Surface finite type formal fibres

Given: A field or complete equicharacteristic Noetherian local ring $A$, an essentially finite-type $A$-algebra $B$, and primes $\mathfrak{q}\subseteq \mathfrak{p}$ of $B$.

def-axiom-of-choice. The Axiom of Choice (AC) is the following statement. > Every family of nonempty sets has a choice function > (def-choice-function). Written out: for every set $\mathcal{F}$ all of whose members are nonempty, there exists a function $g$ with domain $\mathcal{F}$ satisfying $g(S)\in S$ for all $S\in \mathcal{F}$. (The Axiom of Choice)

def-dependent-choice. Let $X$ be a set and let $R\subseteq X\times X$ be a binary relation on $X$. Call $R$ entire on $X$ when $\text{for every }x\in X\text{ there is }y\in X\text{ with }xRy\mathrm{.}$ The Axiom of Dependent Choice, written $\mathrm{D}\mathrm{C}$, is the following statement. (The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain)

lem-ag-local-flatness-regular-parameters. Assume the Axiom of Choice (The Axiom of Choice). Let $(R,\mathfrak{m})\to (S,\mathfrak{n})$ be a local homomorphism of Noetherian local rings and let $M$ be a finite $S$-module. If ${\mathrm{Tor}}_1^R(R\mathrm{/}\mathfrak{m},M)=0$, then $M$ is flat over $R$. The module $M$ is not assumed finite over $R$. (Local flatness criterion by regular parameters)

lem-flat-local-ascent-of-regularity. Assume the Axiom of Choice. For a flat local map $(R,\mathfrak{m})\to (S,\mathfrak{n})$ of nonzero Noetherian local rings: if $R$ and $S\mathrm{/}\mathfrak{m}S$ are regular, then $S$ is regular. Conversely, regularity of $S$ implies regularity of $R$. (flat local ascent of regularity)

lem-surface-complete-equicharacteristic-formal-fibres. Assume AC. For a complete equicharacteristic Noetherian local ring $A$ and primes $\mathfrak{q}\subseteq \mathfrak{p}$, the formal fibre $\widehat{A_{\mathfrak{p}}}{\otimes }_A\kappa (\mathfrak{q})$ is geometrically regular over $\kappa (\mathfrak{q})$. (Surface complete equicharacteristic formal fibres)

lem-surface-completed-polynomial-generic-fibre. Assume AC. Let $A$ be a complete equicharacteristic Noetherian local domain, let $\mathfrak{q}$ be maximal in $A[t]$ over its closed point, and let $\mathfrak{r}$ be a nonzero prime ideal with $\mathfrak{r}\subset \mathfrak{q}$ and $\mathfrak{r}\cap A=0$. Then $\widehat{A[t{]}_{\mathfrak{q}}}{\otimes }_{A[t]}\kappa (\mathfrak{r})$ is geometrically regular over $\kappa (\mathfrak{r})$. (Surface completed polynomial generic fibre)

thm-completion-is-exact-on-finite-modules. Assume the Axiom of Choice. Let $R$ be a Noetherian commutative ring, let $I\subseteq R$ be an ideal, and let $0\to M^{\mathrm{\prime }}\to M\to M^{\mathrm{\prime }\mathrm{\prime }}\to 0$ be a short exact sequence of finitely generated $R$-modules. Then the induced sequence of $I$-adic completions $0\to \widehat{M^{\mathrm{\prime }}}\to \widehat{M}\to \widehat{M^{\mathrm{\prime }\mathrm{\prime }}}\to 0$ is exact. (Adic completion is exact on finite modules over a Noetherian ring)

A regular ring map is a flat map with geometrically regular fibres. Such maps are stable under finite-type base change and localization, compose when the resulting fibres are Noetherian, and descend through a faithfully flat target map. For composition, after a finite purely inseparable fibre-field extension, flat-local ascent from regular base and fibre applies. For descent, flatness descends and each extended fibre has a faithfully flat regular cover, so local regularity descends. These are Stacks Lemmas 15.42.3, 15.42.4 (07QI), and 15.42.7 (07NT), with their stated Noetherian-fibre qualifications.

Generic completed fibres of a power-series ring with polynomial variables are geometrically regular; finite ring extensions have the finite-product completion formula. (Surface generic power series formal fibres, Surface finite completion factors)

Completion base change identifies corresponding closed-fibre local completions. (Completion base change preserves completed local rings on the closed fibre)

The maximal-adic completion of a Noetherian local ring is faithfully flat and has the same residue field. (Completion of a Noetherian local ring is local with the same residue field)