Grothendieck's conjecture on resolution of singularities for reduced quasi-excellent noetherian schemes (EGA IV₂, Remarque 7.9.6)

Grothendieck's Éléments de géométrie algébrique IV (Seconde partie, 1965) introduced excellent and quasi-excellent rings as the expected maximal class of noetherian schemes for which resolution of singularities — a proper birational morphism from a regular scheme — should be possible. Proposition 7.9.5 proves the necessity direction: if every integral scheme finite over a locally noetherian scheme $X$ can be resolved, then the affine rings of $X$ satisfy conditions (ii) and (iii) of Définition 7.8.2, i.e. $X$ is quasi-excellent. Remarque 7.9.6 conjectures the converse: every reduced noetherian scheme whose affine rings have geometrically regular formal fibres and a Nagata-type regular-locus property admits a resolution of singularities, and records a reduction to complete noetherian local integral domains. The conjecture is known when all residue fields have characteristic zero, in every dimension (Hironaka, extended by Temkin to all quasi-excellent $\mathbb{Q}$-schemes), and in every characteristic in dimension at most three (Lipman; Cossart–Piltant). The surviving open core is dimension at least four in positive or mixed residue characteristic.

Background

Resolution of singularities asks to replace a singular scheme by a nonsingular one in a controlled way. A scheme is regular when all of its local rings are regular local rings, meaning that at each point the maximal ideal can be generated by as many elements as the Krull dimension of the local ring (the algebraic counterpart of a smooth point). A morphism $f : X' \to X$ is proper when it is separated, of finite type and universally closed, the algebro-geometric analogue of a map with compact fibres, and birational when it restricts to an isomorphism between dense open subsets, equivalently between the fields of rational functions at corresponding generic points. A resolution of $X$ is a proper birational morphism $X' \to X$ with $X'$ regular; Éléments de géométrie algébrique (EGA) IV, Seconde partie, 7.9.1 calls such a morphism a morphisme résolvant and says that one can resolve $X$. Resolutions matter because much of the cohomological machinery for singular spaces — duality, vanishing, trace formulae — is first built for regular schemes and transported along a resolution; EGA IV₂, Remarque 7.9.6 itself motivates the class of excellent rings by the growing topological and homological importance of resolution.

Hironaka (1964) proved that every algebraic variety over a field of characteristic zero admits a resolution; in positive characteristic this is known only up to dimension three and is one of the central open problems of algebraic geometry. Grothendieck's EGA IV (1964–1967) enlarged the setting from varieties — schemes of finite type over a field — to arbitrary noetherian schemes (noetherian: every descending chain of closed subschemes stabilizes, the finiteness condition under which the theory operates) and asked exactly which schemes can be resolved. The expected answer is: the quasi-excellent ones. EGA IV₂, Définition 7.8.2 defines a noetherian ring $A$ to be excellent when it satisfies three conditions; rings satisfying only the second and third are the quasi-excellent class of the conjecture:

(i) $A$ is universally catenary: every finitely generated $A$-algebra has the property that any two saturated chains of prime ideals between the same two primes have equal length, so that dimension theory is homogeneous. For local rings this is now known to be equivalent to the formally catenary condition of EGA — that the completions of all integral quotients be equidimensional — by Ratliff's theorem.

(ii) For every prime ideal $\mathfrak{p}$ of $A$, the formal fibres of the localization $A_{\mathfrak{p}}$ are geometrically regular: writing $\hat{A}_{\mathfrak{p}}$ for the completion, i.e. the inverse limit of $A_{\mathfrak{p}}$ modulo the powers of its maximal ideal (the ring of formal infinitesimal neighbourhoods of the point), the fibres of $\operatorname{Spec}(\hat{A}_{\mathfrak{p}}) \to \operatorname{Spec}(A_{\mathfrak{p}})$ are regular and remain regular after any finite extension of the residue field; rings with this property are called G-rings. The condition says that formal neighbourhoods see the same geometry as the scheme itself.

(iii) A regular-locus condition of Nagata type: for every integral quotient $B$ of $A$ and every finite radicial extension $K'/K$ of the fraction field $K$ of $B$ (radicial meaning that the finite extension induces purely inseparable field extensions), there exists a finite $B$-subalgebra $B'$ of $K'$ with fraction field $K'$ such that the regular points of $\operatorname{Spec}(B')$ contain a nonempty open subset.

The class is large and stable: it contains all schemes of finite type over a field or over the integers, all complete noetherian local rings (EGA IV₂, 7.8.3(iii)) and all Dedekind domains of characteristic zero (a Dedekind domain being a noetherian, integrally closed ring of Krull dimension one, such as the integers), and it passes to localizations and to finitely generated algebras.

EGA IV₂ proves the necessity direction, Proposition 7.9.5: if every integral scheme finite over a locally noetherian scheme $X$ can be resolved, then the affine rings of $X$ satisfy conditions (ii) and (iii). Grothendieck then conjectures the converse in Remarque 7.9.6, recording that the general problem reduces to $X = \operatorname{Spec}(A)$ with $A$ a complete noetherian local integral domain, and that even a negative answer could only come from restrictive conditions imposed on complete local rings. At the time, the known cases were characteristic zero (Hironaka) and two-dimensional algebraic surfaces over perfect fields. The present state of knowledge is: characteristic zero in all dimensions — Hironaka for varieties, extended by Temkin to all noetherian quasi-excellent $\mathbb{Q}$-schemes; dimension two in any characteristic — Lipman resolved every two-dimensional excellent scheme; and dimension three in arbitrary characteristic, including mixed characteristic (some residue fields of characteristic zero and others of positive characteristic, as for schemes of finite type over the integers), by Cossart–Piltant's resolution of arithmetical threefolds (three-dimensional schemes over excellent Dedekind domains such as the integers). Two weaker substitutes hold with no dimension or characteristic restriction: de Jong's alterations, which produce a proper, surjective, generically finite (that is, finite over a dense open subset) map from a regular scheme at the price of a finite extension of the function field, and Gabber's weak local uniformization theorem (a finite-cover substitute for resolution, proved for arbitrary quasi-excellent noetherian schemes), published in the Astérisque volume edited by Illusie, Laszlo and Orgogozo. Neither produces the birational regular cover that the conjecture demands.

Problem Statement

Conjecture (EGA IV₂, Remarque 7.9.6, the converse of Proposition 7.9.5). Let $X$ be a reduced locally noetherian scheme such that, for every affine open subset $U$ of $X$, the ring $\mathcal{O}_X(U)$ satisfies conditions (ii) and (iii) of EGA IV₂, Définition 7.8.2: (ii) for every prime ideal $\mathfrak{p}$, the formal fibres of the localization $\mathcal{O}_X(U)_{\mathfrak{p}}$ are geometrically regular; (iii) for every integral quotient $B$ of $\mathcal{O}_X(U)$ and every finite radicial extension $K'/K$ of the fraction field $K$ of $B$, there exists a finite $B$-subalgebra $B'$ of $K'$ with fraction field $K'$ such that the regular points of $\operatorname{Spec}(B')$ contain a nonempty open subset. (In modern terminology, $X$ is a reduced quasi-excellent scheme.) Prove, or refute, that $X$ can be resolved in the sense of EGA IV₂, 7.9.1: there exists a proper birational morphism $f : X' \to X$ such that $X'$ is regular. EGA IV₂, Remarque 7.9.6 records an equivalent reduction: it suffices to treat $X = \operatorname{Spec}(A)$ with $A$ a complete noetherian local integral domain, and complete noetherian local rings are automatically excellent (EGA IV₂, 7.8.3(iii)). The conjecture is known when all residue fields of $X$ have characteristic zero (any dimension) and in every characteristic when $\dim X \leq 3$; the open core is $\dim X \geq 4$ in positive or mixed residue characteristic (the residue fields are not all of characteristic zero: either of positive characteristic, or of both characteristics at once, as for schemes of finite type over $\mathbb{Z}$), which contains as its finite-type special case the resolution of singularities of algebraic varieties of dimension at least four over fields of positive characteristic.

References

1. A. Grothendieck, Éléments de géométrie algébrique IV. Étude locale des schémas et des morphismes de schémas (seconde partie), Publications Mathématiques de l'Institut des Hautes Études Scientifiques 24 (1965), 5–231, DOI 10.1007/BF02684322, https://doi.org/10.1007/BF02684322
2. H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, I, Annals of Mathematics. Second Series 79 (1964), 109–203, DOI 10.2307/1970486, https://doi.org/10.2307/1970486; and Part II, ibid. 79 (1964), 205–326, DOI 10.2307/1970547, https://doi.org/10.2307/1970547
3. L. J. Ratliff, Jr., Characterizations of catenary rings, American Journal of Mathematics 93 (1971), 1070–1108, DOI 10.2307/2373746, https://doi.org/10.2307/2373746
4. The Stacks Project Authors, The Stacks Project, Chapter 15 (More on Algebra), Section 15.111 (Formally catenary rings), Proposition 15.111.5 (Ratliff): a Noetherian local ring is universally catenary if and only if it is formally catenary, tag 0AW6, https://stacks.math.columbia.edu/tag/0AW6
5. D. Ferrand and M. Raynaud, Fibres formelles d'un anneau local noethérien, Annales scientifiques de l'École Normale Supérieure. Quatrième série 3 (1970), 295–311, DOI 10.24033/asens.1195, https://www.numdam.org/item/ASENS_1970_4_3_3_295_0/
6. T. Kawasaki, On arithmetic Macaulayfication of Noetherian rings, Transactions of the American Mathematical Society 354 (2002), 123–149, DOI 10.1090/s0002-9947-01-02817-3, https://doi.org/10.1090/s0002-9947-01-02817-3
7. J. Lipman, Desingularization of two-dimensional schemes, Annals of Mathematics. Second Series 107 (1978), 151–207, DOI 10.2307/1971141, https://doi.org/10.2307/1971141
8. M. Temkin, Desingularization of quasi-excellent schemes in characteristic zero, Advances in Mathematics 219 (2008), 488–522, DOI 10.1016/j.aim.2008.05.006, arXiv:math/0703678, https://arxiv.org/abs/math/0703678
9. V. Cossart and O. Piltant, Resolution of singularities of arithmetical threefolds, Journal of Algebra 529 (2019), 268–535, DOI 10.1016/j.jalgebra.2019.02.017, https://doi.org/10.1016/j.jalgebra.2019.02.017; preprint version posted as arXiv:1412.0868 under the title 'Resolution of Singularities of Arithmetical Threefolds II', https://arxiv.org/abs/1412.0868
10. A. J. de Jong, Smoothness, semi-stability and alterations, Publications Mathématiques de l'Institut des Hautes Études Scientifiques 83 (1996), 51–93, DOI 10.1007/BF02698644, https://doi.org/10.1007/BF02698644
11. L. Illusie, Y. Laszlo and F. Orgogozo (eds.), Travaux de Gabber sur l'uniformisation locale et la cohomologie étale des schémas quasi-excellents. Séminaire à l'École Polytechnique 2006–2008, Astérisque 363–364, Société Mathématique de France, 2014, DOI 10.24033/ast.935, https://doi.org/10.24033/ast.935
12. T. Murayama, A uniform treatment of Grothendieck's localization problem, Compositio Mathematica 158 (2022), no. 1, 57–88, DOI 10.1112/S0010437X21007715, arXiv:2004.06737, https://arxiv.org/abs/2004.06737
13. T. D. M. Chau, Cohen-Macaulayness of formal fibers and dimension of local cohomology modules, preprint arXiv:2605.08579 (2026), https://arxiv.org/abs/2605.08579
14. Y. Hu, Universal Characteristic-free Resolution of Singularities, I, preprint arXiv:2507.21400 (2025), https://arxiv.org/abs/2507.21400
15. Y. Hu, Introduction to Universal Equations and Characteristic-free Resolution, preprint arXiv:2608.10272 (2026), https://arxiv.org/abs/2608.10272