Affine blowup standard charts and overlaps

Put $g_i=f_{i}t\in S$, a homogeneous element of degree one, so that $S=R(I)=A[It]$ is generated as an $A$-algebra by $I$, and $S_{+}$ is generated as an ideal by $g_0,\dots ,g_r$; hence no homogeneous prime $\mathfrak{p}$ of $S$ contains all $g_i$ without containing $S_{+}$, and the standard opens $D_{+}(g_i)$ cover $\mathrm{Proj}S={\mathrm{Bl}}_I\mathrm{Spec}A$ by [F1], [F3], [F4]. Moreover $B_i=A[I\mathrm{/}{f}_i]=S_{(g_i)}$ by [F2], so $U_i=D_{+}(g_i)=\mathrm{Spec}{B}_i$ by [F5].

For each pair $i,j$, [F3] applied to the degree-one elements $g_i,g_j$ identifies $D_{+}(g_i)\cap D_{+}(g_j)=D_{+}(g_i{g}_j)$ with $D(u_{ij})\subseteq \mathrm{Spec}{B}_i$, where $u_{ij}=g_j\mathrm{/}{g}_i$, and identifies the two charts through the canonical isomorphisms $B_i[u_{ij}^{-1}]=S_{(g_i)}[(g_j\mathrm{/}{g}_i{)}^{-1}]\cong S_{(g_i{g}_j)}\cong S_{(g_j)}[(g_i\mathrm{/}{g}_j{)}^{-1}]=(B_j{)}_{u_{ji}}$.

The identification of step 2.1 can be checked directly and torsion-safely: the canonical map $B_i[u_{ij}^{-1}]\to (S[g_i^{-1},g_j^{-1}]{)}_0$ is surjective, because a degree-zero fraction $c\mathrm{/}(g_i^m{g}_j^n)$ with $c$ homogeneous of degree $m+n$ equals $\left(c\mathrm{/}{g}_i^{m+n}\right)u_{ij}^{-n}$ with $c\mathrm{/}{g}_i^{m+n}\in B_i$; and it is injective, because vanishing of the image means $g_i^p{g}_j^{q}c=0$ in $S$ for some $p,q\ge 0$, whence $u_{ij}^q\left(c\mathrm{/}{g}_i^m\right)=g_i^{p}c{g}_j^q\mathrm{/}{g}_i^{m+p+q}=0$ in $B_i[u_{ij}^{-1}]$. No cancellation of $g_i$ in $S$ is used. The same identification sends $f_l\mathrm{/}{f}_i=(f_{l}t)\mathrm{/}(f_{i}t)$ to $(f_{l}t)\mathrm{/}(f_{i}t)$ computed in $S_{(g_i{g}_j)}$, which equals $(f_l\mathrm{/}{f}_j)\mathrm{/}(f_i\mathrm{/}{f}_j)$, and sends $u_{ij}=g_j\mathrm{/}{g}_i$ to $u_{ji}^{-1}$.

The identifications satisfy the identity condition (for $i=j$, $u_{ii}=1$ and the transition is the identity) and the cocycle condition: on a triple overlap every transition is induced by the localisation map $S\to S[g_i^{-1},g_j^{-1},g_k^{-1}]$ and taking degree zero, so the three compositions around the cycle coincide with the identity on $S_{(g_i{g}_j{g}_k)}$. They preserve the structural maps to $\mathrm{Spec}A$, because they are isomorphisms of $A$-algebras for the structure maps $A=S_0\to S_{(g)}$ of the charts.

For $I=(x,y)\subseteq A=k[x,y]$ with $f_0=x$, $f_1=y$, one has $B_0=A[I\mathrm{/}x]=k[x,y\mathrm{/}x]=k[x,s]$ with $s=y\mathrm{/}x$ and $y=xs$, and $u_{01}=s$, so the overlap $D(u_{01})=D(s)$ in $\mathrm{Spec}{B}_0$ retains the points with $x=0$ and $s\mathrm{\ne }0$. Localising instead at the base element $f_1=y$ gives $D(y)=D(xs)=D(x)\cap D(s)$, which is strictly smaller than $D(s)$ and omits those points; hence localisation at the base element is not the overlap formula.

A second finite generating family $I=(h_0,\dots ,h_{r^{\mathrm{\prime }}})$ gives the charts $D_{+}(h_{k}t)=\mathrm{Spec}A[I\mathrm{/}{h}_k]$ of the same scheme $\mathrm{Proj}S$, with their own overlap identifications supplied by the same formulas of [F3] applied to the degree-one elements of $S$; on the intersection of a chart of the first family and a chart of the second, $D_{+}(g_i)\cap D_{+}(h_{k}t)=D_{+}(g_i{h}_{k}t)$, the transition is again induced by the canonical localisation of $S$, so the two cover structures are compatible. No bijection between the two chart families is asserted: the charts are indexed by different generating sets and need not correspond individually.

The formulas allow zero divisors and empty charts: step 3.1 never cancels $g_i$ in $S$, and if $f_i$ is nilpotent then $g_i=f_{i}t$ is nilpotent, so $D_{+}(g_i)=\mathrm{\varnothing }$ and $B_i=S_{(g_i)}=0$ by [F3] and [F5]; the same holds for the overlap formula in the degenerate cases. This completes the proof.