf on the verified quasi-compact separated scheme to give $H^1(U,{\mathcal{O}}_U)=0$.

For a section $\sigma$ of $F$, the explicit local splittings in step 1.1 and the quotient description in [F4] give lifts $b_i\in A_{u_i}^3$. On an overlap, their difference has the unique form $b_j-b_i=c_{ij}v$ with $c_{ij}\in A_{u_i{u}_j}$, since it maps to zero in $F$ and $v$ is a split injection there. On the triple overlap, cancellation of the three differences gives $c_{23}-c_{13}+c_{12}=0$. The vanishing of first ordered Čech cohomology proved in step 1.1 yields $c_i\in A_{u_i}$ with $c_{ij}=c_j-c_i$. Consequently $b_i-c_{i}v$ agree on overlaps. The gluing axiom in [F5] glues them to a section of ${\mathcal{O}}_U^3$, and step 1.1 identifies that section with a vector $b\in A^3$ lifting $\sigma$. This proves the surjectivity on sections directly, including the exactness consequence needed from the displayed locally split sequence; it does not infer surjectivity merely from sheaf exactness. The identification $F\cong Q^{\vee }$ then proves the functional assertion.