Common completions of algebraic and holomorphic local rings and their finite-type quotients

AC implies DC. For a nonempty set $Y$ with a serial relation $R$, the successor sets $\{z\in Y:yRz\}$ are nonempty; [A1] chooses one successor for each such set. This defines a self-map $s:Y\to Y$ with $yRs(y)$, and [F10] recursively generates a sequence from any specified starting point.

For each $k\ge 1$, every $s\in P\setminus \mathfrak{m}$ has the form $s=c(1+h)$ with $c=s(0)\mathrm{\ne }0$ and $h\in \mathfrak{m}$. Modulo ${\mathfrak{m}}^k$, $h^k=0$, so $c^{-1}{\sum }_{j=0}^{k-1}(-h{)}^j$ is an inverse for $s$. Thus localization changes no $k$-jet and $A_0\mathrm{/}{\mathfrak{m}}^k{A}_0\cong P\mathrm{/}{\mathfrak{m}}^k$. The quotient $P\mathrm{/}{\mathfrak{m}}^k$ records precisely the coefficients of monomials of total degree less than $k$, and its transition maps discard coefficients of degree at least the new truncation order. In a compatible sequence, the coefficient of each $w^{\alpha }$ is therefore stable at every stage $k>\mathrm{\mid }\alpha \mathrm{\mid }$, giving an array in $T$; conversely, every array in $T$ gives these compatible truncations. The induced addition is pointwise, and multiplication is the finite convolution in the Statement, so this is a ring isomorphism. Hence $\widehat{A_0}\cong T$.

Every germ in $\mathfrak{n}$ has a power series ${\sum }_{\alpha }{c}_{\alpha }{w}^{\alpha }$ with no constant term by [F6], since evaluating at $0$ gives $c_0=0$. Choose positive radii $r_i$ on which [F6] bounds $\mathrm{\mid }{c}_{\alpha }\mathrm{\mid }\le M{\prod }_i{r}_i^{-{\alpha }_i}$. Assign each nonconstant monomial to the first coordinate with positive exponent. After removing the corresponding factor $w_i$, the coefficient at $w^{\nu }$ of the grouped series $g_i$ is either $c_{\nu +e_i}$ or $0$, and is bounded by $(M\mathrm{/}{r}_i){\prod }_j{r}_j^{-{\nu }_j}$. Thus [F7] makes each $g_i$ holomorphic on that polydisc, and absolute convergence permits the finite grouping to give the germ identity $f={\sum }_{i=1}^n{w}_i{g}_i$. Conversely each $w_i$ vanishes at $0$, so $\mathfrak{n}=(w_1,\dots ,w_n)$.

By [F11], every ideal of $\mathbb{C}$ is finitely generated; the DC-dependent ideal characterization in [F11], using DC from step 1.1, makes $\mathbb{C}$ Noetherian. The finite-variable corollary in [F8] then makes $P$ Noetherian, and [F8] makes its localization $A_0$ Noetherian under the same DC. The ring $A_0$ is local: fractions with numerator outside $\mathfrak{m}$ are units, while the residue map $p\mathrm{/}q\mapsto p(0)\mathrm{/}q(0)$ is a surjective homomorphism to $\mathbb{C}$ with kernel $\mathfrak{m}{A}_0$. The ring $H$ is Noetherian by [F5] and local by [F3] when $n=1$ and by [F4] for every $n$. Since $I\subseteq \mathfrak{m}$ and every element of $I$ vanishes at the origin, $I{A}_0$ and $IH$ lie in the respective maximal ideals. Every $q\in P\setminus \mathfrak{m}$ maps to a unit in $H$ by [F4], so $P\to H$ extends to $A_0\to H$. Evaluation at $0$ on $H$ has kernel $\mathfrak{n}$, and its composite with $A_0\to H$ is $p\mathrm{/}q\mapsto p(0)\mathrm{/}q(0)$; hence the preimage of $\mathfrak{n}$ is $\mathfrak{m}{A}_0$. Because $I\subseteq \mathfrak{m}$ and $IH\subseteq \mathfrak{n}$, the induced $A_I\to H_I$ has preimage of $\mathfrak{n}\mathrm{/}IH$ equal to $\mathfrak{m}{A}_0\mathrm{/}I{A}_0$, so it is local. The quotients are Noetherian local rings, with maximal ideals the images of those ambient maximal ideals.

For $k\ge 1$, [F6] and [F7] give each holomorphic germ a unique power-series coefficient array, independent of its representative after restricting to a common sufficiently small polydisc. Truncation to total degree less than $k$ gives a surjection $H\to P\mathrm{/}{\mathfrak{m}}^k$; addition and multiplication agree with the truncated coefficient operations, by addition and multiplication of absolutely convergent series and uniqueness in [F7]. For each multi-index $\alpha$ of degree at least $k$, assign the first monomial divisor $w^{\beta }$ in a fixed ordering among the finitely many divisors of total degree $k$. If $\mathrm{\mid }{c}_{\alpha }\mathrm{\mid }\le M{\prod }_i{r}_i^{-{\alpha }_i}$ is a bound from [F6], the coefficient at $w^{\nu }$ after removing $w^{\beta }$ from its assigned subseries is either $c_{\nu +\beta }$ or $0$, hence bounded by $(M{\prod }_i{r}_i^{-{\beta }_i}){\prod }_i{r}_i^{-{\nu }_i}$. By [F7] this grouped series defines a holomorphic germ $g_{\beta }$. Finite grouping therefore writes the tail as ${\sum }_{\mathrm{\mid }\beta \mathrm{\mid }=k}{w}^{\beta }{g}_{\beta }$, so the kernel is contained in ${\mathfrak{n}}^k$. Conversely, step 1.3 makes ${\mathfrak{n}}^k$ generated by the degree-$k$ coordinate monomials. Choose positive radii whose closed polydisc lies inside a common convergence polydisc for a finite expression of an element of ${\mathfrak{n}}^k$ as a sum of degree-$k$ monomials times germs. By [F6], each germ's expansion has a coefficient bound on these radii. Multiplication by $w^{\beta }$ shifts its coefficients and preserves a bound of the same form, with the constant multiplied by ${\prod }_i{r}_i^{{\beta }_i}$; a finite sum also preserves such a bound. The expansion of the given element has a bound on the same radii by [F6]. The two sums agree there, so [F7] identifies their coefficients and shows that all coefficients of total degree below $k$ vanish. Thus ${\mathfrak{n}}^k$ lies in the kernel and $H\mathrm{/}{\mathfrak{n}}^k\cong P\mathrm{/}{\mathfrak{m}}^k$.

Taking inverse limits in steps 1.2 and 2.2 identifies $\widehat{A_0}$ and $\widehat{H}$ with $T$. Each polynomial variable maps to the same formal variable under both completion maps, so the induced map $\widehat{A_0}\to \widehat{H}$ is the identity of $T$.

Under the assumed AC, apply [F9] to the finitely generated ideals $I{A}_0\subseteq A_0$ and $IH\subseteq H$; these rings are Noetherian by step 2.1. Their images in $T$ are both generated by the same finite polynomial generators of $I$, because the maps in step 3.1 send each $w_i$ to the same formal variable. Thus $\widehat{A_I}\cong T\mathrm{/}IT\cong \widehat{H_I}$, and the induced map on completions is the identity under these identifications. The quotient is by $I$ itself, not by its radical.