Geometrically connected formal fibres and descent of finite étale coverings for general noetherian henselian local rings (EGA IV, Remarques 18.9.6 (i)–(ii))

The final part of Grothendieck–Dieudonné's Éléments de géométrie algébrique IV (EGA IV_4, 1967) proves in §18.9 that for a noetherian henselian local ring $A$ whose formal fibres are geometrically normal — in particular for $A$ excellent — the $\mathfrak m$-adic completion $\hat A$ controls the connected components and the étale coverings of every $A$-scheme, and Remarques 18.9.6 of that section record three questions left open outside this hypothesis. Two of the three are settled in the later literature: the strict henselization of an excellent local ring is excellent (Greco 1976, resolving 18.9.6 (iii)), and the base-change functor on finite étale coverings of an arbitrary $A$-scheme is an equivalence of categories after completion when $A$ is excellent (Artin 1969 in residue characteristic 0; Popescu's general Néron desingularization removes the characteristic restriction). Of question 18.9.6 (ii) — whether the formal fibres of an arbitrary noetherian henselian local ring are geometrically connected — the sufficient form Grothendieck recorded, irreducibility of $\mathrm{Spec}(\hat A)$ for every noetherian henselian local domain, is refuted by Heitmann's completion constructions (1993) as exhibited in the Stacks Project: noetherian henselian local normal domains with reducible completions exist, so the question must be attacked directly. This problem records the surviving open core: (a) 18.9.6 (ii) — whether the formal fibres of an arbitrary noetherian henselian local ring are geometrically connected, now with the irreducibility route known to fail; and (b) the non-excellent part of 18.9.6 (i) — whether, with no excellence-type hypothesis, the functor $Z \mapsto Z \otimes_A \hat A$ on finite étale coverings is an equivalence. The two questions are the two halves of one comparison: what the $\mathfrak m$-adic completion of a noetherian henselian local ring preserves when the geometric normality (excellence) hypothesis is dropped.

Background

EGA IV (Éléments de géométrie algébrique IV), published 1964–1967, develops the local theory of schemes and morphisms of schemes. Its final chapter, Chapter 18 of the fourth part (Compléments sur les morphismes étales, complements on étale morphisms), derives the theory of henselian rings from the theory of étale morphisms: a morphism of schemes is étale when it is locally of finite presentation, flat, and unramified — the algebro-geometric analogue of a local isomorphism in complex analysis. Section 18.9 (Fibres formelles des anneaux noethériens henséliens, formal fibres of noetherian henselian rings) compares a noetherian henselian local ring with its completion, and Remarques 18.9.6 of that section record the questions posed here.

A local ring $(A, \mathfrak m)$ with residue field $k = A/\mathfrak m$ is henselian if it satisfies Hensel's lemma: every factorization of a monic polynomial over $A$ into two coprime monic polynomials modulo $\mathfrak m$ lifts to such a factorization over $A$. Noetherian (a noetherian ring is one in which every ideal is finitely generated) henselian local rings are the local rings for the étale topology: a finite étale $A$-algebra — a finite covering without ramification — over such an $A$ is a product of henselian local rings, and the category of finite étale $A$-algebras is equivalent, via $B \mapsto B \otimes_A k$, to the category of finite products of finite separable field extensions of $k$. The $\mathfrak m$-adic completion $\hat A = \varprojlim_n A/\mathfrak m^n$ is a complete noetherian local ring with the same residue field, and the induced morphism $\mathrm{Spec}(\hat A) \to \mathrm{Spec}(A)$ is faithfully flat. For a prime ideal $\mathfrak p \subset A$ with residue field $\kappa(\mathfrak p)$, the formal fibre of $A$ at $\mathfrak p$ is the fibre $\mathrm{Spec}(\hat A \otimes_A \kappa(\mathfrak p))$ of this morphism. A scheme over a field $F$ is geometrically connected (resp. geometrically irreducible, geometrically normal, geometrically regular) if it stays connected (resp. irreducible, normal, regular) after base change to any field extension of $F$, equivalently to an algebraic closure of $F$; a noetherian ring all of whose formal fibres are geometrically regular is called a G-ring, and a G-ring satisfying the standard regularity-ascending and catenarity conditions is called excellent — every localization of a scheme of finite type over a field or over $\mathbb Z$ is excellent, which is why excellence is the customary blanket hypothesis in local algebraic geometry.

EGA IV §18.9 proves, for $A$ noetherian henselian local with geometrically normal formal fibres (a hypothesis satisfied by every excellent local ring), that the formal fibres are geometrically connected and even geometrically integral (18.9.1), that $\hat A$ is a domain whenever $A$ is (18.9.2), that the fraction field of $\hat A$ is a separable extension of that of $A$ (18.9.3), that connected components correspond under $X \mapsto X \otimes_A \hat A$ for any $A$-scheme $X$ (18.9.4), and that the induced base-change functor on étale $X$-schemes is fully faithful (18.9.5). Remarques 18.9.6 then ask, outside this hypothesis: (i) whether the functor of 18.9.5, restricted to the categories of étale coverings (revêtements étales, i.e. finite étale coverings) of $X$ and of $X' = X \otimes_A \hat A$, is an equivalence when the residue field of $A$ has characteristic 0 — proven there for $A$ excellent, credited to M. Artin via Hironaka's resolution of singularities — with a variant for principal Galois coverings of order prime to the residue characteristic deemed plausible without restriction on the residue characteristic; (ii) whether the formal fibres of an arbitrary noetherian henselian local ring are geometrically connected (voire géométriquement irréductibles, as EGA puts it), for which it would suffice that $\mathrm{Spec}(\hat A)$ be irreducible for every noetherian henselian local domain $A$; and (iii) whether the strict henselization of an excellent local ring is excellent. Here the strict henselization $A^{sh}$ is the essentially unique strictly henselian local ring (henselian with separably closed residue field) obtained as a filtered colimit of étale local $A$-algebras; it plays the role of the local ring of a geometric point in the étale topology.

Later literature settled (iii) and the excellent case of (i). Greco (1976) proved that excellence ascends along absolutely flat (von Neumann regular) ring maps, so in particular the strict henselization of an excellent local ring is excellent (Corollary 5.6); the Stacks Project (Lemma 15.52.8) gives the modern general form: the henselization and strict henselization of a noetherian local ring whose formal fibres have a property $P$ — for $P$ among geometric normality, reducedness, regularity — again have that property. For (i), Artin's approximation theorem (1969, in residue characteristic 0, via resolution of singularities) and Popescu's general Néron desingularization (1985, 1986) together give the equivalence for every excellent henselian local ring in every residue characteristic: Popescu's theorem implies that such a ring has the Artin approximation property, meaning that solutions in $\hat A$ of finite systems of polynomial equations over $A$ can be approximated modulo powers of $\mathfrak m$ by solutions in $A$. The approximation property is known to fail for some non-excellent noetherian local rings — Rotthaus (1977) constructed noetherian universally Japanese (a finiteness condition on integral closures, weaker than excellence) rings that are not excellent, and Brown (1983) connects the failure of Artin approximation to properties of formal fibres, with henselian rings among its topics — so no generic approximation route is available in the general henselian case. Greco and Sankaran (1976), and Di Franco (1986) in greater generality, showed that a henselian pair $(A, I)$ — a ring $A$ with an ideal $I$ contained in its Jacobson radical such that Hensel's lemma holds modulo $I$ — is separably closed in its $I$-adic completion; this is a field-theoretic shadow of the EGA questions, but it does not decide connectedness or irreducibility of the formal fibres. A parallel literature constructs noetherian local rings with prescribed pathological completions: Heitmann (1993) proved that a complete local ring is the completion of a noetherian local unique factorization domain exactly when it is a field, a discrete valuation ring, or has depth at least two (depth: the maximal length of a regular sequence in the maximal ideal) with no element of its prime ring a zero divisor, and the Stacks Project (Examples, Section 110.20) applies this to exhibit a noetherian henselian local normal domain whose completion is $\mathbf{C}[[x,y,z,w]]/(wx,wy)$, reducible — so EGA's proposed sufficient route to question (ii) through irreducibility of completions is closed, and the Stacks Project (More on Algebra, Section 15.110) records that $A^h \to \hat A$ need not induce a bijection on minimal primes. Ferrand–Raynaud (1970) recall Nagata's example of a normal noetherian local domain of dimension 2 whose generic formal fibre is disconnected — a ring that is not henselian — and prove a clopen correspondence for henselian pairs (their Corollary 4.4) that concerns punctured spectra rather than formal fibres; Nishimura (2012) constructs further bad local rings, including a three-dimensional henselian universally Japanese normal local domain. None of this decides the formal fibres of a henselian ring themselves, and what remains open — the subject of this problem — is the regime with no excellence-type hypothesis at all.

Problem Statement

Let $(A, \mathfrak m)$ be a noetherian henselian local ring and let $\hat A = \varprojlim_n A/\mathfrak m^n$ be its $\mathfrak m$-adic completion. EGA IV_4, Remarques 18.9.6, raises three questions; part (iii) is resolved, the excellent case of part (i) is resolved by later literature, and the sufficient form of part (ii) is refuted, leaving the following open core, stated as in the source.

(a) Are the formal fibres of $A$ — the schemes $\mathrm{Spec}(\hat A \otimes_A \kappa(\mathfrak p))$ for $\mathfrak p$ ranging over the prime ideals of $A$, where $\kappa(\mathfrak p)$ is the residue field of $\mathfrak p$ — geometrically connected over $\kappa(\mathfrak p)$, for an arbitrary noetherian henselian local ring $A$? The stronger form EGA also poses — whether the formal fibres are geometrically irreducible — is a strictly stronger claim. EGA's proposed sufficient condition — that $\mathrm{Spec}(\hat A)$ be irreducible for every noetherian henselian local domain $A$ — is false: by Heitmann's completion theorem as applied in the Stacks Project (Examples, Section 110.20), there is a noetherian henselian local normal domain whose completion is $\mathbf{C}[[x,y,z,w]]/(wx,wy)$, the union of a regular 2-dimensional and a regular 3-dimensional component. So the question must be settled directly: no route through irreducibility of completions is available for a positive answer, and it is not known, in either direction, whether the formal fibres of the known pathological examples are geometrically connected. (EGA 18.9.1 settles all of this when the formal fibres are geometrically normal, in particular when $A$ is excellent; the question is the case with no such hypothesis. For context on the boundary: Nagata constructed — as recalled by Ferrand–Raynaud — a normal noetherian local domain of dimension 2 whose generic formal fibre is disconnected, but that ring is not henselian.)

(b) For an arbitrary $A$-scheme $X$, with $X' = X \otimes_A \hat A$, is the base-change functor

$$Z \mapsto Z \otimes_A \hat A$$

from the category of finite étale coverings (revêtements étales) of $X$ to the category of finite étale coverings of $X'$ an equivalence of categories? EGA proves the functor fully faithful (on all étale $X$-schemes) when the formal fibres of $A$ are geometrically normal, and proves the equivalence for $A$ excellent — in residue characteristic 0 by Artin's approximation theorem, in all residue characteristics by Popescu's general Néron desingularization; the question is the case of a noetherian henselian local $A$ with no excellence-type hypothesis, where even full faithfulness is unknown. EGA also flags the variant in which only principal Galois coverings of $X$ — finite étale Galois coverings whose finite automorphism group acts simply transitively on geometric points — of order prime to the residue characteristic are considered, which it deemed plausible without restriction on the residue characteristic. Full faithfulness of the base-change functor is a strictly weaker property than its being an equivalence of categories.

Parts (a) and (b) are the two halves of one comparison problem: what the $\mathfrak m$-adic completion of a noetherian henselian local ring preserves — the connected geometry and the finite étale covering theory of every $A$-scheme — once geometric normality of the formal fibres is no longer assumed. They are not independent: (b) is the stronger statement and implies (a), because if a formal fibre at a prime $\mathfrak p$ were not geometrically connected, then for $X = \mathrm{Spec}(\kappa(\mathfrak p)^{\mathrm{alg}})$ the scheme $X' = X \otimes_A \hat A$ would be that disconnected geometric formal fibre, and one of its components would be a finite étale cover of $X'$ that is not the pullback of any finite étale cover of $X$; hence a proof of (b) settles (a), and any counterexample to (a) is already a counterexample to (b). EGA's own chain 18.9.1 → 18.9.4 → 18.9.5 shows that geometric connectedness of the formal fibres is the mechanism behind both the connected-component correspondence and the full faithfulness, which is what links the two questions into one objective.

References

1. A. Grothendieck and J. A. Dieudonné, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. Quatrième partie, Publications Mathématiques de l'IHÉS 32 (1967), pp. 5–361, DOI 10.1007/BF02732123, https://doi.org/10.1007/BF02732123 (primary source; contains §18.9, including Remarques 18.9.6)
2. S. Greco, Two theorems on excellent rings, Nagoya Mathematical Journal 60 (1976), pp. 139–149, DOI 10.1017/S0027763000017190, https://doi.org/10.1017/S0027763000017190
3. M. F. Artin, Algebraic approximation of structures over complete local rings, Publications Mathématiques de l'IHÉS 36 (1969), pp. 23–58, DOI 10.1007/BF02684596, https://doi.org/10.1007/bf02684596
4. D. Popescu, General Néron desingularization, Nagoya Mathematical Journal 100 (1985), pp. 97–126, DOI 10.1017/S0027763000000246, https://doi.org/10.1017/S0027763000000246
5. D. Popescu, General Néron desingularization and approximation, Nagoya Mathematical Journal 104 (1986), pp. 85–115, DOI 10.1017/S0027763000022698, https://doi.org/10.1017/S0027763000022698
6. D. Popescu, Artin approximation, in Handbook of Algebra, vol. 2, North-Holland/Elsevier, 2000, pp. 321–356, DOI 10.1016/S1570-7954(00)80031-5, https://doi.org/10.1016/S1570-7954(00)80031-5 (an excellent henselian local ring has the Artin approximation property)
7. S. Greco and N. Sankaran, On the separable and algebraic closedness of a Hensel couple in its completion, Journal of Algebra 39 (1976), no. 2, pp. 335–348, DOI 10.1016/0021-8693(76)90042-9, https://doi.org/10.1016/0021-8693(76)90042-9
8. F. Di Franco, A note on the separable closedness of a Hensel pair, Archiv der Mathematik 46 (1986), no. 1, pp. 44–45, DOI 10.1007/BF01197138, https://doi.org/10.1007/BF01197138
9. M. L. Brown, Artin approximation and formal fibres of local rings, Mathematische Zeitschrift 183 (1983), pp. 221–228, DOI 10.1007/BF01214822, https://doi.org/10.1007/BF01214822
10. C. Rotthaus, Nicht ausgezeichnete, universell japanische Ringe, Mathematische Zeitschrift 152 (1977), pp. 107–125, DOI 10.1007/BF01214184, https://doi.org/10.1007/BF01214184
11. The Stacks Project, Section 15.52: Properties of formal fibres (tag 0BIR), https://stacks.math.columbia.edu/tag/0BIR (Lemma 15.52.8: the henselization and strict henselization of a noetherian local P-ring, for P among geometrically normal, reduced, regular, are P-rings)
12. The Stacks Project, Lemma 15.13.2 (tag 09ZL), https://stacks.math.columbia.edu/tag/09ZL (for a henselian pair (A, I) the functor B -> B/IB is an equivalence from finite étale A-algebras to finite étale A/I-algebras)
13. R. C. Heitmann, Characterization of completions of unique factorization domains, Transactions of the American Mathematical Society 337 (1993), no. 1, pp. 379–387, DOI 10.1090/s0002-9947-1993-1102888-9, https://doi.org/10.1090/s0002-9947-1993-1102888-9 (a complete local ring is the completion of a unique factorization domain if and only if it is a field, a discrete valuation ring, or has depth at least two and no element of its prime ring is a zero divisor)
14. The Stacks Project, Examples, Section 110.20: Existence of bad local Noetherian rings (tag 0AL7), https://stacks.math.columbia.edu/tag/0AL7 (applies Heitmann's theorem to obtain a noetherian local unique factorization domain $R$ with completion $\mathbf{C}[[x,y,z,w]]/(wx,wy)$, whose henselization is a normal domain by More on Algebra, Lemma 15.46.6; the spectrum of the completion is the union of a regular 2-dimensional and a regular 3-dimensional component)
15. The Stacks Project, More on Algebra, Section 15.110: Branches of the completion (tag 0C27), https://stacks.math.columbia.edu/tag/0C27 (records that $A^h \to \hat A$ need not induce a bijection on minimal primes, referring to Examples, Section 110.20; Lemma 15.110.1: minimal primes of $\hat A$ surject onto minimal primes of $A^h$)
16. The Stacks Project, More on Algebra, Lemma 15.46.6 (tag 06DI), https://stacks.math.columbia.edu/tag/06DI (for a local ring $R$, being a normal domain is equivalent to its henselization $R^h$ being a normal domain and to its strict henselization $R^{sh}$ being a normal domain)
17. D. Ferrand and M. Raynaud, Fibres formelles d'un anneau local noethérien, Annales scientifiques de l'École Normale Supérieure (4) 3 (1970), no. 3, pp. 295–311, DOI 10.24033/asens.1195, https://doi.org/10.24033/asens.1195 (Corollary 4.4 gives the correspondence of clopen subsets of the punctured spectra for henselian pairs, with no hypothesis on formal fibres; the remark following it recalls Nagata's example of a normal noetherian local domain of dimension 2 whose generic formal fibre is not connected)
18. J. Nishimura, A few examples of local rings, I, Kyoto Journal of Mathematics 52 (2012), no. 1, pp. 51–87, DOI 10.1215/21562261-1503754, https://doi.org/10.1215/21562261-1503754 (the paper's fifteen examples of bad local rings include a three-dimensional henselian Nagata (universally Japanese) normal local domain that is not catenary, and the Stacks Project cites this circle for a two-dimensional Nagata local normal domain whose completion is $\mathbf{C}[[x,y,z]]/(yz)$)