1 Introduction
1.1 Background
Let
$R \to S$
be a cyclically pure map of rings, which we recall means that
$IS \cap R = I$
for every ideal
$I \subseteq R$
[Reference HochsterHoc77, p. 463]. Cyclically pure maps seem to have been first studied by Besserre [Reference BesserreBes62, Reference BesserreBes67], who calls them good ring maps. Examples of cyclically pure maps include inclusions of rings of invariants by linearly reductive group actions, split maps, and faithfully flat maps. By [Reference Hochster and RobertsHR74, Reference KempfKem79, Reference Hochster and HunekeHH95, Reference Heitmann and MaHM18], if S is regular, then R is Cohen–Macaulay.
An interesting question arising from [Reference Hochster and RobertsHR74] and these subsequent results is the following:
Question 1.1. If
$R \to S$
is a cyclically pure map of rings, what properties descend from S to R?
Here we say a property
$\mathbf {P}$
descends from S to R if whenever S satisfies
$\mathbf {P}$
, so does R. Note that the preceding example is not precisely a descending property since the property on S is regularity whereas the property on R is Cohen–Macaulayness (and neither regularity nor Cohen–Macaulayness descend along cyclically pure maps in general [Reference Hochster and RobertsHR74, §2]). On the other hand, Noetherianity and normality descend from S to R [Reference Hochster and RobertsHR74, Proposition 6.15].
Question 1.1 has attracted particular attention for different classes of singularities. For example:
-
(I) Boutot showed that if R and S are essentially of finite type over a field of characteristic zero and S has rational singularities, then R has rational singularities [Reference BoutotBou87, Théorème on p. 65]. Using techniques from this paper, the second author showed Boutot’s result holds more generally for Noetherian
$\mathbb {Q}$
-algebras [Reference MurayamaMur25, Theorem C]. For other extensions of Boutot’s result, see [Reference SmithSmi97, Reference SchoutensSch08, Reference Heitmann and MaHM18]. -
(II) Recently, Zhuang showed that if R and S are essentially of finite type over an algebraically closed field of characteristic zero, and S has klt type singularities, then R has klt type singularities [Reference ZhuangZhu24, Theorem 1.1]. In the appendix to [Reference ZhuangZhu24], Lyu extended Zhuang’s result to maps of excellent
$\mathbb {Q}$
-algebras with dualizing complexes [Reference ZhuangZhu24, Theorem A.1]. These results extend results from [Reference KawamataKaw84, Reference SchoutensSch05, Reference Braun, Greb, Langlois and MoragaBGLM24].
1.2 Main results
In this paper, we answer Question 1.1 for Du Bois singularities by proving the following analogue of Boutot’s theorem.
Theorem A. Let
$R \to S$
be a cyclically pure map of Noetherian
$\mathbb {Q}$
-algebras. If S has Du Bois singularities with respect to the h topology, then R has Du Bois singularities with respect to the h topology.
See Definition 2.4 for our definition of Du Bois singularities, which coincides with the usual notion for rings essentially of finite type over
$\mathbb {C}$
[Reference SteenbrinkSte83, (3.5)] and for quasi-excellent
$\mathbb {Q}$
-algebras [Reference MurayamaMur, Definition 7.3.1]. See Proposition 2.5. For quasi-excellent
$\mathbb {Q}$
-algebras, our definition is also equivalent to the characteristic-free definition of Du Bois singularities due to Huber and Kelly [Reference HuberHub16, Definition 7.19]. In fact, our proof of Theorem A yields interesting results in prime characteristic and even in mixed characteristic. See Theorem 4.3. In prime characteristic, we recover results on descent of F-injectivity under faithfully flat, quasi-finite and pure, or strongly pure maps due to Datta and the second author [Reference Datta and MurayamaDM24, Theorem 3.8 and Proposition 3.9].
Theorem A is new even when
$R \to S$
is faithfully flat. For rings essentially of finite type over
$\mathbb {C}$
, the case when
$R \to S$
splits is due to Kovács [Reference KovácsKov99, Corollary 2.4], who also proves that Du Bois singularities descend along morphisms
$f\colon Y \to X$
of schemes when
$\mathcal {O}_X \to \mathcal {R} f_*\mathcal {O}_Y$
splits in the derived category. In Theorem 4.3, we extend Kovács’s result to morphisms for which
$\mathcal {O}_X \to \mathcal {R} f_*\mathcal {O}_Y$
induces injective maps on local cohomology, and also prove a version for Du Bois pairs.
Theorem A contrasts with the situation in prime characteristic. The prime characteristic analogue of Du Bois singularities is the notion of F-injectivity [Reference SchwedeSch09]. Question 1.1 has a negative answer for F-injectivity [Reference WatanabeWat97, Example 3.3(1)], even if the map
$R \to S$
splits as a ring map [Reference NguyenNgu12, Example 6.6 and Remark 6.7]. However, as mentioned above, Question 1.1 does hold for F-injectivity under the additional hypothesis that
$R \to S$
is either quasi-finite or strongly pure [Reference Datta and MurayamaDM24, Theorem 3.8 and Proposition 3.9].
As a consequence of Theorem A, we prove the following special cases of a question of Zhuang [Reference ZhuangZhu24, Question 2.10] (cf. [Reference SchoutensSch05, (3.14); Reference Braun, Greb, Langlois and MoragaBGLM24, Question 8.5]), who asked whether Question 1.1 holds for log canonical type singularities. Statement (ii) below is the analogue of [Reference ZhuangZhu24, Lemma 2.3] for log canonical type singularities. We say that a normal complex variety X with canonical divisor
$K_X$
has log canonical type singularities if for some effective
$\mathbb {Q}$
-divisor
$\Delta $
, the pair
$(X,\Delta )$
is log canonical.
Corollary B. Let
$R \to S$
be a cyclically pure map of essentially of finite type
$\mathbb {C}$
-algebras.
-
(i) If S is normal and has Du Bois singularities (e.g., if S has log canonical type singularities) and
$K_R$
is Cartier, then R has log canonical singularities. -
(ii) Let
$f\colon Y \to X$
denote the morphism of affine schemes corresponding to
$R \to S$
. Let U be the Cartier locus of
$K_X$
. If Y has log canonical type singularities,
$K_X$
is
$\mathbb {Q}$
-Cartier, and
$Y \setminus f^{-1}(U)$
has codimension at least two in Y, then X has log canonical singularities.
1.3 Outline
To prove Theorem A, we develop a characteristic-free notion of Du Bois singularities that is essentially due to Huber and Kelly [Reference HuberHub16, Definition 7.19] in Section 2. The key idea behind their definition is that although the usual definition of
$\underline {\Omega }^0_X$
appearing in the definition of Du Bois singularities does not make sense without the existence of resolutions of singularities, an alternative description of
$\underline {\Omega }^0_X$
as the derived pushforward of the sheafification of
$\mathcal {O}_X$
with respect to the h topology (in equal characteristic zero) can be made in arbitrary characteristic. This description is originally due to Lee [Reference LeeLee09, Theorem 4.16] (see also [Reference Huber and JörderHJ14, Corollary 6.16 and Theorem 7.12]).
In Section 3, we characterize Du Bois singularities in terms of injectivity of maps on local cohomology modules in equal characteristic zero (Theorem 3.8). The key ingredient is the following version of the key injectivity theorem of Kovács and Schwede [Reference Kovács and SchwedeKS16a, Theorem 3.3; Reference Kovács and SchwedeKS16b, Theorem 3.2; Reference Ma, Schwede and ShimomotoMSS17, Lemma 3.2; Reference Kollár and KovácsKK20, Theorem 6.3 and Corollary 6.5], which is currently known for schemes essentially of finite type over a field of characteristic zero.
We prove a version of their key injectivity theorem for Noetherian schemes of equal characteristic zero that have isolated non-Du Bois points. For varieties, this special case of the key injectivity theorem is also due to Kovács [Reference KovácsKov99, Proof of Lemma 2.2; Reference KovácsKov00, Lemma 1.4] and Schwede [Reference SchwedeSch09, Proposition 5.11]. The reason it is called an injectivity theorem is that the usual statement for complex varieties is the Matlis dual of the surjectivity statement below.
Theorem C. Let X be a separated Noetherian scheme of equal characteristic zero. Let
$x \in X$
be a point such that
$\operatorname {\mathrm {Spec}}(\mathcal {O}_{X,x}) \setminus \{x\}$
is Du Bois with respect to the h topology. Then, for every i, the natural morphism
$$\begin{align*}H^i_x\big(\operatorname{\mathrm{Spec}}(\mathcal{O}_{X,x}),\mathcal{O}_{X,x}\big) \longrightarrow \mathbb{H}^i_x\big(\operatorname{\mathrm{Spec}}(\mathcal{O}_{X,x}), \underline{\Omega}^0_{X,\mathrm{h},x}\big). \end{align*}$$
is surjective.
In fact, we prove a more general version of Theorem C for pairs. See Theorem 3.7 for the full statement. For the proof, we construct an absolute version of the Nagata compactification of X using the notion of the Zariski–Riemann space associated to a pair [Reference Fujiwara and KatoFK06, Reference TemkinTem10, Reference Fujiwara and KatoFK18] (see Definition 3.1). This compactification is a locally ringed space
$\langle X \rangle _{\mathrm {cpt}}$
that contains X as a subspace whose complement is ind-constructible, and is constructed as the inverse limit of proper schemes over
$\mathbb {Q}$
. Using the long exact sequence on local cohomology, we reduce Theorem C to a surjectivity statement for cohomology on
$\langle X \rangle _{\mathrm {cpt}}$
. This last surjectivity statement is a consequence of the degeneration of the Hodge-to-de Rham spectral sequence [Reference DeligneDel74, §8.2; Reference Du BoisDB81, Théorème 4.5(iii)] and Grothendieck’s limit theorem for inverse limits of toposes [SGA42, Exposé VI, Corollaire 8.7.7].
Finally, in Section 4, we prove Theorem A, its generalization to pairs (Theorem 4.3), and Corollary B.
Conventions
All rings are commutative with identity, and all ring maps are unital. A map
$\varphi \colon M \to M'$
of R-modules over a ring R is pure if the base change
$$\begin{align*}\mathrm{id} \otimes_R \varphi\colon N \otimes_R M \longrightarrow N \otimes_R M' \end{align*}$$
is injective for every R-module N [Reference CohnCoh59, p. 383; Reference OlivierOli70, Définition 1.1].
2 Du Bois pairs via Grothendieck topologies
2.1 Grothendieck topologies
To define Du Bois pairs in full generality, we will need to work with the rh, cdh, eh, or h topologies. We define these Grothendieck topologies and other topologies that appear in their definitions following [Reference Gabber and KellyGK15, Definition 2.5].
Definition 2.1. Let S be a separated Noetherian scheme. We consider the following topologies on a small category of schemes that are of finite type over S, constructed as in [Stacks, Tag 020M]. We will consider finite families
$$ \begin{align} {\big\{f_i \colon U_i \longrightarrow X\big\}}_{i=1}^n \end{align} $$
of morphisms in
$\mathrm {Sch}/S$
in the definitions below.
-
(i) The Zariski topology, denoted by
$\mathrm {Zar}$
, is generated by families (2.1) which are jointly surjective and where each
$f_i$
is an open immersion. We allow
$n = 0$
so that the empty family is a covering family for the empty scheme. -
(ii) [Reference NisnevichNis89, §1.1] The Nisnevich topology, denoted by
$\mathrm {Nis}$
, is generated by completely decomposed families (2.1) where each
$f_i$
is étale. Here, a family (2.1) is completely decomposed if for every
$x \in X$
there is an index i and a point
$u \in U_i$
such that
$f_i(u) = x$
and
$[k(u) : k(x)] = 1$
. -
(iii) The étale topology, denoted by
$\mathrm {\acute {E}t}$
, is generated by families (2.1) which are jointly surjective and where each
$f_i$
is a étale. -
(iv) [Reference FultonFul98, Definition 18.3] The cdp topology, denoted by
$\mathrm {cdp}$
, is generated by completely decomposed families (2.1) where each
$f_i$
is proper. Such a family where
$n = 1$
is called a cdp morphism [Reference Huber, Kebekus and KellyHKK17, Definition 2.10]. Note that cdp morphisms are called envelopes in [Reference FultonFul98, Definition 18.3] and proper cdh covers in [Reference Suslin and VoevodskySV00, Definition 5.7]. -
(v) The finite topology, denoted by
$\mathrm {f}$
, is generated by families (2.1) which are jointly surjective and where each
$f_i$
is finite. -
(vi) The proper topology, denoted by
$\mathrm {prop}$
, is generated by families (2.1) which are jointly surjective and where each
$f_i$
is proper. -
(vii) [Reference Goodwillie and LichtenbaumGL01, Definition 1.2] The rh topology, denoted by
$\mathrm {rh}$
, is the coarsest topology finer than both
$\mathrm {Zar}$
and
$\mathrm {cdp}$
. -
(viii) [Reference Suslin and VoevodskySV00, Definition 5.7] The cdh topology, denoted by
$\mathrm {cdh}$
, is the coarsest topology finer than both
$\mathrm {Nis}$
and
$\mathrm {rh}$
. -
(ix) [Reference GeisserGei06, Definition 2.1] The eh topology, denoted by
$\mathrm {eh}$
, is the coarsest topology finer than both
$\mathrm {\acute {E}t}$
and
$\mathrm {rh}$
. -
(x) [Reference VoevodskyVoe96, Definition 3.1.2 and Lemma 3.4.2] The qfh topology, denoted by
$\mathrm {qfh}$
, is the coarsest topology finer than both
$\mathrm {\acute {E}t}$
and
$\mathrm {f}$
. -
(xi) [Reference Huber, Kebekus and KellyHKK17, Definition 6.2] The sdh topology, denoted by
$\mathrm {sdh}$
, is the coarsest topology finer than both
$\mathrm {\acute {E}t}$
and the topology generated by separably decomposed families (2.1). Here, a family (2.1) is separably decomposed if each
$f_i$
is proper and if, for every
$x \in X$
, there is an index i and a point
$u \in U_i$
such that
$f_i(u) = x$
and
$k(u)/k(x)$
is finite separable. -
(xii) [Reference VoevodskyVoe96, Definition 3.1.2; Reference Goodwillie and LichtenbaumGL01, Definition 1.1 and Theorem 4.1] The h topology, denoted by
$\mathrm {h}$
, is the coarsest topology finer than both
$\mathrm {Zar}$
and
$\mathrm {prop}$
. Note that this is what [Reference Goodwillie and LichtenbaumGL01] and [Stacks, Tag 0DBC and Tag 0ETQ] call the ph topology.
These topologies are related in the following manner, where the arrows below point to finer topologies [Reference Gabber and KellyGK15, (6); Reference Ertl and MillerEM19, p. 5288]:
2.2 Du Bois pairs
We define Du Bois pairs following [Reference KovácsKov11, §§3.C–3.D; Reference Kovács and SchwedeKS16b, §2C], adapted to working with Grothendieck topologies instead of hyperresolutions.
Definition 2.2. A (generalized) pair
$(X,\Sigma )$
consists of a separated Noetherian scheme together with a closed subscheme
$\Sigma \subseteq X$
. A morphism of pairs
is a morphism of schemes
$f\colon X \to Y$
such that
$f(\Sigma _X) \subseteq \Sigma _Y$
.
Let 
be a morphism of pairs. Letting
$\tau \in \{\mathrm {rh},\mathrm {cdh},\mathrm {eh},\mathrm {sdh},\mathrm {h}\}$
, we have the commutative diagram
of morphisms of sites, where the schemes in the bottom half of the diagram are thought of as their associated Zariski sites. This commutative diagram induces the commutative diagram
where each row is an exact triangle. Here, we set
and similarly for Y. We also set
Remark 2.3. Let X be a separated Noetherian scheme. If X is quasi-excellent and of equal characteristic zero, then the natural morphisms
$$\begin{align*}\mathcal{R}\rho_{X,\mathrm{eh}*}\mathcal{O}_{X_{\mathrm{eh}}} \longrightarrow \mathcal{R}\rho_{X,\mathrm{sdh}*}\mathcal{O}_{X_{\mathrm{sdh}}} \longrightarrow \mathcal{R}\rho_{X,\mathrm{h}*}\mathcal{O}_{X_{\mathrm{h}}} \end{align*}$$
are quasi-isomorphisms. The composition is a quasi-isomorphism by the proof of [Reference Huber and JörderHJ14, Proposition 6.1] ([Reference Huber and JörderHJ14] uses resolutions of singularities, which exist by [Reference TemkinTem08, Theorem 1.1]). The second morphism is a quasi-isomorphism since the sdh and h topologies coincide in equal characteristic zero [Reference Huber, Kebekus and KellyHKK17, Remark 6.4].
We now define Du Bois singularities with respect to Grothendieck topologies. When
$\Sigma = \emptyset $
, this definition has appeared before. The definition using the h topology is used to characterize Du Bois singularities for complex varieties in [Reference LeeLee09, Theorem 4.16]. In arbitrary characteristic, the definition using the cdh topology is due to Huber and Kelly [Reference HuberHub16, Definition 7.19], and the definition using the eh topology appears in [Reference Kawakami and WitaszekKW, Definition 2.13].
Definition 2.4. Let
$(X,\Sigma )$
be a pair and let
$\tau \in \{\mathrm {rh},\mathrm {cdh},\mathrm {eh},\mathrm {sdh},\mathrm {h}\}$
. We say that the pair
$(X,\Sigma )$
has Du Bois singularities with respect to the
$\tau $
topology if the morphism
$$\begin{align*}\mathcal{I}_{\Sigma \subseteq X} \longrightarrow \underline{\Omega}^0_{X,\Sigma,\tau} \end{align*}$$
defined in (2.2) is a quasi-isomorphism. If
$(X,\emptyset )$
is a Du Bois pair with respect to the
$\tau $
topology, we say that X has Du Bois singularities with respect to the
$\tau $
topology. The Du Bois defect of the pair
$(X,\Sigma )$
with respect to the
$\tau $
topology is the mapping cone
For quasi-excellent separated Noetherian schemes of equal characteristic zero, this definition matches the usual definition of Du Bois singularities and pairs using the 0-th graded piece of the Deligne–Du Bois complex. The statement for the h topology is essentially due to Lee [Reference LeeLee09, Theorem 4.16] and Huber and Jörder [Reference Huber and JörderHJ14, Corollary 6.16 and Theorem 7.12]. The statement for the cdh topology is essentially due to Cortiñas, Haesemeyer, Walker, and Weibel [Reference Cortiñas, Haesemeyer, Walker and WeibelCHWW11, Lemma 2.1].
Proposition 2.5. Let X be a quasi-excellent separated Noetherian scheme of equal characteristic zero. Let
$\pi _\bullet \colon X_\bullet \to X_{\mathrm {red}}$
be a simplicial or (possibly iterated) cubical hyperresolution. Let
$\tau \in \{\mathrm {cdh},\mathrm {eh},\mathrm {sdh},\mathrm {h}\}$
. Then, we have
$$\begin{align*}\mathcal{R}\pi_{\bullet\ast}\mathcal{O}_{X_{\bullet}} \cong \mathcal{R}\rho_{X,\tau\ast}\mathcal{O}_{X_{\tau}}. \end{align*}$$
As a consequence, we have the following.
-
(i) Suppose that
$(X,\Sigma )$
is a pair where X is essentially of finite type over a field of equal characteristic zero. Then,
$(X,\Sigma )$
has Du Bois singularities in the sense of [Reference KovácsKov11, Definition 3.13; Reference Kovács and SchwedeKS16b, Definition 2.8] if and only if
$(X,\Sigma )$
has Du Bois singularities with respect to the
$\tau $
topology. -
(ii) Suppose that
$\Sigma = \emptyset $
. Then, X has Du Bois singularities in the sense of [Reference SteenbrinkSte83, (3.5); Reference MurayamaMur, Definition 7.3.1] if and only if X has Du Bois singularities with respect to the
$\tau $
topology.
Proof. By [Reference MurayamaMur, Corollary 7.2.8], the left-hand side is the same regardless of whether
$X_\bullet $
is a simplicial or a (possibly iterated) cubical hyperresolution. By Remark 2.3, it suffices to show the cases when
$\tau \in \{\mathrm {cdh},\mathrm {h}\}$
.
We claim we may replace X by the spectrum of one of its local rings to assume that X is the spectrum of a quasi-excellent local
$\mathbb {Q}$
-algebra, which in particular is of finite Krull dimension. Since the left-hand side is local, it suffices to show that the cdh and h topologies are compatible with localization. This follows from Nayak’s version of Nagata compactification for separated essentially of finite type morphisms [Reference NayakNay09, Theorem 4.1], which implies that standard cdh covers from [Reference Suslin and VoevodskySV00, Proposition 5.9] and the standard h covers from [Reference Goodwillie and LichtenbaumGL01, Corollary 3.9; Stacks, Tag 0DBD] defined over a localization of X can be extended to standard cdh or h covers of X.
For the case when
$\tau = \mathrm {h}$
, the proof of [Reference Huber and JörderHJ14, Corollary 6.16] applies. While the proof is stated for varieties over a field of characteristic zero, the results from [Reference GeisserGei06] used in [Reference Huber and JörderHJ14] hold in our situation: The required strong form of resolutions of singularities [Reference GeisserGei06, Definition 2.4] holds by [Reference HironakaHir64, Chapter I, §3, Main Theorem I
$(n)$
], and the cited result [Reference GrosGro85, Chapitre IV, Théorème 1.2.1] (see also [Reference Guillén and Navarro AznarGNA02, Proposition 3.3]) holds for the structure sheaves
$\mathcal {O}$
.
For the case when
$\tau = \mathrm {cdh}$
, the proof of [Reference Cortiñas, Haesemeyer, Walker and WeibelCHWW11, Lemma 2.1] applies because simplicial hyperresolutions exist in our situation [Reference MurayamaMur, Corollary 4.5.5]. The result [Reference Cortiñas, Haesemeyer and WeibelCHW08, Corollary 2.5] cited in [Reference Cortiñas, Haesemeyer, Walker and WeibelCHWW11] can be replaced by the earlier result [Reference Cortiñas, Haesemeyer, Schlichting and WeibelCHSW08, Proposition 6.3] to avoid discussing differential forms on X.
Finally, statements (i) and (ii) follow by comparing the definitions in Definition 2.4 and in [Reference SteenbrinkSte83, (3.5); Reference KovácsKov11, Definition 3.13; Reference Kovács and SchwedeKS16b, Definition 2.8; Reference MurayamaMur, Definition 7.3.1].
2.3
$\tau $
-injective pairs
In the next section, we will characterize Du Bois pairs in equal characteristic zero in terms of injectivity of maps on local cohomology. This definition is inspired by the proofs of [Reference KovácsKov11, Theorem 5.4] and [Reference KovácsKov12, Theorem 2.5], and this injectivity condition is used to characterize Du Bois singularities in [Reference KovácsKov99, Lemma 2.2; Reference Bhatt, Schwede and TakagiBST17, Theorem 4.8] when X is essentially of finite type over a field and
$\Sigma = \emptyset $
.
When X is of prime characteristic
$p> 0$
and
$\Sigma = \emptyset $
, this injectivity condition for
$\tau = \mathrm {h}$
characterizes F-injectivity by [Reference Bhatt, Schwede and TakagiBST17, Theorem 4.8] and its proof (see [Reference Bhatt, Schwede and TakagiBST17, Theorem 3.3; Reference Bhatt and ScholzeBS17, Theorem 4.1(i); Stacks, Tag 0EVW]). A similar definition for
$\tau = \mathrm {eh}$
and
$\Sigma = \emptyset $
is called pseudo-Du Bois in [Reference Kawakami and WitaszekKW, Definition 3.4].
Definition 2.6. Let
$(X,\Sigma )$
be a pair and let
$\tau \in \{\mathrm {rh},\mathrm {cdh},\mathrm {eh},\mathrm {sdh},\mathrm {h}\}$
. We say that the pair
$(X,\Sigma )$
is
$\tau $
-injective at a point
$x \in X$
if the natural morphism
$$\begin{align*}H^i_x\big(\operatorname{\mathrm{Spec}}(\mathcal{O}_{X,x}),\mathcal{I}_{\Sigma,x}\big) \longrightarrow \mathbb{H}^i_x\big(\operatorname{\mathrm{Spec}}(\mathcal{O}_{X,x}), \underline{\Omega}^0_{X,\Sigma,\tau,x}\big) \end{align*}$$
induced by (2.2) is injective. We say that the pair
$(X,\Sigma )$
is
$\tau $
-injective if it is
$\tau $
-injective at every point
$x \in X$
.
Remark 2.7. Let
$\tau ,\tau ' \in \{\mathrm {rh},\mathrm {cdh},\mathrm {eh},\mathrm {sdh},\mathrm {h}\}$
be a pair of topologies such that
$\tau '$
is finer than
$\tau $
, or in other words, there is an arrow
$\tau \to \tau '$
in the diagram from Definition 2.1. Then, we have the factorization
Thus,
$\tau '$
-injectivity implies
$\tau $
-injectivity. If X is quasi-excellent of equal characteristic zero, then cdh-, eh-, sdh-, and h-injectivity all coincide by Remark 2.3 and Proposition 2.5.
3 The key injectivity theorem
In this section, we prove our version (Theorem C) of the key injectivity theorem of Kovács and Schwede [Reference Kovács and SchwedeKS16a, Theorem 3.3; Reference Kovács and SchwedeKS16b, Theorem 3.2; Reference Ma, Schwede and ShimomotoMSS17, Lemma 3.2; Reference Kollár and KovácsKK20, Theorem 6.3 and Corollary 6.5]. As in [Reference MurayamaMur25, Reference MurayamaMur], the idea is that we want to use Noetherian approximation to reduce Theorem C to the case when X is of finite type over a field.
A new idea in this paper is that we will use Zariski–Riemann spaces associated to pairs [Reference Fujiwara and KatoFK06, Reference TemkinTem10, Reference Fujiwara and KatoFK18] (see Definition 3.1) to construct an analogue of the Nagata compactification of a separated finite type morphism of schemes. For a quasi-compact quasi-separated scheme X over a Noetherian ring
$\mathbf {k}$
, our construction results in a locally ringed space
$\langle X \rangle _{\mathrm {cpt}}$
that is an inverse limit of proper
$\mathbf {k}$
-schemes such that
$\langle X \rangle _{\mathrm {cpt}} \setminus X$
is ind-constructible (in fact, a union of closed subsets). We can then prove a version (Theorem 3.6) of the Hodge-theoretic input for
$\langle X \rangle _{\mathrm {cpt}}$
that is used in the proofs in [Reference Kovács and SchwedeKS16a, Reference Kovács and SchwedeKS16b, Reference Ma, Schwede and ShimomotoMSS17] using Grothendieck’s limit theorem for inverse limits of toposes [SGA42, Exposé VI]. For varieties, this Hodge-theoretic statement is a consequence of the
$E_1$
degeneration of the Hodge-to-de Rham spectral sequence [Reference DeligneDel74, Reference Du BoisDB81] (see Theorem 3.4). We then use the long exact sequence on local cohomology to deduce Theorem C from Theorem 3.6.
Finally, we use our key injectivity theorem to prove that in equal characteristic zero, having Du Bois singularities with respect to the h topology is the same thing as being h-injective (Theorem 3.8). For quasi-excellent schemes of equal characteristic zero, the same statement holds for the cdh, eh, and sdh topologies.
3.1 Zariski–Riemann spaces
We define Zariski–Riemann spaces associated to pairs. These are called relative Riemann–Zariski spaces in [Reference TemkinTem10].
Definition 3.1 [Reference Fujiwara and KatoFK06, Definition 5.3 and Definition 5.9; Reference TemkinTem10, §3.3; Reference Fujiwara and KatoFK18, Chapter II, Definition E.2.2].
Let X be a quasi-compact quasi-separated scheme. Let
$\mathcal {I} \subseteq \mathcal {O}_X$
be a quasi-coherent ideal sheaf of finite type such that
$U = X \setminus V(\mathcal {I})$
is a dense open subset of S. Denote by
$\operatorname {AId}_{(X,U)}$
the set of quasi-coherent ideal sheaves
$\mathcal {J} \subseteq \mathcal {O}_X$
of finite type such that
$\lvert V(\mathcal {J}) \rvert \subseteq \lvert V(\mathcal {I}) \rvert $
. The Zariski–Riemann space associated to the pair
$(X,U)$
is the inverse limit
computed in the category of locally ringed spaces. Because the blowup morphisms are isomorphisms along U, the projection morphism
$\langle X \rangle _U \to X$
induces an isomorphism between an open subspace of
$\langle X \rangle _U$
and U.
We can use Zariski–Riemann spaces to define canonical compactifications.
Definition 3.2 [Reference Fujiwara and KatoFK18, Definition F.2.1].
Let
$f\colon X \to Y$
be a separated finite type morphism of quasi-compact quasi-separated schemes. By Nagata’s compactification theorem [Reference NagataNag63, Reference ConradCon07], there exists a commutative diagram
where j is a dense open immersion and
$\bar {f}$
is proper. The canonical compactification of X over Y
is
Remark 3.3. We have defined
$\langle X \rangle _{\mathrm {cpt}}$
using Nagata’s compactification theorem as input. In [Reference Fujiwara and KatoFK18, Chapter II, Definition F.2.9], Fujiwara and Kato define the canonical compactification
$\langle X \rangle _{\mathrm {cpt}}$
without assuming Nagata’s compactification theorem and use the notion to give an alternative proof of Nagata’s compactification theorem [Reference Fujiwara and KatoFK18, Chapter II, Theorem F.1.1].
3.2 The Hodge-theoretic input
Our goal in this section is to prove a version of the surjectivity in (3.1) below for separated schemes that are not proper or even of finite type over a field.
Theorem 3.4 [Reference DeligneDel74, §8.2; Reference Du BoisDB81, Théorème 4.5(iii) and Proof of Théorème 4.6; Reference KovácsKov11, Theorem 4.1 and Corollary 4.2].
Let
$(X,\Sigma )$
be a pair such that X is proper over
$\mathbb {C}$
. Consider the Deligne–Du Bois complex
$(\underline {\Omega }_{X,\Sigma }^\bullet ,F)$
and set 
. Then, the spectral sequence
$$\begin{align*}E_1^{p,q} = \mathbb{H}^q\big(X,\operatorname{Gr}^p_F \underline{\Omega}_{X,\Sigma}^\bullet \big) = \mathbb{H}^q\big(X,\underline{\Omega}_{X,\Sigma}^p\big) \Rightarrow H^{p+q}_c(U^{\mathrm{an}},\mathbb{C}) \end{align*}$$
of the filtration F degenerates on the
$E_1$
page and abuts to the Hodge filtration on the mixed Hodge structure
$H^{p+q}_c(U^{\mathrm {an}},\mathbb {C})$
. As a consequence, the natural morphisms
$$ \begin{align} H^i(X,\mathcal{I}_\Sigma) \longrightarrow \mathbb{H}^i\big(X,\underline{\Omega}_{X,\Sigma}^0\big) \end{align} $$
are surjective for every i.
To state our version of the surjectivity (3.1), we define our version of the canonical compactification
$\langle X \rangle _{\mathrm {cpt}}$
for quasi-compact separated schemes over Noetherian rings.
Setup 3.5. Let X be a quasi-compact separated scheme over a Noetherian ring
$\mathbf {k}$
. Consider a closed subscheme
$Z \subseteq X$
such that
$\mathcal {I}_Z$
is of finite type. By absolute Noetherian approximation [Reference Thomason and TrobaughTT90, Theorem C.9], we can write
$$\begin{align*}X \cong \varprojlim_{\lambda \in \Lambda} X_\lambda \end{align*}$$
as an inverse limit of separated schemes of finite type over
$\mathbf {k}$
with affine, scheme-theoretically dominant transition morphisms
$u_{\lambda \mu }\colon X_\mu \to X_\lambda $
and with projection morphisms
$u_\lambda \colon X \to X_\lambda $
.
By [Reference Grothendieck and DieudonnéEGAIV3, Proposition 8.6.3], there exists an index
$\lambda _0 \in \Lambda $
such that
$\Sigma = u_{\lambda _0}^{-1}(\Sigma _{\lambda _0})$
for a closed subscheme
$\Sigma _{\lambda _0} \subseteq X_{\lambda _0}$
. Choose a Nagata compactification
$X_{\lambda _0} \hookrightarrow \bar {X}_{\lambda _0}$
over
$\mathbf {k}$
and let
$\bar {\Sigma }_{\lambda _0}$
be the closure of
$\Sigma _{\lambda _0}$
in
$\bar {X}_{\lambda _0}$
. Denote by
$$\begin{align*}\pi_{\lambda_0}\colon \langle X_{\lambda_0} \rangle_{\operatorname{cpt}} \xrightarrow{v_{\lambda_0,\mathcal{J}}} \operatorname{Bl}_{\mathcal{J}} \bar{X}_{\lambda_0} \xrightarrow{\pi_{\lambda_0,\mathcal{J}}} \bar{X}_{\lambda_0} \end{align*}$$
the projection morphisms for
$\mathcal {J} \in \operatorname {AId}_{(\bar {X}_{\lambda _0},X_{\lambda _0})}$
, and let
$\pi _{\lambda _0}$
be their composition. We then set
For each
$\lambda \ge \lambda _0$
, choose a Nagata compactification
$X_\lambda \hookrightarrow \bar {X}_\lambda $
over
$\mathbf {k}$
that fits into the commutative diagram
where we denote
$$\begin{align*}\pi_{\lambda}\colon \langle X_{\lambda} \rangle_{\operatorname{cpt}} \xrightarrow{v_{\lambda,\mathcal{J}}} \operatorname{Bl}_{\mathcal{J}} \bar{X}_{\lambda} \xrightarrow{\pi_{\lambda,\mathcal{J}}} \bar{X}_{\lambda} \end{align*}$$
as before for
$\mathcal {J} \in \operatorname {AId}_{(\bar {X}_{\lambda },X_{\lambda })}$
. By the universal property of blowups [Stacks, Tag 0806], the morphisms
$\langle u_{\lambda _0\lambda } \rangle $
of locally ringed spaces exist and fit into the commutative diagram above. Using the universal property of blowups again, we obtain the inverse system
$$\begin{align*}{\Big\{\langle u_{\lambda\mu} \rangle\colon \langle X_\mu \rangle_{\mathrm{cpt}} \longrightarrow \langle X_\lambda \rangle_{\mathrm{cpt}}\Big\}}_{\mu \ge \lambda \ge \lambda_0} \end{align*}$$
of locally ringed spaces. We then set
where the inverse limit is computed in the category of locally ringed spaces with projection morphisms
$$\begin{align*}\langle u_\lambda \rangle \colon \langle X \rangle_{\mathrm{cpt}} \longrightarrow \langle X_\lambda \rangle_{\mathrm{cpt}}. \end{align*}$$
Choosing a Nagata compactification
$X_\lambda \hookrightarrow \bar {X}_\lambda $
for each
$\lambda $
, we also set
where the inverse limit is computed as sites as in [SGA42, Exposé VI, Théorème 8.2.3]. This definition does not depend on the choice of compactification
$\bar {X}_\lambda $
because for any other choice of compactification
$\bar {X}^{\prime }_\lambda $
, the inverse systems of blowups
$\operatorname {Bl}_{\mathcal {J}}\bar {X}_{\lambda }$
and
$\operatorname {Bl}_{\mathcal {J}}\bar {X}_{\lambda }'$
appearing in the inverse limit above are coinitial by [Reference Raynaud and GrusonRG71, Première partie, Corollaire 5.7.12] (see also [Reference ConradCon07, Theorem 2.11]).
Finally, for each
$\lambda \ge \lambda _0$
, we set
We also let
$\mathcal {I}_{\bar {\Sigma }_\lambda ,\mathrm {h}}$
,
$\mathcal {I}_{\langle \Sigma _\lambda \rangle ,\mathrm {h}}$
, and
$\mathcal {I}_{\langle \Sigma \rangle ,\mathrm {h}}$
be their h-sheafifications. Note that
$\mathcal {I}_{\bar {\Sigma }_\lambda ,\mathrm {h}}$
,
$\mathcal {I}_{\langle \Sigma _\lambda \rangle ,\mathrm {h}}$
, and
$\mathcal {I}_{\langle \Sigma \rangle ,\mathrm {h}}$
are ideal sheaves in
$\mathcal {O}_{\bar {X}_{\lambda ,\mathrm {h}}}$
,
$\mathcal {O}_{\langle X_\lambda \rangle _{\mathrm {cpt},\mathrm {h}}}$
, and
$\mathcal {O}_{\langle X \rangle _{\mathrm {cpt},\mathrm {h}}}$
since sheafification is exact [SGA41, Exposé II, Théorème 4.1(1); Stacks, Tag 00WJ].
We can now state and prove our version of the surjectivity (3.1) in Theorem 3.4.
Theorem 3.6. Let X be a quasi-compact separated scheme of equal characteristic zero, and let
$\Sigma \subseteq X$
be a closed subscheme such that
$\mathcal {I}_\Sigma $
is of finite type. Fix notation as in Setup 3.5 where
$\mathbf {k} = \mathbb {Q}$
. Then, the morphism
$$\begin{align*}H^i\big(\langle X \rangle_{\mathrm{cpt}},\mathcal{I}_{\langle \Sigma \rangle}\big) \longrightarrow H^i\big(\langle X \rangle_{\mathrm{cpt},\mathrm{h}}, \mathcal{I}_{\langle \Sigma \rangle,\mathrm{h}}\big) \end{align*}$$
is surjective for every i.
Proof. We have the commutative diagram
We claim that the top horizontal map in (3.3) is surjective. We first note that
$$\begin{align*}H^i\Bigg( \big(\operatorname{Bl}_{\mathcal{J}}\bar{X}_\lambda \big)_{\mathrm{h}},\Big( \pi_{\lambda,\mathcal{J}}^{-1}\mathcal{I}_{\bar{\Sigma}_\lambda}\cdot \mathcal{O}_{\operatorname{Bl}_{\mathcal{J}}\bar{X}_\lambda}\Big)_{\mathrm{h}} \Bigg) \cong \mathbb{H}^i\Big(\operatorname{Bl}_{\mathcal{J}} \bar{X}_\lambda,\underline{\Omega}^0_{\operatorname{Bl}_{\mathcal{J}} \bar{X}_\lambda,\bar{\Sigma}_\lambda}\Big) \end{align*}$$
by the definition of
$\underline {\Omega }^0_{X,\Sigma ,\mathrm {h}}$
, Proposition 2.5, and the definition of
$\underline {\Omega }^0_{X,\Sigma }$
. Since the formation of
$\underline {\Omega }^0_{X,\Sigma }$
is compatible with regular base change [Reference Du BoisDB81, Corollary 7.2.7] and by flat base change, it suffices to show that the top horizontal map is surjective after base change to
$\mathbb {C}$
. After this reduction, we see that the top horizontal map in (3.3) is surjective by Theorem 3.4.
To finish the proof, our goal is to apply Grothendieck’s limit theorem [SGA42, Exposé VI, Corollaire 8.7.7 and Remarque 8.7.8] to the maps in the top row of (3.3) to obtain the surjectivity of the bottom horizontal map in (3.3). We note that the h-topology over a separated Noetherian scheme is coherent in the sense of [SGA42, Exposé VI, Définition 2.3] by [Reference Goodwillie and LichtenbaumGL01, §3; Reference Gabber and KellyGK15, Remark 2.4] and that the morphisms of the associated toposes in (3.2) are coherent in the sense of [SGA42, Exposé VI, Définition 3.1] by [SGA42, Exposé VI, Corollaire 3.3]. Thus, taking cohomology is compatible with the inverse limit in (3.2) by Grothendieck’s limit theorem for inverse limits of toposes [SGA42, Exposé VI, Corollaire 8.7.7 and Remarque 8.7.8]. We now note that
$$ \begin{align*} \mathcal{I}_{\langle\Sigma\rangle} &\cong \varinjlim_{\lambda \ge \lambda_0} \varinjlim_{\mathcal{J} \in \operatorname{AId}_{(\bar{X}_\lambda,X_\lambda)}} \langle v_{\lambda,\mathcal{J}} \rangle^{-1} \langle \pi_{\lambda,\mathcal{J}} \rangle^{-1} \mathcal{I}_{\bar{\Sigma}_{\lambda}} \cdot \mathcal{O}_{\langle X \rangle_{\mathrm{cpt}}} \end{align*} $$
which also implies
$$ \begin{align*} \mathcal{I}_{\langle\Sigma\rangle,\mathrm{h}} &\cong \varinjlim_{\lambda \ge \lambda_0} \varinjlim_{\mathcal{J} \in \operatorname{AId}_{(\bar{X}_\lambda,X_\lambda)}} \Big( \langle v_{\lambda,\mathcal{J}} \rangle^{-1} \langle \pi_{\lambda,\mathcal{J}} \rangle^{-1} \mathcal{I}_{\bar{\Sigma}_{\lambda}} \cdot \mathcal{O}_{\langle X \rangle_{\mathrm{cpt}}} \Big)_{\mathrm{h}} \end{align*} $$
by the fact that h-sheafification commutes with direct limits [Stacks, Tag 00WI]. Thus, Grothendieck’s limit theorem says that the morphisms
$$ \begin{align*} \varinjlim_{\lambda \ge \lambda_0} \varinjlim_{\mathcal{J} \in \operatorname{AId}_{(\bar{X}_\lambda,X_\lambda)}} H^i\Big(\operatorname{Bl}_{\mathcal{J}}\bar{X}_\lambda, \pi_{\lambda,\mathcal{J}}^{-1}\mathcal{I}_{\bar{\Sigma}_\lambda}\cdot \mathcal{O}_{\operatorname{Bl}_{\mathcal{J}}\bar{X}_\lambda}\Big) &\overset{\sim}{\longrightarrow} H^i\big(\langle X \rangle_{\mathrm{cpt}},\mathcal{I}_{\langle\Sigma\rangle}\big)\\ \varinjlim_{\lambda \ge \lambda_0} \varinjlim_{\mathcal{J} \in \operatorname{AId}_{(\bar{X}_\lambda,X_\lambda)}} H^i\Bigg( \big(\operatorname{Bl}_{\mathcal{J}}\bar{X}_\lambda \big)_{\mathrm{h}},\Big( \pi_{\lambda,\mathcal{J}}^{-1}\mathcal{I}_{\bar{\Sigma}_\lambda}\cdot \mathcal{O}_{\operatorname{Bl}_{\mathcal{J}}\bar{X}_\lambda}\Big)_{\mathrm{h}} \Bigg) &\overset{\sim}{\longrightarrow} H^i\big(\langle X \rangle_{\mathrm{cpt},\mathrm{h}}, \mathcal{I}_{\langle\Sigma\rangle,\mathrm{h}}\big) \end{align*} $$
coming from taking direct limits of the vertical maps in (3.3) are isomorphisms. Combined with the surjectivity proved in the previous paragraph, this shows that the bottom horizontal map in (3.3) is surjective.
3.3 The key injectivity theorem
We now prove our version of the key injectivity theorem of Kovács and Schwede. When X is essentially of finite type over a field of characteristic zero, a stronger statement which removes the assumption that
$X \setminus \{x\}$
is Du Bois is known. This stronger injectivity theorem is proved in [Reference Kovács and SchwedeKS16a, Theorem 3.3] when
$\Sigma = \emptyset $
, in [Reference Kovács and SchwedeKS16b, Theorem 3.2] for reduced pairs, and in [Reference Ma, Schwede and ShimomotoMSS17, Lemma 3.2] in general. See also [Reference Kollár and KovácsKK20, Theorem 6.3 and Corollary 6.5]. Theorem C is the special case of statement (i) when
$\Sigma = \emptyset $
.
Theorem 3.7. Let X be a separated Noetherian scheme of equal characteristic zero and let
$\Sigma \subseteq X$
be a closed subscheme. Let
$\tau \in \{\mathrm {cdh},\mathrm {eh},\mathrm {sdh},\mathrm {h}\}$
. For every
$x \in X$
, consider the natural morphism
$$ \begin{align} H^i_x\big(\operatorname{\mathrm{Spec}}(\mathcal{O}_{X,x}),\mathcal{I}_{\Sigma,x}\big) \longrightarrow \mathbb{H}^i_x\big(\operatorname{\mathrm{Spec}}(\mathcal{O}_{X,x}), \underline{\Omega}^0_{X,\Sigma,\tau,x}\big) \end{align} $$
induced by (2.2). Suppose one of the following conditions holds.
-
(i)
$\tau = \mathrm {h}$
. -
(ii) X is quasi-excellent.
Suppose that
$\operatorname {\mathrm {Spec}}(\mathcal {O}_{X,x}) \setminus \{x\}$
is Du Bois with respect to the
$\tau $
topology. Then, (3.4) is surjective for every
$x \in X$
and every i.
Proof. By Remark 2.3 and Proposition 2.5, it suffices to show (i). Since the question is local, we may assume that
$X = \operatorname {\mathrm {Spec}}(R)$
is the spectrum of a Noetherian local
$\mathbb {Q}$
-algebra
$(R,\mathfrak {m})$
and that
$x = \mathfrak {m}$
. After possibly replacing
$\lambda _0$
by a larger index in
$\Lambda $
, we may assume that
$\mathfrak {m} = u_{\lambda _0}^{-1}(Z_{\lambda _0})$
for a closed subset
$Z_{\lambda _0} \subseteq X_{\lambda _0}$
. Set 
for every
$\lambda \ge \lambda _0$
.
With notation as in Setup 3.5, let 
and 
. We then have the commutative diagram
with exact columns, where
$\rho _{\langle X \rangle _{\mathrm {cpt}},\mathrm {h}}\colon \langle X \rangle _{\mathrm {cpt},\mathrm {h}} \to \langle X \rangle _{\mathrm {cpt}}$
is the projection morphism from the inverse limit of h sites to the inverse limit of Zariski sites. The third horizontal map is surjective by Theorem 3.6.
We now show that the top and bottom horizontal maps in (3.5) are isomorphisms. We claim we have the chain of isomorphisms
$$ \begin{align*} H^{i}_{\langle X \rangle_{\mathrm{cpt}} / (\Phi \cup \{\mathfrak{m}\})}\big(\langle X \rangle_{\mathrm{cpt}},\mathcal{I}_{\langle \Sigma \rangle}\big) &\cong \varinjlim_{\lambda \ge \lambda_0} H^{i}\Big(\langle X \rangle_{\mathrm{cpt}} \setminus \big( \langle u_{\lambda} \rangle^{-1}(\Phi_\lambda) \cup \{\mathfrak{m}\}\big),\mathcal{I}_{\langle \Sigma \rangle}\Big)\\ &\cong \varinjlim_{\lambda \ge \lambda_0} \varinjlim_{\mu \ge \lambda} H^{i}\Big(\langle X_\mu \rangle_{\mathrm{cpt}} \setminus \big( \langle u_{\lambda\mu} \rangle^{-1}(\Phi_\lambda \cup \overline{Z_\lambda})\big),\mathcal{I}_{\langle \Sigma_\mu \rangle}\Big)\\ &\cong \varinjlim_{\mu \ge \lambda_0} H^{i}\Big(\langle X_\mu \rangle_{\mathrm{cpt}} \setminus ( \Phi_\mu \cup \overline{Z_\mu}),\mathcal{I}_{\langle \Sigma_\mu \rangle}\Big)\\ &= \varinjlim_{\mu \ge \lambda_0} H^{i}\big(X_\mu \setminus Z_\mu,\mathcal{I}_{\Sigma_\mu}\big)\\ &\cong H^{i}\big(X \setminus \{\mathfrak{m}\},\mathcal{I}_{\Sigma}\big). \end{align*} $$
The first isomorphism holds by [Reference HartshorneHar66, Motif D on p. 221], and also holds for the complexes
$\mathcal {R}\rho _{\langle X \rangle _{\mathrm {cpt}},\mathrm {h}*} \mathcal {I}_{\langle \Sigma \rangle ,\mathrm {h}}$
. The second isomorphism holds by Grothendieck’s limit theorem [SGA42, Exposé VI, Corollaire 8.7.7 and Remarque 8.7.8] where the transition morphisms are those induced by the morphisms
of locally ringed spaces. The third isomorphism holds since the two direct systems are cofinal. The fourth equality holds since
$X_\mu = \langle X_\mu \rangle _{\mathrm {cpt}} \setminus \Phi _\mu $
. The last isomorphism holds by Grothendieck’s limit theorem [SGA42, Exposé VI, Corollaire 8.7.7 and Remarque 8.7.8]. The analogous chain holds for the corresponding sheaves on the h topology. In both topologies, we use the fact that the ideal sheaves on
$\langle X \rangle _{\mathrm {cpt}}$
and
$\langle X \rangle _{\mathrm {cpt},\mathrm {h}}$
are direct limits of the corresponding ideal sheaves on
$\langle X_\lambda \rangle _{\mathrm {cpt}}$
and
$\langle X_\lambda \rangle _{\mathrm {cpt},\mathrm {h}}$
(see the last paragraph of the proof of Theorem 3.6). Thus, the top and bottom horizontal maps in (3.5) are isomorphisms by the assumption that
$(X,\Sigma )$
is Du Bois away from
$\mathfrak {m}$
.
By the four lemma, we see that the second horizontal map in (3.5) is surjective. Our next step is to realize the map (3.4) as a direct summand of the second horizontal map in (3.5). By [Reference HartshorneHar66, Motif D on p. 219], the Mayer–Vietoris sequence [Reference HartshorneHar75, Chapter II, Proof of Proposition 4.2], and the fact that
$\mathfrak {m} \notin Z$
, the second horizontal map can be identified with the direct sum
and hence the maps on each direct summand are also surjective. By Excision [Reference GrothendieckGro67, Proposition 1.3], the fact that
$$\begin{align*}X_{\mathrm{h}} = \varprojlim_{\lambda \in \Lambda} X_{\lambda,\mathrm{h}} \end{align*}$$
as sites by [Reference HeHe24, Lemma 3.6], and the fact that the h topology on a scheme is compatible with localization, the surjective map on local cohomology with support in
$\mathfrak {m}$
can be identified with the surjection
Under the isomorphism
$$\begin{align*}\underline{\Omega}^0_{X,\Sigma,\mathrm{h}} \cong \mathcal{R}\rho_{X,\mathrm{h}*}\mathcal{I}_{\Sigma,\mathrm{h}}, \end{align*}$$
we obtain the desired surjection in Theorem C.
3.4 A characterization of Du Bois pairs
We can now show the following characterization of Du Bois pairs inspired by the proofs of [Reference KovácsKov11, Theorem 5.4] and [Reference KovácsKov12, Theorem 2.5]. See also [Reference KovácsKov99, Lemma 2.2; Reference KovácsKov00, Corollary 1.5; Reference Bhatt, Schwede and TakagiBST17, Theorem 4.8] for related statements when
$\Sigma _X = \emptyset $
.
Theorem 3.8. Let X be a separated Noetherian scheme of equal characteristic zero. Consider a closed subscheme
$\Sigma _X \subseteq X$
. The following are equivalent:
-
(i)
$(X,\Sigma _X)$
has Du Bois singularities with respect to the h topology. -
(ii) The natural morphism
$\mathcal {I}_{\Sigma _X} \to \underline {\Omega }^0_{X,\Sigma _X,\mathrm {h}}$
admits a left inverse in
$D(X)$
. -
(iii)
$(X,\Sigma _X)$
is h-injective, that is, for every
$x \in X$
, the natural morphism (3.6)is injective.
$$ \begin{align} H^i_x\big(\operatorname{\mathrm{Spec}}(\mathcal{O}_{X,x}),\mathcal{I}_{\Sigma_X,x}\big) \longrightarrow \mathbb{H}^i_x\big(\operatorname{\mathrm{Spec}}(\mathcal{O}_{X,x}), \underline{\Omega}^0_{X,\Sigma_X,\mathrm{h},x}\big) \end{align} $$
If X is quasi-excellent, then these conditions are also equivalent to:
-
(iv)
$(X,\Sigma _X)$
is
$\tau $
-injective for some (resp. every)
$\tau \in \{\mathrm {cdh},\mathrm {eh},\mathrm {sdh},\mathrm {h}\}$
.
Proof. (i)
$ \Rightarrow $
(ii) follows by the definition of a Du Bois pair, and (ii)
$ \Rightarrow $
(iii) holds because the morphism (3.6) also admits a left inverse.
(iii)
$ \Rightarrow $
(i). Suppose that
$(X,\Sigma _X)$
is not Du Bois. Let
$x \in X$
be a point that is maximal among all points at which
$(X,\Sigma _X)$
is not Du Bois. After localizing at x, we may assume that
$(X \setminus \{x\},\Sigma _X\rvert _{X \setminus \{x\}})$
is Du Bois and that
$X = \operatorname {\mathrm {Spec}}(R)$
is the spectrum of a local ring
$(R,\mathfrak {m})$
where
$x = \mathfrak {m}$
. By the hypothesis in (iii) and by Theorem 3.7, we know that (3.6) is an isomorphism for all i.
We now base change to the completion to show (i). This is necessary to ensure the existence of dualizing complexes, in which case we can apply local duality. Let
$\hat {\mathcal {I}}_{\Sigma _X} = \mathcal {I}_{\Sigma _X} \otimes _{R} \hat {R}$
and
$\hat {\Omega } = \underline {\Omega }^0_{X,\Sigma _X,\mathrm {h},x} \otimes _{R} \hat {R}$
. Denote by
$\hat {\omega }^\bullet $
a dualizing complex on
$\hat {R}$
. Applying Grothendieck local duality [Reference HartshorneHar66, Chapter III, Theorem 6.2] on
$\hat {R}$
, we see that
$$\begin{align*}\operatorname{\mathcal{RH}\mathit{om}}_{\hat{R}}( \hat{\Omega},\hat{\omega}^\bullet) \longrightarrow \operatorname{\mathcal{RH}\mathit{om}}_X\big( \hat{\mathcal{I}}_{\Sigma_X},\hat{\omega}^\bullet\big) \end{align*}$$
is a quasi-isomorphism. Since
$\hat {\omega }^\bullet $
is a dualizing complex and
$R \to \hat {R}$
is faithfully flat, we see that
$\mathcal {I}_{\Sigma _X} \to \underline {\Omega }^0_{X,\Sigma _X}$
is a quasi-isomorphism, that is,
$(X,\Sigma _X)$
has Du Bois singularities.
Finally, when X is quasi-excellent, (iii)
$ \Leftrightarrow $
(iv) by Remark 2.3 and Proposition 2.5 (or by repeating the proof in the previous paragraph using the quasi-excellent case in Theorem 3.7).
4 Proofs of main theorems
We prove a version of [Reference KovácsKov99, Corollary 2.4; Reference Kollár and KovácsKK10, Theorem 1.6; Reference KovácsKov12, Theorem 3.3] that replaces splitting conditions with conditions on injectivity of maps on local cohomology. Note that [Reference Kollár and KovácsKK10, Theorem 1.6; Reference KovácsKov12, Theorem 3.3] assume that f is proper, which we do not need in the statement below.
Proposition 4.1. Let
$f\colon Y \to X$
be a morphism between separated Noetherian schemes. Let
${\tau \in \{\mathrm {cdh},\mathrm {eh},\mathrm {sdh},\mathrm {h}\}}$
. Consider a closed subscheme
$\Sigma _X \subseteq X$
and set 
. Assume one of the following conditions holds.
-
(i) For every
$x \in X$
, there exists a maximal point
$y \in f^{-1}(x)$
such that the natural morphism is injective.
$$\begin{align*}H^i_x\big(\operatorname{\mathrm{Spec}}(\mathcal{O}_{X,x}),\mathcal{I}_{\Sigma_X,x}\big) \longrightarrow H^i_y\big(\operatorname{\mathrm{Spec}}(\mathcal{O}_{Y,y}), \mathcal{I}_{\Sigma_Y,y}\big) \end{align*}$$
-
(ii) Y is of equal characteristic zero,
$\tau = \mathrm {h}$
, and for every
$x \in X$
, the natural morphism (4.1)is injective.
$$ \begin{align} H^i_x\big(\operatorname{\mathrm{Spec}}(\mathcal{O}_{X,x}),\mathcal{I}_{\Sigma_X,x}\big) \longrightarrow \mathbb{H}^i_x\big(\operatorname{\mathrm{Spec}}(\mathcal{O}_{X,x}), (\mathcal{R} f_*\mathcal{I}_{\Sigma_{Y}})_x\big) \end{align} $$
If
$(Y,\Sigma _{Y})$
is
$\tau $
-injective, then
$(X,\Sigma _X)$
is
$\tau $
-injective. In particular, if Y is of equal characteristic zero and
$(Y,\Sigma _Y)$
has Du Bois singularities with respect to the h topology, then:
-
•
$(X,\Sigma _X)$
has Du Bois singularities with respect to the h topology; and -
• X has Du Bois singularities with respect to the h topology if and only if
$\Sigma _X$
has Du Bois singularities with respect to the h topology.
Proof. Under each respective assumption, we have the commutative diagrams
and
respectively for every i, where the right vertical arrow is injective (resp. an isomorphism) by the assumption that
$(Y,\Sigma _Y)$
is
$\tau $
-injective (resp. the assumption that
$(Y,\Sigma _Y)$
is
$\tau $
-injective and by Theorem 3.8), and the bottom horizontal arrow is injective by hypothesis. By the commutativity of the diagrams, we see that the left vertical arrow is injective for every i. The statements for Du Bois singularities now follow from Theorem 3.8 and by considering the exact triangle of Du Bois defects (see [Reference KovácsKov11, Definition 3.11])
$$\begin{align*}\underline{\Omega}_{X,\Sigma_X}^\times \longrightarrow \underline{\Omega}_X^\times \longrightarrow \underline{\Omega}_{\Sigma_X}^\times \xrightarrow{+1}.\\[-42pt] \end{align*}$$
Using Proposition 4.1, we can prove a more general version of Theorem A for pairs. The statement (iii) below is a version of [Reference KovácsKov99, Corollary 2.4; Reference Kollár and KovácsKK10, Theorem 1.6; Reference KovácsKov12, Theorem 3.3].
By [Reference Bhatt, Schwede and TakagiBST17, Theorem 4.8] and its proof (see [Reference Bhatt, Schwede and TakagiBST17, Theorem 3.3; Reference Bhatt and ScholzeBS17, Theorem 4.1(i); Stacks, Tag 0EVW]), statements (i) and (ii) give a new proof that F-injectivity descends under faithfully flat maps or maps that localize to pure maps [Reference Datta and MurayamaDM24, Theorem 3.8 and Proposition 3.9]. We will use the following terminology in statement (v) below.
Definition 4.2 (cf. [Reference Chakraborty, Gurjar and MiyanishiCGM16, p. 38]).
Let
$f\colon Y \to X$
be a morphism of locally Noetherian schemes. For a point
$x \in X$
, we say that f is partially pure at x
if there exists a point
$y \in f^{-1}(x)$
such that
$\mathcal {O}_{X,x} \to \mathcal {O}_{Y,y}$
is pure.
Theorem 4.3. Let
$f\colon Y \to X$
be a surjective morphism between Noetherian schemes. Let
$\tau \in \{\mathrm {rh},\mathrm {cdh},\mathrm {eh},\mathrm {sdh},\mathrm {h}\}$
. Consider a closed subscheme
$\Sigma _X \subseteq X$
and set 
. Assume one of the following conditions holds.
-
(i) f is faithfully flat.
-
(ii)
$\Sigma _X = \emptyset $
and for every
$x \in X$
, there is a maximal point
$y \in f^{-1}(x)$
such that
$\mathcal {O}_{X,x} \to \mathcal {O}_{Y, y}$
is pure (cf. [Reference YamaguchiYam25, Definition 5.7]). -
(iii) Y is of equal characteristic zero,
$\tau = \mathrm {h}$
, and the natural morphism
$\mathcal {I}_{\Sigma _X} \to \mathcal {R} f_*\mathcal {I}_{\Sigma _Y}$
admits a left inverse in
$D(X)$
. -
(iv) Y is of equal characteristic zero,
$\tau = \mathrm {h}$
, f is affine, and for every affine open subset
$U \subseteq X$
, the
$H^0(U,\mathcal {O}_X)$
-module map is pure.
$$\begin{align*}H^0\big(U,\mathcal{I}_{\Sigma_X}\big) \longrightarrow H^0\big(f^{-1}(U),\mathcal{I}_{\Sigma_Y}\big) \end{align*}$$
-
(v) Y is of equal characteristic zero,
$\tau = \mathrm {h}$
,
$\Sigma _X = \emptyset $
, and f is partially pure at every
$x \in X$
.
If
$(Y,\Sigma _{Y})$
is
$\tau $
-injective, then
$(X,\Sigma _X)$
is
$\tau $
-injective. In particular, if
$(Y,\Sigma _Y)$
is of equal characteristic zero and has Du Bois singularities with respect to the h topology and one of the conditions (i)–(v) holds, then:
-
•
$(X,\Sigma _X)$
has Du Bois singularities with respect to the h topology; and -
• X has Du Bois singularities with respect to the h topology if and only if
$\Sigma _X$
has Du Bois singularities with respect to the h topology.
Proof. For (i) and (ii), we show that the hypothesis in Proposition 4.1(i) is satisfied. For (i), let
$x \in X$
be a point and let
$y \in Y$
such that
$f(y) = x$
. The morphism
$\operatorname {\mathrm {Spec}}(\mathcal {O}_{Y,y}) \to \operatorname {\mathrm {Spec}}(\mathcal {O}_{X,x})$
is faithfully flat, and hence
$$\begin{align*}\mathcal{I}_{\Sigma_X,x} \longrightarrow \mathcal{I}_{\Sigma_X,x} \otimes_{\mathcal{O}_{X,x}} \mathcal{O}_{Y,y} \simeq \mathcal{I}_{\Sigma_Y,y} \end{align*}$$
is pure by [Reference Hochster and RobertsHR74, p. 136]. For (ii), we choose y as in the assumption. In either situation, the associated map on local cohomology is injective by [Reference KempfKem79, Corollary 3.2(a)].
For (iii) and (iv), we show that the hypothesis in Proposition 4.1 (ii) is satisfied. For (iii), it suffices to note that the morphism (4.1) admits a left inverse. For (iv), the morphism (4.1) is
$$\begin{align*}H^i_x\Big(\operatorname{\mathrm{Spec}}(\mathcal{O}_{X,x}),H^0\big(U,\mathcal{I}_{\Sigma_X}\big)_x\Big) \longrightarrow H^i_x\Big(\operatorname{\mathrm{Spec}}(\mathcal{O}_{X,x}),H^0\big(f^{-1}(U), \mathcal{I}_{\Sigma_Y}\big)_x\Big) \end{align*}$$
where U is an affine open containing x. This map is injective by [Reference KempfKem79, Corollary 3.2(a)].
For (v), let
$x \in X$
be a point. By assumption, there exists
$y \in Y$
such that
$f(y) = x$
and the map
$\mathcal {O}_{X,x} \to \mathcal {O}_{Y,y}$
is pure, in which case (iv) applies to the morphism
$\operatorname {\mathrm {Spec}}(\mathcal {O}_{Y,y}) \to \operatorname {\mathrm {Spec}}(\mathcal {O}_{X,x})$
.
We single out a special case of Theorem 4.3, which generalizes [Reference KovácsKov99, Corollary 2.5]. See [Reference Grothendieck and DieudonnéEGAIV3, Définition 13.2.2 and Définition 13.3.2; Reference Grothendieck and DieudonnéEGAIV4, ErrIV, 34 and 35] for the definition of locally equidimensional morphisms. Finite dominant morphisms of integral schemes are locally equidimensional.
Corollary 4.4. Let
$f\colon Y \to X$
be a locally equidimensional surjective morphism between integral Noetherian schemes of equal characteristic zero. Suppose that Y has Du Bois singularities with respect to the h topology and that X is normal. Then, X has Du Bois singularities with respect to the h topology.
Proof. Let
$x \in X$
be a point. Replacing X by an affine open neighborhood of x and shrinking Y, we may assume that
$X = \operatorname {\mathrm {Spec}}(R)$
and
$Y = \operatorname {\mathrm {Spec}}(S)$
are affine. Since R is a normal
$\mathbb {Q}$
-algebra, the inclusion
$R \hookrightarrow S$
is pure [Reference ZhuangZhu24, Proof of Corollary A.2]. We can now apply Theorem 4.3(iii).
Finally, we show Theorem A and Corollary B.
Proof of Theorem A.
Set
$$\begin{align*}(X,\Sigma_X) = \big(\operatorname{\mathrm{Spec}}(R),\emptyset\big) \qquad \text{and} \qquad (Y,\Sigma_Y) = \big(\operatorname{\mathrm{Spec}}(S),\emptyset\big). \end{align*}$$
Since
$R \to S$
is cyclically pure, the map
$Y \to X$
is surjective. Next, S is reduced since it has Du Bois singularities with respect to the h topology, and hence the subring R of S is reduced. As a consequence,
$R \to S$
is pure by [Reference HochsterHoc77, Proposition 1.4 and Theorem 1.7]. Thus, Theorem 4.3(iv) applies.
Proof of Corollary B.
For (i), note that if S has log canonical type singularities, then it is normal (by definition) and Du Bois (by [Reference Kollár and KovácsKK10, Theorem 1.4]). It therefore suffices to consider the case when S is normal and Du Bois. In this case, R is normal (by [Reference Hochster and RobertsHR76, Proposition 6.15(b)]) and Du Bois (by Theorem A). Finally, since
$K_R$
is Cartier, we see that R being Du Bois implies that R has log canonical singularities [Reference KovácsKov99, Theorem 3.3].
For (ii), we note that R is reduced since it is a subring of S, and hence
$R\to S$
is pure by [Reference HochsterHoc77, Proposition 1.4 and Theorem 1.7]. Denote by
$f\colon Y \to X$
the morphism of affine schemes associated to
$R \to S$
. We adapt the proof of [Reference ZhuangZhu24, Lemma 2.3]. We want to show that X has log canonical singularities at every closed point
$x \in X$
. Since the statement is local at x, we can replace X with an affine open neighborhood of x to assume that
$\mathcal {O}_X(rK_X) \cong \mathcal {O}_X$
, where r is the Cartier index of
$K_X$
at x.
Let
$s\in H^0(X,\mathcal {O}_X(rK_X))$
be a nowhere vanishing section, and let
$\pi \colon X' \to X$
be the corresponding index 1 cover as in [Reference Kollár and MoriKM98, Definition 5.19]. Let
$Y'$
be the normalization of the components of
$X' \times _X Y$
dominating
$X'$
, and denote by
$\pi '$
the composition
$Y' \to X' \times _X Y \to Y$
. Let U be the Cartier locus of
$K_X$
, let
$V = f^{-1}(U)$
, and let
$U'$
(resp.
$V'$
) be the preimage of U (resp. V) in
$X'$
(resp.
$Y'$
). Then,
$\pi $
and
$\pi '$
are étale over U and V, respectively (see [Reference Kollár and MoriKM98, Definition 2.49]), and
$$ \begin{align*} \pi_*\mathcal{O}_{U'} &= \left. \Bigg( \bigoplus_{m \in \mathbb{N}} \mathcal{O}_U(mK_U) \cdot t^m\Bigg)\middle/ (st^r-1)\right.\\ \pi'_*\mathcal{O}_{V'} &= \left. \Bigg( \bigoplus_{m \in \mathbb{N}} \mathcal{O}_V(mf^*K_U) \cdot t^m\Bigg)\middle/ (f^*s\,t^r-1)\right. \end{align*} $$
Here, in the second equality,
$f^*s$
denotes the pullback of s to
$H^0(V,\mathcal {O}_V(rf^*K_U))$
. By assumption, the complement of U (resp. V) in X (resp. Y) has codimension at least two. We therefore have
$$\begin{align*} H^0(X',\mathcal{O}_{X'}) &= H^0(U,\pi_*\mathcal{O}_{U'}) = \left. \Bigg( \bigoplus_{m \in \mathbb{N}} H^0\big(U,\mathcal{O}_U(mK_U)\big) \cdot t^m\Bigg)\middle/ (st^r-1)\right.\\ H^0(Y',\mathcal{O}_{Y'}) &= H^0(V,\pi'_*\mathcal{O}_{V'}) = \left. \Bigg( \bigoplus_{m \in \mathbb{N}} H^0\big(V,\mathcal{O}_V(mf^*K_U)\big) \cdot t^m\Bigg)\middle/ (f^*s\,t^r-1)\right. \end{align*}$$
By [Reference ZhuangZhu24, Lemma 2.2], the ring map
$H^0(X',\mathcal {O}_{X'}) \to H^0(Y',\mathcal {O}_{Y'})$
is pure. Since
$Y' \to Y$
is étale in codimension one (it is étale over V), we know
$Y'$
has log canonical type singularities [Reference Kollár and MoriKM98, Proposition 5.20]. By (i), this implies
$X'$
has log canonical singularities. Finally, since
$X' \to X$
is étale in codimension one (it is étale over U), we see that X has log canonical singularities [Reference Kollár and MoriKM98, Proposition 5.20].
Acknowledgments
We would like to thank Donu Arapura, Rankeya Datta, János Kollár, Sándor J. Kovács, Linquan Ma, Kenji Matsuki, Joaquín Moraga, Karl Schwede, and Farrah Yhee for helpful discussions. We also thank the referee for their helpful comments.
Competing interests
The authors have no competing interests to declare.
Funding statement
The first author was partially supported by the University of Washington Department of Mathematics Graduate Research Fellowship, and by the NSF grant DMS-1440140, administered by the Mathematical Sciences Research Institute, while in residence at MSRI during the program Birational Geometry and Moduli Spaces.
The second author was supported by the National Science Foundation under Grant Nos. DMS-1701622, DMS-1902616, and DMS-2201251.
Open access
