Ramified coverings of formal schemes beyond the normality, normal-crossings, and tameness hypotheses of Grothendieck–Murre

Grothendieck and Murre (Lecture Notes in Mathematics 208, 1971) defined tamely ramified finite coverings of a locally noetherian normal formal scheme with respect to a finite family of regular divisors with normal crossings, showed that these coverings form a Galois category, and proved a specialization theorem: if the special fibre (reduction modulo an ideal of definition) is normal and the inverse-image divisors are regular with normal crossings, then base change is an equivalence between the tame coverings of the formal scheme and those of its special fibre. This theorem drives their identification of tame fundamental groups of formal neighbourhoods of divisors. The whole development is contingent on its hypotheses — normality of the base and special fibre, regularity and normal crossings of the divisors, and invertibility of ramification indices (tameness) — and the book does not treat the failure of these conditions. The problem recorded here is to construct the corresponding theory of ramified finite coverings of formal schemes (covering class, Galois-category and fundamental-group structure, specialization comparison, formal-neighbourhood fundamental groups) without these hypotheses, allowing non-normal bases, divisors without normal crossings, and wild or bounded ramification in place of tameness, or else to establish the precise obstructions and determine which Grothendieck–Murre statements survive. Later literature settled the scheme-level definitional question by intrinsic notions of tameness (Wiesend; Kerz–Schmidt; Hübner–Schmidt) and built partial wild-ramification frameworks (Kumar's formal orbifolds; the abelian theory with modulus of Kerz–Saito), while work at the 2026 frontier still imposes normal-crossing-type hypotheses on formal or rigid models; the formal-scheme comparison theorem beyond the Grothendieck–Murre hypotheses remains open.

Background

In algebraic geometry, a finite étale morphism is the analogue of a finite unramified covering, and Grothendieck's theory (developed in the SGA 1 seminar) shows that the finite étale covers of a connected scheme form a Galois category — a category equivalent to finite sets equipped with a continuous action of a profinite group, the étale fundamental group $\pi_1$ . To allow controlled ramification along a specified locus one passes to tamely ramified covers: an extension of discrete valuation rings $B/A$ is tamely ramified if its ramification index is prime to the residue characteristic and the residue field extension is separable, and a finite morphism $Z\to X$ of normal schemes is tamely ramified along a divisor $D$ (a divisor being a finite formal integer combination of codimension-one subvarieties) if it is étale over $X\setminus D$ and tamely ramified at the codimension-one points of $D$ . The corresponding tame fundamental group $\pi_1^t$ is a quotient of $\pi_1$ . Two local tools underpin this theory: Kummer coverings, which are covers locally of the form $A\hookrightarrow A[T]/(T^n-a)$ with $n$ invertible in $A$ (adjoining an $n$ -th root), and Abhyankar's lemma, which asserts that a tame cover is locally dominated by such radical covers. A divisor with (strict) normal crossings on a regular scheme is one that, étale-locally (respectively locally), is cut out by part of a regular system of parameters (in a regular local ring, a minimal system of generators of the maximal ideal); this is precisely the condition under which Kummer coverings give a sufficient local model for tame covers at all points simultaneously. A formal scheme is a ringed space locally isomorphic to $\operatorname{Spf} A$ , the formal spectrum of a ring $A$ complete and separated for the $I$ -adic topology of an ideal $I$ ; an ideal of definition $\mathcal{J}$ plays the role of $I$ on every affine piece, and the special fibre is the ordinary scheme $X_0=(\mathfrak{X},\mathcal{O}_{\mathfrak{X}}/\mathcal{J})$ . Formal schemes arise as formal neighbourhoods: the completion of a scheme $X$ along a closed subscheme $Y$ records the geometry of $X$ to all infinitesimal orders along $Y$ . Because one cannot generally invert elements in a complete ring, the étale theory of formal schemes does not reduce to the schematic theory. In this setting Grothendieck and Murre developed the theory of tamely ramified coverings of formal schemes in order to compute tame fundamental groups of punctured formal neighbourhoods of divisors and, in their final chapter, of punctured spectra of two-dimensional complete local rings. Their covering theory assumes: the formal scheme $\mathfrak{X}$ is locally noetherian and normal (all stalks are normal domains); the divisors $D_i$ form a finite family of regular effective divisors ('effective' meaning that the coefficients are non-negative) with normal crossings, with $D=\sum_i D_i$ ; and, for the specialization theorem (Theorem 4.3.2), the ideal of definition $\mathcal{J}$ is such that the special fibre $X_0$ is normal and the inverse images $D_{i,0}$ exist, are regular, and have normal crossings. The conclusion is that base change induces an equivalence of categories $\operatorname{Rev}^{D}(\mathfrak{X})\to\operatorname{Rev}^{D_0}(X_0)$ between the finite covers of the formal scheme tamely ramified along $D$ and the finite covers of the special fibre tamely ramified along $D_0$ . Combined with the earlier chapters this identifies the tame fundamental group of a punctured formal neighbourhood with a quotient of the tame fundamental group of the complement of the divisor. Every hypothesis is used: normality enters through normalization arguments and the Galois-category formalism, normal crossings is what makes Kummer coverings a sufficient local model, and invertibility of indices is the definition of tameness — the wild regime is expressly out of scope. The book does not treat what happens when these hypotheses fail, and subsequent applications have had to remain within them or route around them; formulating and removing this restriction is the problem recorded here.

Problem Statement

Construct the theory of ramified finite coverings of formal schemes without the standing Grothendieck–Murre hypotheses, or establish precisely where such a theory cannot exist. For locally noetherian formal schemes $\mathfrak{X}$ , the components of the sought theory are: (1) a class of finite coverings playing the role of 'tamely ramified along a divisor' when $\mathfrak{X}$ or its special fibre $X_0$ is non-normal, or when the divisors $D_i$ or their inverse images on $X_0$ fail to be regular with normal crossings, together with the local structure theory replacing the Kummer-covering and Abhyankar-lemma description; (2) the Galois-category structure of this class of coverings, the associated fundamental group, and its identification for formal neighbourhoods of divisors, generalizing the Grothendieck–Murre identifications; (3) a specialization theorem generalizing Grothendieck–Murre's Theorem 4.3.2: an equivalence between the category of such coverings of $\mathfrak{X}$ and the corresponding category for the special fibre modulo an ideal of definition, valid under the weakened hypotheses; and (4) the non-invertible (wild) regime: the analogous theory for coverings whose ramification is bounded by appropriate data, so that tameness becomes the special case of a trivial bound. Where an extension is impossible, the required answer is instead the precise obstruction: explicit examples of formal schemes, ideals of definition, divisors, and coverings for which the naive analogue of a Grothendieck–Murre statement fails, and a characterization of which statements survive the removal of each hypothesis. All these components are tied by one scientific objective: a covering theory of formal schemes with specialization comparison and fundamental-group identification that recovers Grothendieck–Murre's Theorem 4.3.2 as the normal, normal-crossings, tame special case. Components (1)–(4) are interdependent: they constitute one theory and are posed jointly, and a component that admits no construction is replaced by the corresponding explicit obstruction; in the negative direction the characterization must cover every hypothesis removed.

References

A. Grothendieck and J. P. Murre, The Tame Fundamental Group of a Formal Neighbourhood of a Divisor with Normal Crossings on a Scheme, Lecture Notes in Mathematics 208, Springer-Verlag, Berlin, 1971; chapter 'Tamely ramified coverings of formal schemes', pp. 52–68. Book DOI 10.1007/BFb0069608; chapter DOI 10.1007/BFb0069612, https://doi.org/10.1007/BFb0069612
A. Grothendieck and M. Raynaud, Revêtements Étales et Groupe Fondamental (SGA 1, Séminaire de Géométrie Algébrique du Bois Marie 1960–61), Lecture Notes in Mathematics 224, Springer, 1971. DOI 10.1007/BFb0058656, https://doi.org/10.1007/BFb0058656
G. Wiesend, Tamely ramified covers of varieties and arithmetic schemes, Forum Mathematicum 20 (2008), no. 3, 515–522. DOI 10.1515/forum.2008.026, https://doi.org/10.1515/forum.2008.026
M. Kerz and A. Schmidt, On different notions of tameness in arithmetic geometry, Mathematische Annalen 346 (2010), no. 3, 641–668. DOI 10.1007/s00208-009-0409-6, https://doi.org/10.1007/s00208-009-0409-6 (arXiv:0807.0979)
A. Schmidt, Tame coverings of arithmetic schemes, Mathematische Annalen 322 (2002), no. 1, 1–18. DOI 10.1007/s002080100262, https://doi.org/10.1007/s002080100262
K. Hübner and A. Schmidt, The tame site of a scheme, Inventiones mathematicae 223 (2021), no. 2, 379–443. DOI 10.1007/s00222-020-00993-4, https://doi.org/10.1007/s00222-020-00993-4 (arXiv:2004.05468)
H. Esnault and L. Kindler, Lefschetz theorems for tamely ramified coverings, Proceedings of the American Mathematical Society 144 (2016), no. 12, 5071–5080. DOI 10.1090/proc/13151, https://doi.org/10.1090/proc/13151 (arXiv:1509.08160)
V. Drinfeld, On a Conjecture of Deligne, Moscow Mathematical Journal 12 (2012), no. 3, 515–542. DOI 10.17323/1609-4514-2012-12-3-515-542, https://doi.org/10.17323/1609-4514-2012-12-3-515-542
M. Kerz and S. Saito, Lefschetz theorem for abelian fundamental group with modulus, Algebra & Number Theory 8 (2014), no. 3, 689–701. DOI 10.2140/ant.2014.8.689, https://doi.org/10.2140/ant.2014.8.689
M. Kumar, Ramification theory and formal orbifolds in arbitrary dimension, Proceedings — Mathematical Sciences 129 (2019), no. 3, Paper No. 38. DOI 10.1007/s12044-019-0493-9, https://doi.org/10.1007/s12044-019-0493-9 (arXiv:1604.05531)
H. Esnault, M. Shusterman, and V. Srinivas, Finite presentation of the tame fundamental group, Selecta Mathematica 28 (2022), no. 2, article 37. DOI 10.1007/s00029-021-00732-4, https://doi.org/10.1007/s00029-021-00732-4 (arXiv:2102.13424)
P. Achinger, K. Hübner, M. Lara, and J. Stix, Tame fundamental groups of rigid spaces, arXiv:2608.12601 (2026). DOI 10.48550/arXiv.2608.12601, https://arxiv.org/abs/2608.12601