Conditions 3.1 Fix a full subcategory ${\mathscr {C}}$ of the category of noetherian rings, and let $\mathbf {R}$ be a property of noetherian local rings. We consider the following conditions on the property $\mathbf {R}$.

• $({\rm R}_0)$ The property $\mathbf {R}$ holds for every field $k$.

• $(\text {R}_{\text {I}}^{\mathscr {C}})$ (Ascent via geometrically regular homomorphisms) For every geometrically regular local homomorphism $\varphi \colon A \to B$ of noetherian local rings in ${\mathscr {C}}$, if $A$ satisfies $\mathbf {R}$, then $B$ satisfies $\mathbf {R}$.

• $(\text {R}_{\text {II}}^{\mathscr {C}})$ (Descent) For every flat local homomorphism $\varphi \colon A \to B$ of noetherian local rings such that $B$ is in ${\mathscr {C}}$, if $B$ satisfies $\mathbf {R}$, then $A$ satisfies $\mathbf {R}$.

• $({\rm R}_{{\rm III}})$ (Openness) For every noetherian complete local ring $C$, the locus $U_\mathbf {R}({\rm Spec}(C))$ is open.

• $(\text {R}_{\text {IV}}^{\mathscr {C}})$ (Lifting from Cartier divisors) For every noetherian local ring $A$ in ${\mathscr {C}}$ and for every nonzerodivisor $t$ in its maximal ideal, if $A/tA$ satisfies $\mathbf {R}$, then $A$ satisfies $\mathbf {R}$.

• $(\text {R}_{\text {V}}^{\mathscr {C}})$ (Localization) If a noetherian local ring $A$ in ${\mathscr {C}}$ satisfies $\mathbf {R}$, then $A_\mathfrak {p}$ satisfies $\mathbf {R}$ for every prime ideal $\mathfrak {p} \subseteq A$.

In addition, we consider the following variant of $({\rm R}_{\rm {I}}^{{\mathscr {C}}})$.

• $(\text {R}^{{\mathscr {C}}\prime }_{\text {I}})$ (Ascent via geometrically $\mathbf {R}$ homomorphisms) For every local geometrically $\mathbf {R}$ homomorphism $\varphi \colon A \to B$ of noetherian local rings in ${\mathscr {C}}$, if $A$ satisfies $\mathbf {R}$, then $B$ satisfies $\mathbf {R}$.

We also consider the following conditions for $\mathbf {R}$ that affect geometrically $\mathbf {R}$ homomorphisms.

1. $(\text {P}_{\text {I}}^{\mathscr {C}})$ If $\varphi \colon A \to B$ and $\psi \colon B \to C$ are a geometrically $\mathbf {R}$ homomorphism and a geometrically regular homomorphism of noetherian rings, respectively, and $B$ and $C$ are in ${\mathscr {C}}$, then $\psi \circ \varphi$ is a geometrically $\mathbf {R}$ homomorphism.

2. $(\text {P}_{\text {II}}^{\mathscr {C}})$ (Descent) If $\varphi \colon A \to B$ and $\psi \colon B \to C$ are two homomorphisms of noetherian rings such that $\psi \circ \varphi$ is geometrically $\mathbf {R}$, $\psi$ is faithfully flat, and $C$ is in ${\mathscr {C}}$, then $\varphi$ is geometrically $\mathbf {R}$.

3. $({\rm P}_{{\rm III}})$ Every field $k$ is geometrically $\mathbf {R}$ over itself.

4. $(\text {P}_{\text {IV}}^{\mathscr {C}})$ (Stability under finitely generated ground field extensions) If a noetherian ring $A$ in ${\mathscr {C}}$ is geometrically $\mathbf {R}$ over a field $k$, then, for all finitely generated field extensions $k \subseteq k'$, the ring $A \otimes _k k'$ is geometrically $\mathbf {R}$ over $k'$.

In addition, we consider the following variant of $({\rm {P}}_{{\rm I}}^{{\mathscr {C}}})$.

• $({\rm P}_{\rm {I}}^{{\mathscr {C}}})$ If $\varphi \colon A \to B$ and $\psi \colon B \to C$ are geometrically $\mathbf {R}$ homomorphisms of noetherian rings such that $B$ and $C$ are in ${\mathscr {C}}$, then $\psi \circ \varphi$ is also geometrically $\mathbf {R}$.

We drop ${\mathscr {C}}$ from our notation if ${\mathscr {C}}$ is the entire category of noetherian rings.

Proof. (i) is clear from definition of being geometrically $\mathbf {R}$.

To show (ii) and (iii), it suffices to consider the case when $A$ is a field $k$ by transitivity of fibers; see [Reference Grothendieck and DieudonnéEGAIV₂, Remarque 7.3.5(ii)]. Note that for (ii) (respectively, (iii)), we use the hypothesis on ${\mathscr {C}}$ to guarantee that after this reduction, $B$ and $C$ (respectively, $C$) are still in ${\mathscr {C}}$. Consider the composition

\[ k \overset{\varphi}{\longrightarrow} B \overset{\psi}{\longrightarrow} C \]

of homomorphisms of noetherian rings, and consider the base change

(2)\begin{equation} k' \xrightarrow{\varphi \otimes_k \mathrm{id}_{k'}} B \otimes_k k' \xrightarrow{\psi \otimes_k \mathrm{id}_{k'}} C \otimes_k k' \end{equation}

for a finite field extension $k \subseteq k'$. For (ii), we first note that since $B \to B \otimes _k k'$ is a module-finite homomorphism, it induces finite field extensions on residue fields. Thus, the base change $\psi \otimes _k \mathrm {id}_{k'}$ of $\psi$ is geometrically $\mathbf {R}'$. Since the local rings of $B \otimes _k k'$ satisfy $\mathbf {R}$ by assumption, we see that the local rings of $C \otimes _k k'$ also satisfy $\mathbf {R}$. For (iii), we note that in (2), the local rings of $C \otimes _k k'$ satisfy $\mathbf {R}$ by assumption. Thus, the local rings of $B \otimes _k k'$ also satisfy $\mathbf {R}$ by $({\rm R}_{{\rm II}}^{\mathscr {C}})$, since $\psi \otimes _k \mathrm {id}_{k'}$ is faithfully flat by base change.

For (iv), let $k \subseteq k'$ be a finitely generated field extension. By [Reference Datta and MurayamaDM20, Lemma 4.9], there exist a finite extension $k \subseteq k_1$ and a diagram

of finitely generated field extensions, where $k_1 \subseteq k_2 := (k' \otimes _k k_1)_\mathrm {red}$ is a separable field extension. Since $A$ is geometrically $\mathbf {R}$ over $k$, the local rings of $A \otimes _k k_1$ satisfy $\mathbf {R}$. We therefore see that the local rings of $A \otimes _k k_2$ also satisfy $\mathbf {R}$, since $k_1 \to k_2$ is regular and both $A \otimes _k k_1$ and $A \otimes _k k_2$ are in ${\mathscr {C}}$. Finally, the local rings of $A \otimes _k k'$ satisfy $\mathbf {R}$ by $({\rm R}_{{\rm II}}^{{\mathscr {C}}})$.

For (v), we note that $\varphi \otimes _A \mathrm {id}_C$ is flat by base change, and hence it suffices to show that, for every prime ideal $\mathfrak {q} \subseteq C$, the fiber $B \otimes _A C \otimes _C \kappa (\mathfrak {q})$ is geometrically $\mathbf {R}$. Letting $\mathfrak {p} = \mathfrak {q} \cap A$, the field extension $\kappa (\mathfrak {p}) \subseteq \kappa (\mathfrak {q})$ is finitely generated since $A \to C$ is essentially of finite type [Reference Grothendieck and DieudonnéEGAI_new, Proposition 6.5.10]. Since

\[ B \otimes_A C \otimes_C \kappa(\mathfrak{q}) \simeq B \otimes_A \kappa(\mathfrak{q}) \simeq B \otimes_A \kappa(\mathfrak{p}) \otimes_{\kappa(\mathfrak{p})} \kappa(\mathfrak{q}), \]

we see that $B \otimes _A C \otimes _C \kappa (\mathfrak {q})$ is geometrically $\mathbf {R}$ by $({\rm P}_{{\rm IV}}^{{\mathscr {C}}})$.

Proof Proof of Theorem 3.4

We want to show that, for every prime ideal $\mathfrak {p} \subseteq A$, the $\kappa (\mathfrak {p})$-algebra $B \otimes _A \kappa (\mathfrak {p})$ is geometrically $\mathbf {R}$ over $\kappa (\mathfrak {p})$. By noetherian induction, it suffices to show that if $\mathfrak {p}_0 \subseteq A$ is a prime ideal and $B \otimes _A \kappa (\mathfrak {p})$ is geometrically $\mathbf {R}$ over $\kappa (\mathfrak {p})$ for every prime ideal $\mathfrak {p} \supsetneq \mathfrak {p}_0$, then $B \otimes _A \kappa (\mathfrak {p}_0)$ is geometrically $\mathbf {R}$ over $\kappa (\mathfrak {p}_0)$. Replacing $A$ by $A/\mathfrak {p}_0$ and $B$ by $B/\mathfrak {p}_0 B$, we may therefore assume that $A$ is a domain and that $B \otimes _A \kappa (\mathfrak {p})$ is geometrically $\mathbf {R}$ over $\kappa (\mathfrak {p})$ for every non-zero prime ideal $\mathfrak {p} \subseteq A$. We note that $A/\mathfrak {p}_0$ is quasi-excellent by [Reference Grothendieck and DieudonnéEGAIV₂, Proposition 7.3.15(i)], and that $B/\mathfrak {p}_0 B$ appears in ${\mathscr {C}}$ by (ii).

We now reduce to showing that the local rings of $B \otimes _A K$ satisfy $\mathbf {R}$, where $K := {\rm Frac}(A)$. We want to show that, for every finite field extension $K \subseteq K'$, the local rings of $B \otimes _A K'$ satisfy $\mathbf {R}$. Let $A \subseteq A'$ be a module-finite extension such that $A'$ is a domain and such that $K' = {\rm Frac}(A')$. Then $A'$ is a semi-local ring, and setting $B' := B \otimes _A A'$, we have $B \otimes _A K' \simeq B' \otimes _{A'} K'$. Note that every local ring of $B \otimes _A K'$ is a local ring of $B'_\mathfrak {q} \otimes _{A'_\mathfrak {p}} K'$ for some prime ideals $\mathfrak {p} \subseteq A'$ and $\mathfrak {q} \subseteq B'$. Moreover, denoting by $k'$ the residue field of $A'_\mathfrak {p}$, we have

\[ B \otimes_A k' \simeq B \otimes_A A' \otimes_{A'} k' \simeq B' \otimes_{A'} k'. \]

Thus, $B'_\mathfrak {q} \otimes _{A'_\mathfrak {p}} k'$ is a localization of $B \otimes _A k'$, and hence $B'_\mathfrak {q} \otimes _{A'_\mathfrak {p}} k'$ is geometrically $\mathbf {R}$ over $k'$. We may therefore replace $A$ by $A'_\mathfrak {p}$, $B$ by $B'_\mathfrak {q}$, and $K$ by $K'$, in which case it suffices to show that the local rings of $B \otimes _A K$ satisfy $\mathbf {R}$. We note that $A'_\mathfrak {p}$ is quasi-excellent by [Reference Grothendieck and DieudonnéEGAIV₂, Proposition 7.3.15(i) and Théorème 7.7.2], and that $B'_\mathfrak {q}$ appears in ${\mathscr {C}}$ by (ii).

We now apply Gabber's weak local uniformization theorem. Since $X:= {\rm Spec}(A)$ is quasi-excellent by (i), Theorem 2.9 implies there exists a finite covering $\{f_i\colon Y_i \to X\}_{i=1}^m$ of $X$ in the alteration topology, where $Y_i$ is regular and integral for every $i$. Note that $X$ is irreducible by our reduction in the first paragraph and by construction of $A'$ in the previous paragraph. Thus, Theorem 2.8 implies there exist a proper surjective morphism $\pi \colon V \to X$ and a Zariski open covering $V = \bigcup _{i=1}^m V_i$ fitting into a commutative diagram

for every $i$. By base change along the morphism ${\rm Spec}(\varphi )\colon X' \to X$, where $X' := {\rm Spec}(B)$, we obtain the commutative diagram

(3)

with cartesian squares for every $i$. We now consider a point $\eta ' \in X'$ lying over the generic point of $X$. We want to show that $\mathcal {O}_{X',\eta '}$ satisfies $\mathbf {R}$. Since $\pi '$ is surjective, there exists a point $\xi ' \in V'$ such that $\pi '(\xi ') = \eta '$. Since $\pi '$ is closed, the specialization $\eta ' \rightsquigarrow \mathfrak {n}$ in $X' = {\rm Spec}(B)$ then lifts to a specialization $\xi ' \rightsquigarrow v'$ in $V'$ [Sta20, Tag 0066]. Since $V' = \bigcup _{i=1}^m V'_i$ is a Zariski open covering, there exists an index $i_0 \in \{1,2,\ldots,m\}$ such that $v' \in V_{i_0}'$, and since open sets are stable under generization, we have $\xi ' \in V_{i_0}'$ as well. We claim that it suffices to show that $\mathcal {O}_{Y_{i_0}',h_{i_0}'(\xi ')}$ satisfies $\mathbf {R}$. Since the morphism $f_{i_0}$ is maximally dominating, base-changing the bottom square in (

) for $i = i_0$ along the morphism ${\rm Spec}(K) \to X = {\rm Spec}(A)$, localizing at the generic point of $Y_{i_0}$, and taking global sections yields the cocartesian square

of rings, where $L_{i_0}$ is the function field of $Y_{i_0}$. The bottom horizontal arrow is faithfully flat; thus, the top horizontal arrow is also faithfully flat by base change. After localizing, we therefore obtain a faithfully flat homomorphism

\[ \mathcal{O}_{X',\eta'} \simeq (B \otimes_A K)_{\eta'} \longrightarrow (B \otimes_A L_{i_0})_{h_{i_0}'(\xi')} \simeq \mathcal{O}_{Y_{i_0}',h_{i_0}'(\xi')}, \]

and $({\rm R}_{\rm II}^{{\mathscr {C}}})$ implies that if $\mathcal {O}_{Y_{i_0}',h_{i_0}'(\xi ')}$ satisfies $\mathbf {R}$, then $\mathcal {O}_{X',\eta '}$ satisfies $\mathbf {R}$. Here we use the fact that $(B \otimes _A L_{i_0})_{h_{i_0}'(\xi ')}$ is in ${\mathscr {C}}$ by (ii).

It remains to show that $\mathcal {O}_{Y_{i_0}',h_{i_0}'(\xi ')}$ satisfies $\mathbf {R}$. Setting $y' := h_{i_0}'(v')$ and $y := g_{i_0}(y')$, the residue field extension $k \subseteq \kappa (y)$ is finitely generated since $f_{i_0}$ is of finite type. Thus, the closed fiber of the flat homomorphism $\psi$ in the commutative diagram

is geometrically $\mathbf {R}$ by $({\rm P}_{{\rm IV}}^{{\mathscr {C}}})$, since $B \otimes _A k$ is in ${\mathscr {C}}$ by (ii). Since $\mathcal {O}_{Y_{i_0},y}$ is regular by construction and $\mathcal {O}_{Y'_{i_0},y'}$ appears in ${\mathscr {C}}$ by (ii), we can apply Lemma 3.5 (which uses $({\rm R}_{{\rm IV}}^{{\mathscr {C}}})$) to deduce that $\mathcal {O}_{Y'_{i_0},y'}$ satisfies $\mathbf {R}$. Finally, $({\rm R}_{{\rm V}}^{{\mathscr {C}}})$ implies that $\mathcal {O}_{Y_{i_0}',h_{i_0}'(\xi ')}$ satisfies $\mathbf {R}$, since the specialization $\xi ' \rightsquigarrow v'$ in $V'$ maps to the specialization $h_{i_0}'(\xi ') \rightsquigarrow h_{i_0}'(v') = y'$ by continuity.

Proof. We have the commutative square

where $\sigma$ and $\tau$ are the canonical $\mathfrak {m}$-adic completion homomorphisms. By (i), the homomorphism $\sigma$ is geometrically $\mathbf {R}$. We claim that $\varphi ^*$ is geometrically $\mathbf {R}$. By [Reference Grothendieck and DieudonnéEGAI_new, Chapitre 0, Lemme 6.8.3.1], the ring $B^*$ is a noetherian local ring, and we have

\[ B^* \otimes_{\hat{A}} k \simeq B \otimes_A k. \]

Theorem 3.4 therefore implies that $\varphi ^*$ is geometrically $\mathbf {R}$, where we use the fact that the complete local ring $\hat {A}$ is excellent by [Reference Grothendieck and DieudonnéEGAIV₂, Scholie 7.8.3(iii)]. The composition $\tau \circ \varphi = \varphi ^* \circ \sigma$ is geometrically $\mathbf {R}$ by $({\rm P}_{{\rm I}}^{{\mathscr {C}}'})$, and therefore $\varphi$ is also by $({\rm P}_{{\rm II}}^{{\mathscr {C}}})$ (which holds by $({\rm R}_{{\rm II}}^{{\mathscr {C}}})$ and Lemma 3.3(iii)) since $\tau$ is faithfully flat [Reference MatsumuraMat89, Theorem 8.14].

Proof. It suffices to show that the hypotheses in Theorem B imply those in Corollary 3.6 when ${\mathscr {C}}$ is the entire category of noetherian rings. Note that (i) is already a hypothesis in Theorem B and that (ii) is vacuously true. It therefore suffices to note that $({\text {R}'_{\text {I}}})$ implies $({\text {P}'_{\text {I}}})$ by Lemma 3.3(ii), and that $({\text {R}_{\text {I}}})$ and $({\text {R}_{\text {II}}})$ imply $({\text {P}_{\text {IV}}})$ by Lemma 3.3(iv). Here, $({\text {R}_{\text {I}}})$ holds by $({\text {R}'_{\text {I}}})$ and the assumption that regular local rings satisfy $\mathbf {R}$.

Proof. We have the commutative square

where $\sigma$ and $\tau$ are the canonical $\mathfrak {m}$-adic and $\mathfrak {n}$-adic completion homomorphisms, respectively. By (i), the homomorphism $\sigma$ is geometrically $\mathbf {R}$. We claim that $\hat {\varphi }$ is geometrically $\mathbf {R}$. Note that $\hat {\varphi }$ is flat by [Reference MatsumuraMat89, Theorem 22.4(i)], and that $B \otimes _A k$ has geometrically $\mathbf {R}$ formal fibers by (i). Thus, the composition

\[ k \longrightarrow B \otimes_A k \longrightarrow \hat{B} \otimes_{\hat{A}} k \]

is geometrically $\mathbf {R}$ by applying $({\text {P}'_{\text {I}}})$ (which holds by $({\text {R}'_{\text {I}}})$ and Lemma 3.3(ii)). Since this composition is equal to $\hat {\varphi } \otimes _{\hat {A}} \mathrm {id}_k$, Theorem 3.4 implies that $\hat {\varphi }$ is geometrically $\mathbf {R}$, where we use the fact that the complete local ring $\hat {A}$ is excellent by [Reference Grothendieck and DieudonnéEGAIV₂, Scholie 7.8.3(iii)].

The composition $\tau \circ \varphi = \varphi ^* \circ \sigma$ is geometrically $\mathbf {R}$ by $({\text {P}'_{\text {I}}})$, and therefore $\varphi$ is also by $({\rm P}_{{\rm I}}')$ (which holds by $({\rm R}_{{\rm II}}^{{\mathscr {C}}})$ and Lemma 3.3(iii)) since $\tau$ is faithfully flat [Reference MatsumuraMat89, Theorem 8.14].

Proof. Let $\xi \in Y$ be an arbitrary point, and let $\eta = f(\xi )$. We want to show that $\mathcal {O}_{Y,\xi }$ is geometrically $\mathbf {R}$ over $\kappa (\eta )$. By [Sta20, Tag 02IL], the point $\xi$ specializes to a closed point $y \in Y$. Since $f$ maps closed points to closed points, the point $x = f(y)$ is closed in $X$. After localization, we then obtain a flat local homomorphism

\[ \varphi\colon \mathcal{O}_{X,x} \longrightarrow \mathcal{O}_{Y,y} \]

whose closed fiber is geometrically $\mathbf {R}$ by assumption. Finally, we can apply Theorem 3.4, Corollary 3.6, and Corollary 3.7 to the homomorphism $\varphi$ under the assumptions in (i), (ii), and (iii), respectively, to conclude that $\mathcal {O}_{Y,\xi }$ is geometrically $\mathbf {R}$ over $\kappa (\eta )$.

Proof. Consider a specialization $\eta \rightsquigarrow x$ in $X$, and suppose that $f^{-1}(x)$ is geometrically $\mathbf {R}$ over $\kappa (x)$. We want to show that $f^{-1}(\eta )$ is geometrically $\mathbf {R}$ over $\kappa (\eta )$. It suffices to show that, for every $\xi \in f^{-1}(\eta )$, the local ring $\mathcal {O}_{Y,\xi }$ is geometrically $\mathbf {R}$ over $\kappa (\eta )$. Since $f$ is closed, the specialization $\eta \rightsquigarrow x$ lifts to a specialization $\xi \rightsquigarrow y$ [Sta20, Tag 0066]. After localization, we then obtain a flat local homomorphism

\[ \varphi\colon \mathcal{O}_{X,x} \longrightarrow \mathcal{O}_{Y,y} \]

whose closed fiber is geometrically $\mathbf {R}$ by assumption. Finally, we can apply Theorem 3.4, Corollary 3.6, and Corollary 3.7 to the homomorphism $\varphi$ under the assumptions in (i), (ii), and (iii), respectively, to conclude that $\mathcal {O}_{Y,\xi }$ is geometrically $\mathbf {R}$ over $\eta$.

Proof. As in the proof of Theorem B, it suffices to note that the hypotheses in (i) and (ii) imply those in Propositions 3.8(ii) and 3.9(ii), respectively, when ${\mathscr {C}}$ is the entire category of noetherian rings.

Proof. Since a noetherian semi-local ring is Nagata if and only if it has geometrically reduced formal fibers (Remark 2.5), it suffices to verify the hypotheses in Theorem 3.10 for $\mathbf {R}' = \mathbf {R} +$ ‘reduced’. Both (i) and (ii) hold by assumption. Theorem B implies (iii) holds, since complete local rings have geometrically $\mathbf {R}$ formal fibers by the assumption that regular local rings satisfy $\mathbf {R}$.

Proof. The $I$-adic completion $\hat {B}$ of $B$ is a noetherian semi-local ring that is $I\hat {B}$-adically complete by [Reference BourbakiBou98, Chapter III, § 3, ${\rm n}^{{\rm o}}$ 4, Proposition 8]. Thus, by Theorem C, it suffices to show that $\hat {B}/I\hat {B} \simeq B/I$ is a Nagata ring with geometrically $\mathbf {R}$ formal fibers. But $B/I$ has geometrically $\mathbf {R}$ formal fibers by [Reference Grothendieck and DieudonnéEGAIV₂, Proposition 7.3.15(i)], and is also Nagata by Definition 2.4.

Proof. We apply Theorem 3.4 when ${\mathscr {C}}$ is the category of excellent rings, which is stable under homomorphisms essentially of finite type by [Reference Grothendieck and DieudonnéEGAIV₂, Scholie 7.8.3(ii)]. To verify the hypotheses of Theorem 3.4, it suffices to note that excellent local rings are homomorphic images of Cohen–Macaulay rings [Reference KawasakiKaw02, Corollary 1.2], and hence $({\rm R}_{{\rm IV}}^{{\mathscr {C}}})$, $({\rm R}_{{\rm V}}^{{\mathscr {C}}})$, and $({\rm P}_{{\rm IV}}^{{\mathscr {C}}})$ hold by Table 2.

Proof. Throughout this proof, if $R$ is a local ring, then we denote by $\mathfrak {m}_R$ and $k_R$ the maximal ideal and residue field of $R$, respectively. By replacing $A$ with its reduction $A_\mathrm {red}$, it suffices to consider the case when $A$ is reduced.

We first show that $B \to C$ is module-finite, for which it suffices to show that $A \to C$ is module-finite. By Kollár's finiteness theorem [Reference KollárKol17, Theorem 2], $(5) \Rightarrow (1)$, it suffices to show that for every $\mathfrak {p} \in {\rm Ass}_A(B)$, the local hull

\[ \Gamma\bigl(U,(A/\mathfrak{p})^\sim\rvert_U\bigr) \]

is a finitely generated $A$-module. Since $A$ and $B$ are reduced and ${\rm Spec}(B) \to {\rm Spec}(A)$ is a homeomorphism, we see that ${\rm Ass}_A(A) = {\rm Ass}_A(B)$. Thus, applying Kollár's finiteness theorem [Reference KollárKol17, Theorem 2], $(1) \Rightarrow (5)$ to $F = \mathcal {O}_{{\rm Spec}(A)}$, it suffices to show that the local hull $\Gamma (U,\tilde {A}\rvert _U)$ is a finitely generated $A$-module. But this is automatic since the natural homomorphism $A \to \Gamma (U,\tilde {A}\rvert _U)$ is an isomorphism by [Reference KollárKol17, Lemma 14], using the condition ${\rm depth}(A) \geqslant 2$.

We now show that ${\rm Spec}(C) \to {\rm Spec}(B)$ is a universal homeomorphism. Let $A' := A^\mathrm {sh}$ be a strict Henselization of $A$, and set $U' := {\rm Spec}(A') - \{\mathfrak {m}_{A'}\}$. Denoting $B' := B \otimes _A A^\mathrm {sh}$ and $C' := C \otimes _A A^\mathrm {sh}$, it suffices to show that the morphism

(7)\begin{equation} {\rm Spec}(C') \longrightarrow {\rm Spec}(B') \end{equation}

is a universal homeomorphism by [Reference Grothendieck and DieudonnéEGAIV₂, Corollaire 2.6.4(iv)] since $A \to A'$ is faithfully flat. Setting

\[ U' := {\rm Spec}(A') - \{\mathfrak{m}_{A'}\} \quad \text{and} \quad V' := {\rm Spec}(B') - \{\mathfrak{m}_{B'}\}, \]

we see that (7) is an isomorphism over $V'$, and hence it suffices to show that there is only one prime ideal in $C'$ lying over $\mathfrak {m}_{B'}$, and that the residue field extension $k_{B'} \to k_{C'}$ is purely inseparable [Reference Grothendieck and DieudonnéEGAIV₄, Corollaire 18.12.11]. First, since ${\rm depth}(A') \geqslant 2$ by [Reference Bruns and HerzogBH98, Proposition 1.2.16(a)], Hartshorne's connectedness theorem [Reference Grothendieck and DieudonnéEGAIV₂, Théorème 5.10.7] implies that the punctured spectrum $U'$ is connected, and thus, $V'$ is also connected by the fact that ${\rm Spec}(B') \to {\rm Spec}(A')$ is a homeomorphism. By the Henselian property of $A' = A^\mathrm {sh}$, the semi-local ring $C'$ is a direct product of local rings [Reference Grothendieck and DieudonnéEGAIV₄, Proposition 18.5.9${\rm (ii)}$]. But the fact that $V'$ is connected forces there to be only one factor in this decomposition, since the morphism (7) is an isomorphism over $V'$, and the support of

(8)\begin{equation} C' \simeq \Gamma\bigl(U',\widetilde{B'}\rvert_{U'}\bigr) \end{equation}

is the closure of the inverse image of $V'$ under (7) by [Reference KollárKol17, Definition 1, $(3')$]. Thus, $C'$ is a local ring, and there is therefore only one prime ideal in $C'$ lying over $\mathfrak {m}_{B'}$. Note that the isomorphism (8) follows from flat base change for local hulls [Reference KollárKol17, (13)] since the preimage of $\{\mathfrak {m}_B\}$ in ${\rm Spec}(B')$ is $\{\mathfrak {m}_{B'}\}$. Finally, since $k_{A'}$ is separably closed, the residue field extension $k_{B'} \to k_{C'}$ is purely inseparable, and hence (7) is a universal homeomorphism.

We now note that

(9)\begin{equation} {\rm depth}_{\mathfrak{m}_C}(C) = {\rm depth}_{\mathfrak{m}_A}(C) \geqslant 2, \end{equation}

where the first equality is [Reference Grothendieck and DieudonnéEGAIV₁, Chapitre 0, Proposition 16.4.8], and the second inequality follows from [Reference KollárKol17, Lemma 14] since $C$ is module-finite over $A$. Moreover, the morphism ${\rm Spec}(C) \to {\rm Spec}(B)$ is birational since it induces an isomorphism between ${\rm Spec}(B) - \{\mathfrak {m}_B\}$ and ${\rm Spec}(C) - \{\mathfrak {m}_C\}$, which are dense in ${\rm Spec}(B)$ and ${\rm Spec}(C)$, respectively. It remains to show that $C$ is reduced. Since ${\rm Spec}(C) \to {\rm Spec}(B)$ is birational, we see that $C$ satisfies $(R_0)$, and satisfies $(S_1)$ everywhere except possibly at $\mathfrak {m}_C$. But (9) implies

\[ {\rm depth}_{\mathfrak{m}_C}(C) \geqslant 2 \geqslant 1 = \min\bigl\{1,\dim(C)\bigr\}. \]

Thus, $C$ satisfies $(S_1)$ at $\mathfrak {m}_C$, and $C$ is therefore reduced.

Proof. Throughout this proof, if $R$ is a local ring, then we denote by $\mathfrak {m}_R$ the maximal ideal of $R$. We note that $A$ is reduced since reducedness lifts from Cartier divisors [Reference Grothendieck and DieudonnéEGAIV₂, Proposition 3.4.6].

Set $\bar {A} := A/tA$ and $\bar {C} := C/tC$. We have

\[ {\rm depth}(\bar{A}) \geqslant \min\bigl\{1,\dim(\bar{A})\bigr\} \geqslant 1 \]

since $A/tA$ satisfies $(S_1)$ [Reference Grothendieck and DieudonnéEGAIV₂, Proposition 5.8.5]. Thus, we have ${\rm depth}(A) \geqslant 2$ by [Reference Bruns and HerzogBH98, Proposition 1.2.10(d)], and hence ${\rm Spec}(C) \to {\rm Spec}(B)$ is a partial weak normalization by Lemma 4.8. Since $g \circ h$ is an integral universal homeomorphism by base change, to show that $g \circ h$ is a partial weak normalization, it suffices to show that $g \circ h$ is birational and that $\bar {C}$ is reduced.

We start by showing that $g \circ h$ is birational. Since $g \circ h$ is a universal homeomorphism and $\bar {A}$ is reduced, to show that $g \circ h$ is birational, it suffices to show that $\bar {A} \to \bar {C}$ localizes to an isomorphism $\bar {A}_{\bar {\mathfrak {p}}} \overset {\sim }{\to } \bar {C}_{\bar {\mathfrak {p}}}$ at every minimal prime $\bar {\mathfrak {p}} \subseteq \bar {A}$. Let $\mathfrak {p} \subseteq A$ be the prime in $A$ corresponding to such a minimal prime $\bar {\mathfrak {p}}$. Since regularity lifts from Cartier divisors [Reference Grothendieck and DieudonnéEGAIV₁, Chapitre 0, Corollaire 17.1.8] and $\bar {A}$ is reduced, we see that $A_\mathfrak {p}$ is regular, in which case the homomorphism $A \to C$ induces an isomorphism $A_\mathfrak {p} \overset {\sim }{\to } C_\mathfrak {p}$ by the fact that regular rings are weakly normal; see (6). Thus, $g \circ h$ is birational.

We now show that $\bar {C}$ is reduced. By our assumption that ${\rm Spec}(B) \to {\rm Spec}(A)$ is an isomorphism along $V(t) - \{\mathfrak {m}_A\}$, we know that ${\rm Spec}(C) \to {\rm Spec}(B) \to {\rm Spec}(A)$ is an isomorphism along $V(t) - \{\mathfrak {m}_A\}$ by [Reference KollárKol17, Definition 1, $(4')$]. Since $\bar {A}$ is reduced and $g \circ h$ is birational, we see that $\bar {A} \to \bar {C}$ is injective by [Reference Grothendieck and DieudonnéEGAI_new, Corollaire 1.2.6]. Thus, the homomorphism $\bar {A} \to \bar {C}$ restricts to an isomorphism

\[ \tilde{\bar{A}}\rvert_{\bar{U}} \overset{\sim}{\longrightarrow} \tilde{\bar{C}}\rvert_{\bar{U}} \]

over $\bar {U} := {\rm Spec}(\bar {A}) - \{\mathfrak {m}_{\bar {A}}\}$. Since $\bar {A}$ is reduced, this shows that $\bar {C}$ satisfies $(R_0)$, and satisfies $(S_1)$ everywhere except possibly at $\mathfrak {m}_{\bar {C}}$. It suffices to show ${\rm depth}(\bar {C}) \geqslant 1$, since this would imply

\[ {\rm depth}(\bar{C}) \geqslant \min\bigl\{1,\dim(\bar{C})\bigr\} = 1 \]

by the assumption that $\dim (\bar {A}) = \dim (\bar {C}) \geqslant 1$. Note that ${\rm depth}(C) \geqslant 2$ by Lemma 4.8, and hence it suffices to show that $t$ is a nonzerodivisor on $C$ by [Reference Bruns and HerzogBH98, Proposition 1.2.10(d)]. Consider the commutative diagram

where the isomorphism of total rings of fractions on the bottom holds by the birationality of $g \circ h$ and the two vertical homomorphisms are injective since $A$ and $C$ are reduced. Since $A \to {\rm Frac}(A)$ is flat, it maps $t$ to a nonzerodivisor on ${\rm Frac}(A)$. By the commutativity of the diagram, $t$ is therefore a nonzerodivisor on $C$.

Proof. We want to show that, for every module-finite $A$-algebra $B$ such that ${\rm Spec}(B) \to {\rm Spec}(A)$ is a partial weak normalization, the homomorphism $A \to B$ is an isomorphism. We will show that $A_\mathfrak {p} \to B_\mathfrak {p}$ is an isomorphism for every prime ideal $\mathfrak {p} \subseteq A$ containing $t$, which would suffice since $\mathfrak {m}$ is a prime ideal containing $t$.