Saturation detected on projective charts

Chart membership is a divisibility condition. Let $h\in B$ be homogeneous of degree $d$ and fix $i$. Because a homogeneous ideal is generated by its homogeneous elements, the condition $h\mathrm{/}{x}_i^d\in (I[x_i^{-1}]{)}_0$ means that there are $k\ge 0$ and $c\in I$ homogeneous of degree $k$ with $h\mathrm{/}{x}_i^d=c\mathrm{/}{x}_i^k$ in $B[x_i^{-1}]$, and equality of these fractions means that $x_i^N(x_i^{k}h-x_i^{d}c)=0$ for some $N\ge 0$. Multiplication by the polynomial variable $x_i$ is injective on $B=A[x_0,\dots ,x_n]$ for every coefficient ring $A$, because it shifts the $A$-basis of monomials injectively; hence $x_i^{k}h=x_i^{d}c$. If $k\ge d$, cancellation gives $x_i^{k-d}h=c\in I$; if $k<d$, it gives $h=x_i^{d-k}c\in I$. Conversely either containment $x_i^{k^{\mathrm{\prime }}}h\in I$ represents $h\mathrm{/}{x}_i^d$ by an element of $(I[x_i^{-1}]{)}_0$. Thus chart membership is equivalent to $x_i^{k^{\mathrm{\prime }}}h\in I$ for some $k^{\mathrm{\prime }}\ge 0$, without assuming $A$ is a domain.

From chart conditions to one power of $\mathfrak{b}$. Suppose $h$ is homogeneous of degree $d$ and $x_i^{r_i}h\in I$ for every $i$ with $0\le r_i<\mathrm{\infty }$. Put $r=r_0+\cdots +r_n$. Every monomial $x_0^{a_0}\cdots x_n^{a_n}$ of total degree $r$ has some $a_i\ge r_i$: otherwise $a_i\le r_i-1$ for all $i$ would give $r=\sum a_i\le \sum (r_i-1)=r-(n+1)<r$. For such an $i$ the product equals $x_i^{a_i-r_i}\cdot (x_i^{r_i}h)\in I$, since $x_i^{r_i}h\in I$ and $I$ is an ideal; a monomial with total degree $r$ times $h$ is precisely one of these products. As ${\mathfrak{b}}^r$ is spanned by the monomials of total degree $r$, we get ${\mathfrak{b}}^{r}h\subseteq I$, hence $h\in I^{\mathrm{s}\mathrm{a}\mathrm{t}}$.

The saturation criterion. If $h\in I^{\mathrm{s}\mathrm{a}\mathrm{t}}$ then ${\mathfrak{b}}^{r}h\subseteq I$ for some $r$, and in particular $x_i^{r}h\in I$ for every $i$, so step 1.1 gives $h\mathrm{/}{x}_i^d\in (I[x_i^{-1}]{)}_0$ for every $i$. Conversely, if $h\mathrm{/}{x}_i^d\in (I[x_i^{-1}]{)}_0$ for every $i$, then step 1.1 provides exponents $r_i$ with $x_i^{r_i}h\in I$, and step 1.2 gives ${\mathfrak{b}}^{r}h\subseteq I$ for $r={\sum }_i{r}_i$, that is $h\in I^{\mathrm{s}\mathrm{a}\mathrm{t}}$. This is claim (1) as stated for homogeneous $h$. Since $I$ and $I^{\mathrm{s}\mathrm{a}\mathrm{t}}$ are homogeneous ideals, their membership for an arbitrary polynomial can also be checked componentwise.

Same chart ideals. Since $I\subseteq I^{\mathrm{s}\mathrm{a}\mathrm{t}}$, we have $(I[x_i^{-1}]{)}_0\subseteq ((I^{\mathrm{s}\mathrm{a}\mathrm{t}})[x_i^{-1}]{)}_0$. Conversely let $g\mathrm{/}{x}_i^k\in ((I^{\mathrm{s}\mathrm{a}\mathrm{t}})[x_i^{-1}]{)}_0$ with $g\in I^{\mathrm{s}\mathrm{a}\mathrm{t}}$ homogeneous of degree $k$; then $x_i^{s}g\in I$ for some $s\ge 0$ by definition of $I^{\mathrm{s}\mathrm{a}\mathrm{t}}$, so $g\mathrm{/}{x}_i^k=(x_i^{s}g)\mathrm{/}{x}_i^{k+s}\in (I[x_i^{-1}]{)}_0$, giving the reverse inclusion and claim (2).

Saturated ideals with equal charts are equal. Let $I,J$ be saturated homogeneous ideals with $(I[x_i^{-1}]{)}_0=(J[x_i^{-1}]{)}_0$ for all $i$, and let $h\in I$ be homogeneous. By step 2.1 applied with $I^{\mathrm{s}\mathrm{a}\mathrm{t}}=I$, the element $h$ satisfies $h\mathrm{/}{x}_i^d\in (I[x_i^{-1}]{)}_0=(J[x_i^{-1}]{)}_0$ for all $i$ of the same degree $d=\mathrm{deg}h$, so step 2.1 applied with $J^{\mathrm{s}\mathrm{a}\mathrm{t}}=J$ gives $h\in J$. Hence $I\subseteq J$, and symmetrically $J\subseteq I$; this is claim (3).

Conclusion. Steps 1.1, 1.2 and 2.1 prove the chartwise saturation criterion (1), step 3.1 identifies the chart ideals of $I$ and $I^{\mathrm{s}\mathrm{a}\mathrm{t}}$, and step 3.2 shows that a saturated homogeneous ideal is determined by its chart ideals. For $n=0$ the index set is $\{0\}$, the pigeonhole count in step 1.2 reduces to $r=r_0$, and all statements read $x_0^{r}h\in I$ on the single chart $D_{+}(x_0)=\mathrm{Spec}A$ of [F1], so the case is included. The Axiom of Choice [A1] is inherited from the Proj construction and is not used again. [A1, F1, step 1.2, step 2.1, step 3.1, step 3.2, cases: n=0] \qed