1 Introduction
The notion of Du Bois singularities was introduced by Steenbrink based on work of Du Bois [Reference Du BoisDBoi81], which itself was an attempt to localize Deligne’s Hodge theory of singular varieties [Reference DeligneDel74]. Steenbrink studied them initially because families with Du Bois fibers have remarkable basechange properties [Reference SteenbrinkSte81]. On the other hand, Du Bois singularities have recently found new connections across several areas of mathematics [Reference Kollár and KovácsKK10, Reference Kovács and SchwedeKS16a, Reference LeeLee09, Reference Cortiñas, Haesemeyer, Walker and WeibelCHWW11, Reference Huber and JörderHJ14]. In this paper we find numerous applications of Du Bois singularities especially to questions of local cohomology. Our key observation is as follows.
Key Lemma (Lemma 3.3). Suppose that $(S,\mathfrak{m})$ is a local ring essentially of finite type over $\mathbb{C}$ and that $S_{\text{red}}=S/\sqrt{0}$ has Du Bois singularities. Then
is surjective for every $i$ .
In fact, we obtain this surjectivity for Du Bois pairs, but we leave the statement simple in the introduction. Utilizing this lemma and related methods, we prove several results.
Theorem A (Corollary 3.9).
Let $X$ be a reduced scheme and let $H\subseteq X$ be a Cartier divisor. If $H$ is Du Bois and $X\backslash H$ is Cohen–Macaulay, then $X$ is Cohen–Macaulay and hence so is $H$ .
This first consequence answers a question of Kovács and the second author and should be viewed as a generalization of several results in [Reference Kollár and KovácsKK10, §7]. In particular, we do not need a projective family. This result holds if one replaces Cohen–Macaulay with $\text{S}_{k}$ and also generalizes to the context of Du Bois pairs as in [Reference Kovács and SchwedeKS16b]. Theorem A should also be viewed as a characteristic0 analog of [Reference Fedder and WatanabeFW89, Proposition 2.13] and [Reference Horiuchi, Miller and ShimomotoHMS14, Corollary A.4]. Theorem A also implies that if $X\setminus H$ has rational singularities and $H$ has Du Bois singularities, then $X$ has rational singularities (see Corollary 3.10), generalizing [Reference Kovács and SchwedeKS16b, Theorem E].
Our next consequence of the key lemma is the following.
Theorem B (Proposition 4.1).
Let $(R,\mathfrak{m})$ be a Gorenstein local ring essentially of finite type over $\mathbb{C}$ , and let $I\subseteq R$ be an ideal such that $R/I$ has Du Bois singularities. Then the natural map $\operatorname{Ext}_{R}^{j}(R/I,R)\rightarrow H_{I}^{j}(R)$ is injective for every $j$ .
This gives a partial characteristic $0$ answer to [Reference Eisenbud, Mustaţă and StillmanEMS00, Question 6.2], asking when such maps are injective. A special case of the analogous characteristic $p>0$ result was obtained by Singh–Walther [Reference Singh and WaltherSW07]. This result leads to an answer of [Reference De Stefani and NúñezBetancourtDN16, Question 7.5] on the bounds on the projective dimension of Du Bois and log canonical singularities. In the graded setting, we can prove a much stronger result on the injectivity of Ext.
Theorem C (Theorem 4.5).
Let $(R,\mathfrak{m})$ be a reduced Noetherian $\mathbb{N}$ graded $(R_{0}=\mathbb{C})$ algebra with $\mathfrak{m}$ the unique homogeneous maximal ideal. Suppose that $R_{P}$ is Du Bois for all $P\neq \mathfrak{m}$ . Write $R=A/I$ , where $A=\mathbb{C}[x_{1},\ldots ,x_{n}]$ is a polynomial ring with $\deg x_{i}=d_{i}>0$ and $I$ is a homogeneous ideal. Then the natural degreepreserving map $\operatorname{Ext}_{A}^{j}(R,A)\rightarrow H_{I}^{j}(A)$ induces an injection
for every $j$ , where $d=\sum d_{i}$ .
Theorem C leads to new vanishing for local cohomology for $\mathbb{N}$ graded isolated nonDu Bois singularities (Theorem 4.8), a generalization of the Kodaira vanishing theorem for Cohen–Macaulay Du Bois projective varieties.
Our next consequence of the key lemma answers a longstanding question, but first we give some background. Du Bois singularities are closely related to the notion of $F$ injective singularities (rings where the Frobenius acts injectively on local cohomology of maximal ideals). In particular, it is conjectured that $X$ has Du Bois singularities if and only if its reduction to characteristic $p>0$ has $F$ injective singularities for a Zariski dense set of primes $p\gg 0$ (this is called dense $F$ injective type). This conjecture is expected to be quite difficult, as it is closely related to asking for infinitely many ordinary characteristic $p>0$ reductions for smooth projective varieties over $\mathbb{C}$ [Reference Bhatt, Schwede and TakagiBST13]. On the other hand, several recent papers have studied how Du Bois and $F$ injective singularities deform [Reference Kovács and SchwedeKS16a, Reference Horiuchi, Miller and ShimomotoHMS14]. We know that if a Cartier divisor $H\subseteq X$ has Du Bois singularities, then $X$ also has Du Bois singularities near $H$ . However, the analogous statement for $F$ injective singularities in characteristic $p>0$ is open and has been since [Reference FedderFed83]. In fact, $F$ injective singularities were introduced because it was observed that Cohen–Macaulay $F$ injective singularities deform in this way but $F$ pure singularitiesFootnote ^{1} do not. We show that at least the property of having dense $F$ injective type satisfies such a deformation result, in other words that $F$ injectivity deforms when $p$ is large.
Theorem D (Theorem 5.3).
Let $(R,\mathfrak{m})$ be a local ring essentially of finite type over $\mathbb{C}$ and let $x$ be a nonzero divisor on $R$ . Suppose that $R/xR$ has dense $F$ injective type. Then, for infinitely many $p>0$ , the Frobenius action $x^{p1}F$ on $H_{\mathfrak{m}_{p}}^{i}(R_{p})$ is injective for every $i$ , where $(R_{p},\mathfrak{m}_{p})$ is the reduction mod $p$ of $R$ . In particular, $R$ has dense $F$ injective type.
Our final result is a characteristic $0$ analog of a strengthening of the main result of [Reference Hochster and RobertsHR76], and we can give a characteristic $p>0$ generalization as well.
Theorem E (Corollary 4.13).
Let $R$ be a Noetherian $\mathbb{N}$ graded $k$ algebra, with $\mathfrak{m}$ the unique homogeneous maximal ideal. Suppose that $R$ is equidimensional and Cohen–Macaulay on $\operatorname{Spec}R\{\mathfrak{m}\}$ . Assume one of the following:

(a) $k$ has characteristic $p>0$ and $R$ is $F$ injective;

(b) $k$ has characteristic $0$ and $R$ is Du Bois.
Then $[H^{r}(\text{}\underline{x},R)]_{0}\cong H_{\mathfrak{m}}^{r}(R)$ for every $r<n=\dim R$ and every homogeneous system of parameters $\text{}\underline{x}=x_{1},\ldots ,x_{n}$ , where $H^{r}(\text{}\underline{x},R)$ denotes the $r$ th Koszul cohomology of $\text{}\underline{x}$ . In other words, it is not necessary to take a direct limit when computing the local cohomology.
In fact, we prove a more general result (Theorem 4.12), which does not require any $F$ injective or Du Bois conditions, from which Theorem E follows immediately thanks to our injectivity and vanishing results (Theorems 4.5 and 4.8).
2 Preliminaries
Throughout this paper, all rings will be Noetherian and all schemes will be Noetherian and separated. When in equal characteristic 0, we will work with rings and schemes that are essentially of finite type over $\mathbb{C}$ . Of course, nearly all of our results also hold over all other fields of characteristic $0$ by base change.
2.1 Du Bois singularities
We give a brief introduction to Du Bois singularities and pairs. For more details, see for instance [Reference Kovács, Schwede, Friedman, Hunsicker, Libgober and MaximKS11] and [Reference KollárKol13]. Frequently we will work in the setting of pairs in the Du Bois sense.
Definition 2.1. Suppose that $X$ and $Z$ are schemes of essentially finite type over $\mathbb{C}$ . By a pair we will mean the combined data of $(X,Z)$ . We will call the pair reduced if both $X$ and $Z$ are reduced.
Suppose that $X$ is a scheme essentially of finite type over $\mathbb{C}$ . Associated to $X$ is an object $\text{}\underline{\unicode[STIX]{x1D6FA}}_{X}^{0}\in D_{\text{coh}}^{b}(X)$ with a map ${\mathcal{O}}_{X}\rightarrow \text{}\underline{\unicode[STIX]{x1D6FA}}_{X}^{0}$ functorial in the following way. If $f:Y\rightarrow X$ is a map of schemes, then we have a commutative square
If $X$ is nonreduced, then $\text{}\underline{\unicode[STIX]{x1D6FA}}_{X}^{0}=\text{}\underline{\unicode[STIX]{x1D6FA}}_{X_{\text{red}}}^{0}$ by definition. To define $\text{}\underline{\unicode[STIX]{x1D6FA}}_{X}^{0}$ in general, let $\unicode[STIX]{x1D70B}:X_{\boldsymbol{\cdot }}\rightarrow X$ be a hyperresolution and set $\text{}\underline{\unicode[STIX]{x1D6FA}}_{X}^{0}=\mathbf{R}\unicode[STIX]{x1D70B}_{\ast }{\mathcal{O}}_{X_{\boldsymbol{\cdot }}}$ (alternatively, see [Reference SchwedeSch07]).
If $Z\subseteq X$ is a closed subscheme, then we define $\text{}\underline{\unicode[STIX]{x1D6FA}}_{X,Z}^{0}$ as the object completing the following distinguished triangle:
We observe that there is an induced map $\mathscr{I}_{Z\subseteq X}\rightarrow \text{}\underline{\unicode[STIX]{x1D6FA}}_{X,Z}^{0}$ . We also note that by this definition and the fact that $\text{}\underline{\unicode[STIX]{x1D6FA}}_{X}^{0}=\text{}\underline{\unicode[STIX]{x1D6FA}}_{X_{\text{red}}}^{0}$ , we have $\text{}\underline{\unicode[STIX]{x1D6FA}}_{X,Z}^{0}=\text{}\underline{\unicode[STIX]{x1D6FA}}_{X_{\text{red}},Z_{\text{red}}}^{0}$ .
Definition 2.2 (Du Bois singularities).
We say that $X$ has Du Bois singularities if the map ${\mathcal{O}}_{X}\rightarrow \text{}\underline{\unicode[STIX]{x1D6FA}}_{X}^{0}$ is a quasiisomorphism. If $Z\subseteq X$ is a closed subscheme, we say that $(X,Z)$ has Du Bois singularities if the map $\mathscr{I}_{Z\subseteq X}\rightarrow \text{}\underline{\unicode[STIX]{x1D6FA}}_{X,Z}^{0}$ is a quasiisomorphism.
Remark 2.3. It is clear that Du Bois singularities are reduced. In general, a pair $(X,Z)$ being Du Bois does not necessarily imply that $X$ or $Z$ is reduced. However, if a pair $(X,Z)$ is Du Bois and $X$ is reduced, then so is $Z$ [Reference Kovács and SchwedeKS16b, Lemma 2.9].
2.2 Cyclic covers of nonreduced schemes
In this paper we will need to take cyclic covers of nonreduced schemes. We will work in the following setting. We assume what follows is well known to experts but we do not know a reference in the generality we need (see [Reference Kollár and MoriKM98, §2.4] or [Reference de Fernex, Ein and MustaţădFEM, §2.1.1]). Note also that the reason we are able to work with nonreduced schemes is because our $\mathscr{L}$ is actually locally free and not just a reflexive rank1 sheaf.
Setting 2.4. Suppose that $X$ is a scheme of finite type over $\mathbb{C}$ (typically projective). Suppose also that $\mathscr{L}$ is a line bundle (typically semiample). Choose a (typically general) global section $s\in H^{0}(X,\mathscr{L}^{n})$ for some $n>0$ (typically $n\gg 0$ ) and form the sheaf of rings
where, for $i,j<n$ and $i+j>n$ , the multiplication $\mathscr{L}^{i}\otimes \mathscr{L}^{j}\rightarrow \mathscr{L}^{ij+n}$ is performed by the formula $a\otimes b\mapsto abs$ . We define $\unicode[STIX]{x1D708}:Y=Y_{\mathscr{L},s}=\mathbf{Spec}\,\mathscr{R}(X,\mathscr{L},s)\rightarrow X$ . Note we did not assume that $s$ was nonzero or even locally a nonzero divisor.
Now let us work locally on an affine chart $U$ trivializing $\mathscr{L}$ , where $U=\operatorname{Spec}R\subseteq X$ with corresponding $\unicode[STIX]{x1D708}^{1}(U)=\operatorname{Spec}S$ . Fix an isomorphism $\mathscr{L}_{U}\cong {\mathcal{O}}_{U}$ and write
where $t$ is a dummy variable used to help keep track of the degree. The choice of the section $s_{U}\in H^{0}(X,\mathscr{L}^{n})$ is the same as the choice of map ${\mathcal{O}}_{X}\rightarrow \mathscr{L}^{n}$ (send $1$ to the section $s$ ). If $s$ is chosen to be general and $\mathscr{L}^{n}$ is very ample, then this map is an inclusion, but it is not injective in general. Working locally where we have fixed $\mathscr{L}_{U}={\mathcal{O}}_{U}$ , we also have implicitly chosen $\mathscr{L}^{n}={\mathcal{O}}_{U}$ and hence we have just specified that $t^{n}=s_{U}\in \unicode[STIX]{x1D6E4}(U,\mathscr{L}^{n})=\unicode[STIX]{x1D6E4}(U,{\mathcal{O}}_{U})$ . In other words,
In particular, it follows that $\unicode[STIX]{x1D708}$ is a finite map.
Lemma 2.5. The map $\unicode[STIX]{x1D708}$ is functorial with respect to taking closed subschemes of $X$ . In particular, if $Z\subseteq X$ is a closed subscheme and $\mathscr{L}$ and $s$ are as above, then we have the following commutative diagram.
Furthermore, $W_{\mathscr{L}_{Z},s_{Z}}=\unicode[STIX]{x1D70B}^{1}(Z)$ is the schemetheoretic preimage of $Z$ .
Proof. The first statement follows since $\mathscr{R}(X,\mathscr{L},s)={\mathcal{O}}_{X}\oplus \mathscr{L}^{1}\oplus \cdots \oplus \mathscr{L}^{n+1}$ surjects onto $\mathscr{R}(Z,\mathscr{L}_{Z},s_{Z})={\mathcal{O}}_{Z}\oplus \mathscr{L}_{Z}^{1}\oplus \cdots \oplus \mathscr{L}_{Z}^{n+1}$ in the obvious way. For the second statement, we simply notice that
Lemma 2.6. Suppose that we are working in Setting 2.4 and that $X$ is projective and also reduced (respectively normal), $\mathscr{L}^{n}$ is globally generated and $s$ is chosen to be general. Then $Y=Y_{\mathscr{L},s}$ is also reduced (respectively normal).
Proof. To show that $Y$ is reduced, we will show that it is $\text{S}_{1}$ and $\text{R}_{0}$ . We work locally on an open affine chart $U=\operatorname{Spec}R\subseteq X$ .
To show that it is $\text{R}_{0}$ , it suffices to consider the case where $R=K$ is a field. Because $s$ was picked generally, we may assume that the image of $s$ in $K$ is nonzero (we identify $s$ with its image). Then we need to show that $K[t]/\langle t^{n}s\rangle$ is a product of fields. But this is obvious since we are working in characteristic $0$ .
Next, we show that it is $\text{S}_{1}$ . Indeed, if $(R,\mathfrak{m})$ is a local ring of depth ${\geqslant}1$ , then obviously $S=R[t]/\langle t^{n}s\rangle$ has depth ${\geqslant}1$ as well since it is a free $R$ module. Finally, the depth condition is preserved after localizing at the maximal ideals of the semilocal ring of $S$ .
This proves the reduced case. The normal case is well known but we sketch it. The $\text{R}_{1}$ condition follows from [Reference Kollár and MoriKM98, Lemma 2.51] and the fact that $s$ is chosen generally, utilizing Bertini’s theorem. The $\text{S}_{2}$ condition follows in the same way that $\text{S}_{1}$ followed above.◻
2.3 $F$ pure and $F$ injective singularities
Du Bois singularities are conjectured to be the characteristic $0$ analog of $F$ injective singularities [Reference SchwedeSch09, Reference Bhatt, Schwede and TakagiBST13]. In this short subsection we collect some definitions about singularities in positive characteristic. Our main focuses are $F$ pure and $F$ injective singularities.
A local ring $(R,\mathfrak{m})$ is called $F$ pure if the Frobenius endomorphism $F$ : $R\rightarrow R$ is pure.Footnote ^{2} Under mild conditions, for example when $R$ is $F$ finite, which means that the Frobenius map $R\xrightarrow[{}]{F}R$ is a finite map, $R$ being $F$ pure is equivalent to the condition that the Frobenius endomorphism $R\xrightarrow[{}]{F}R$ is split [Reference Hochster and RobertsHR76, Corollary 5.3]. The Frobenius endomorphism on $R$ induces a natural Frobenius action on each local cohomology module $H_{\mathfrak{m}}^{i}(R)$ and we say that a local ring is $F$ injective if this natural Frobenius action on $H_{\mathfrak{m}}^{i}(R)$ is injective for every $i$ [Reference FedderFed83]. This holds if $R$ is $F$ pure [Reference Hochster and RobertsHR76, Lemma 2.2]. For some other basic properties of $F$ pure and $F$ injective singularities, see [Reference Hochster and RobertsHR76, Reference FedderFed83, Reference Enescu and HochsterEH08].
3 The Cohen–Macaulay property for families of Du Bois pairs
We begin with a lemma, which we assume is well known to experts; indeed, it is explicitly stated in [Reference KollárKol13, Corollary 6.9]. However, because many of the standard references also include implicit reducedness hypotheses, we include a careful proof that deduces it from the reduced case. Of course, one could also go back to first principles but we believe the path we take below is quicker.
Lemma 3.1. Let $X$ be a projective scheme over $\mathbb{C}$ and let $Z\subseteq X$ be a closed subscheme ( $X,Z$ are not necessarily reduced). Then the natural map
is surjective for every $i\in \mathbb{Z}$ .
Proof. Note that if the result holds for $\mathbb{C}$ , it certainly also holds for other fields of characteristic $0$ . The isomorphism in the statement of the lemma follows from the definition. For the surjectivity, we consider the following commutative diagram, where we let $U=X\backslash Z$ . Note that the rows are not exact.
The composite map in the top horizontal line is a surjection by [Reference Kovács and SchwedeKS16b, Lemma 2.17] or [Reference KovácsKov11, Corollary 4.2]. The vertical isomorphism on the left holds because the constant sheaf $\mathbb{C}$ does not see the nonreduced structure. Likewise for the vertical isomorphism on the right, where $\text{}\underline{\unicode[STIX]{x1D6FA}}_{X,Z}^{0}$ does not see the nonreduced structure. The diagram then shows that $H^{i}(X,\mathscr{I}_{Z})\rightarrow \mathbb{H}^{i}(X_{\text{red}},\text{}\underline{\unicode[STIX]{x1D6FA}}_{X_{\text{red}},Z_{\text{red}}}^{0})$ is a surjection.◻
Next, we prove the key injectivity lemma for possibly nonreduced pairs. The proof is essentially the same as in [Reference Kovács and SchwedeKS16b] or [Reference Kovács and SchwedeKS16a]. We reproduce it here carefully because we need it in the nonreduced setting.
Lemma 3.2. Let $X$ be a scheme of essentially finite type over $\mathbb{C}$ and $Z\subseteq X$ a closed subscheme ( $X$ and $Z$ are not necessarily reduced). Then the natural map
is injective for every $j\in \mathbb{Z}$ , where $\text{}\underline{\unicode[STIX]{x1D714}}_{X,Z}^{\boldsymbol{\cdot }}=\mathbf{R}\,\mathscr{H}\!\!\operatorname{om}_{{\mathcal{O}}_{X}}(\text{}\underline{\unicode[STIX]{x1D6FA}}_{X,Z}^{0},\unicode[STIX]{x1D714}_{X}^{\boldsymbol{\cdot }})$ .
Proof. The question is local and compatible with restricting to an open subset; hence, we may assume that $X$ is projective with ample line bundle $\mathscr{L}$ . Let $s\in H^{0}(X,\mathscr{L}^{n})$ be a general global section for some $n\gg 0$ and let $\unicode[STIX]{x1D702}$ : $Y=\mathbf{Spec}\,\bigoplus _{i=0}^{n1}\mathscr{L}^{i}\rightarrow X$ be the $n$ th cyclic cover corresponding to $s$ . Set $W=\unicode[STIX]{x1D702}^{1}(Z)$ . Then, for $n\gg 0$ and $s$ general, the restriction of $\unicode[STIX]{x1D702}$ to $W$ is the cyclic cover $W=\mathbf{Spec}\,\bigoplus _{i=0}^{n1}\mathscr{L}_{Z}^{i}\rightarrow Z$ by Lemma 2.5. Likewise, $\unicode[STIX]{x1D702}$ also induces the corresponding cyclic covers of the closed subschemes $Y_{\text{red}}\rightarrow X_{\text{red}}$ and $W_{\text{red}}\rightarrow Z_{\text{red}}$ by Lemmas 2.5 and 2.6. We have $\unicode[STIX]{x1D702}_{\ast }\mathscr{I}_{W}=\bigoplus _{i=0}^{n1}\mathscr{I}_{Z}\otimes \mathscr{L}^{i}$ and
where the second isomorphism is [Reference Kovács and SchwedeKS16b, Lemma 3.1] (see also [Reference Kovács and SchwedeKS16a, Lemma 3.1]). Since Lemma 3.1 implies that $H^{j}(Y,\mathscr{I}_{W}){\twoheadrightarrow}\mathbb{H}^{j}(Y,\text{}\underline{\unicode[STIX]{x1D6FA}}_{Y,W}^{0})$ is surjective for every $j\in \mathbb{Z}$ , we know that $H^{j}(X,\mathscr{I}_{Z}\otimes \mathscr{L}^{i}){\twoheadrightarrow}\mathbb{H}^{j}(X,\text{}\underline{\unicode[STIX]{x1D6FA}}_{X,Z}^{0}\otimes \mathscr{L}^{i})$ is surjective for every $i\geqslant 0$ and $j\in \mathbb{Z}$ .
Applying Grothendieck–Serre duality, we obtain an injection
for all $i\geqslant 0$ and $j\in \mathbb{Z}$ . Since $\mathscr{L}$ is ample, for $i\gg 0$ the spectral sequence computing the above hypercohomology degenerates. Hence, for $i\gg 0$ , we get
But again since $\mathscr{L}$ is ample, the above injection for $i\gg 0$ implies the injection $h^{j}(\text{}\underline{\unicode[STIX]{x1D714}}_{X,Z}){\hookrightarrow}h^{j}(\mathbf{R}\,\mathscr{H}\!\!\operatorname{om}_{{\mathcal{O}}_{X}}(\mathscr{I}_{Z},\unicode[STIX]{x1D714}_{X}^{\boldsymbol{\cdot }}))$ .◻
Next, we prove the key lemma stated in the introduction (and we do the generalized pair version). It follows immediately from the above injectivity, Lemma 3.2. For simplicity, we will use $(S,S/J)$ to denote the pair $(\operatorname{Spec}S,\operatorname{Spec}S/J)$ .
Lemma 3.3. Let $(S,\mathfrak{m})$ be a local ring essentially of finite type over $\mathbb{C}$ and let $J\subseteq S$ be an ideal. Suppose further that $S^{\prime }=S/N$ , where $N\subseteq S$ is an ideal contained in the nilradical (for instance, $N$ could be the nilradical and then $S^{\prime }=S_{\text{red}}$ ). Suppose that $(S^{\prime },S^{\prime }/JS^{\prime })$ is a Du Bois pair. Then the natural map $H_{\mathfrak{m}}^{i}(J)\rightarrow H_{\mathfrak{m}}^{i}(JS^{\prime })$ is surjective for every $i$ . In particular, if $S_{\text{red}}$ is Du Bois, then $H_{\mathfrak{m}}^{i}(S)\rightarrow H_{\mathfrak{m}}^{i}(S_{\text{red}})$ is surjective for every $i$ .
Proof. We consider the following commutative diagram.
Here the left vertical map is surjective by the Matlis dual of Lemma 3.2 applied to $X=\operatorname{Spec}S$ and $Z=\operatorname{Spec}S/J$ , the right vertical map is an isomorphism because $(S^{\prime },S^{\prime }/JS^{\prime })$ is a Du Bois pair and the bottom map is an isomorphism because $S_{\text{red}}=S_{\text{red}}^{\prime }$ . Chasing the diagram shows that $H_{\mathfrak{m}}^{i}(J)\rightarrow H_{\mathfrak{m}}^{i}(JS^{\prime })$ is surjective. The last sentence is the case that $J$ is the unit ideal (i.e., the nonpair version).◻
Remark 3.4. The characteristic $p>0$ analog of the above lemma (in the case $J=S$ , i.e., the nonpair version) holds if $S_{\text{red}}$ is $F$ pure. We give a short argument here. Without loss of generality, we may assume that $S$ and $S_{\text{red}}$ are complete, so $S=A/I$ and $S_{\text{red}}=A/\sqrt{I}$ , where $A$ is a complete regular local ring. We may pick $e\gg 0$ such that $(\sqrt{I})^{[p^{e}]}\subseteq I$ . We have a composite map
We know from [Reference LyubeznikLyu06, Lemma 2.2] that the image of the composite map is equal to $\operatorname{span}_{A}\langle f^{e}(H_{\mathfrak{m}}^{i}(S_{\text{red}}))\rangle$ , where $f^{e}$ denotes the natural $e$ th Frobenius action on $H_{\mathfrak{m}}^{i}(S_{\text{red}})$ . In particular, the Frobenius action on $H_{\mathfrak{m}}^{i}(S_{\text{red}})/\text{im}\,\unicode[STIX]{x1D719}$ is nilpotent. However, since $\operatorname{im}\unicode[STIX]{x1D719}$ is an $F$ stable submodule of $H_{\mathfrak{m}}^{i}(S_{\text{red}})$ and $S_{\text{red}}$ is $F$ pure, [Reference MaMa14, Theorem 3.7] shows that the Frobenius acts injectively on $H_{\mathfrak{m}}^{i}(S_{\text{red}})/\text{im}\,\unicode[STIX]{x1D719}$ . Hence, we must have $H_{\mathfrak{m}}^{i}(S_{\text{red}})=\operatorname{im}\unicode[STIX]{x1D719}$ , that is, $H_{\mathfrak{m}}^{i}(S)\rightarrow H_{\mathfrak{m}}^{i}(S_{\text{red}})$ is surjective.
We next give an example showing that the analog of Lemma 3.3 for $F$ injectivity fails in general. The example is a modification of [Reference Enescu and HochsterEH08, Example 2.16].
Example 3.5. Let $K=k(u,v)$ , where $k$ is an algebraically closed field of characteristic $p>0$ , and let $L=K[z]/(z^{2p}+uz^{p}+v)$ be as in [Reference Enescu and HochsterEH08, Example 2.16]. Now let $R=K+(x,y)L[[x,y]]$ with $\mathfrak{m}=(x,y)L[[x,y]]$ . Then it is easy to see that $(R,\mathfrak{m})$ is a local ring of dimension $2$ and we have a short exact sequence:
The long exact sequence of local cohomology gives
Moreover, one can check that the Frobenius action on $H_{\mathfrak{m}}^{1}(R)$ is exactly the Frobenius action on $L/K$ . The Frobenius action on $L/K$ is injective since $L^{p}\cap K=K^{p}$ , and the Frobenius action on $H_{\mathfrak{m}}^{2}(L[[x,y]])$ is injective because $L[[x,y]]$ is regular. Hence, the Frobenius actions on both $H_{\mathfrak{m}}^{1}(R)$ and $H_{\mathfrak{m}}^{2}(R)$ are injective. This proves that $R$ is $F$ injective.
Write $R=A/I$ for a regular local ring $(A,\mathfrak{m})$ . One checks that the Frobenius action $F:H_{\mathfrak{m}}^{1}(R)\rightarrow H_{\mathfrak{m}}^{1}(R)$ is not surjective up to $K$ span (and hence not surjective up to $R$ span, since the residue field of $R$ is $K$ ) because $L\neq L^{p}[K]$ by our choice of $K$ and $L$ (see [Reference Enescu and HochsterEH08, Example 2.16] for a detailed computation on this). But now [Reference LyubeznikLyu06, Lemma 2.2] shows that $H_{\mathfrak{m}}^{1}(A/I^{[p]})\rightarrow H_{\mathfrak{m}}^{1}(A/I)$ is not surjective. Therefore, we can take $S=A/I^{[p]}$ with $S_{\text{red}}=R$ that is $F$ injective, but $H_{\mathfrak{m}}^{1}(S)\rightarrow H_{\mathfrak{m}}^{1}(S_{\text{red}})$ is not surjective.
In view of Remark 3.4 and Example 3.5 and the relation between $F$ injective and Du Bois singularities, it is tempting to try to define a more restrictive variant of $F$ injective singularities for local rings. In particular, if $(R,\mathfrak{m})$ is a local ring such that $R_{\text{red}}$ has these more restrictive $F$ injective singularities, then it should follow that $H_{\mathfrak{m}}^{i}(R)\rightarrow H_{\mathfrak{m}}^{i}(R_{\text{red}})$ surjects for all $i$ .
Theorem 3.6. Suppose that $(X,Z)$ is a reduced pair and that $H\subseteq X$ is a Cartier divisor such that $H$ does not contain any irreducible components of either $X$ or $Z$ . If $(H,Z\cap H)$ is a Du Bois pair, then for all $i$ and all points $\unicode[STIX]{x1D702}\in H\subseteq X$ , the following segment of the long exact sequence:
is exact for all $i$ . Dually, in the special case that $\mathscr{I}_{Z}={\mathcal{O}}_{X}$ , we can also phrase this as saying that
is exact for all $i$ .
Proof. Localizing at $\unicode[STIX]{x1D702}$ , we may assume that $X=\operatorname{Spec}R$ , $Z=\operatorname{Spec}R/I$ , $H=\operatorname{div}(x)$ with $(R,\mathfrak{m})$ a local ring and $\unicode[STIX]{x1D702}=\{\mathfrak{m}\}$ . Moreover, the hypotheses imply that $x$ is a nonzero divisor on both $R$ and $R/I$ . It is enough to show that the segment of the long exact sequence
induced by $0\rightarrow I\xrightarrow[{}]{\cdot x}I\rightarrow I/xI\rightarrow 0$ is exact.
Consider $S=R/x^{n}R$ and $J=I(R/x^{n}R)$ . The pair $(S^{\prime }=R/xR,S^{\prime }/IS^{\prime })=(H,Z\cap H)$ is Du Bois by hypothesis (note that we do not assume that $H$ is reduced). Thus, Lemma 3.3 implies that
is surjective for every $i$ , $n$ . Since $x$ is a nonzero divisor on $R/I$ , $x^{n}I=I\cap x^{n}R$ . Thus,
Hence, (3.6.2) shows that
is surjective for every $i$ , $n$ . The long exact sequence of local cohomology induced by $0\rightarrow I/x^{n1}I\xrightarrow[{}]{\cdot x}I/x^{n}I\rightarrow I/xI\rightarrow 0$ tells us that
is injective for every $i$ , $n$ . Hence, after taking a direct limit, we obtain that
is injective for every $i$ . Here the last two isomorphisms come from the fact that $x$ is a nonzero divisor on $I$ (since it is a nonzero divisor on $R$ ) and a simple computation using the local cohomology spectral sequence.
Claim 3.7. The $\unicode[STIX]{x1D719}_{i}$ are exactly the connection maps in the long exact sequence of local cohomology induced by $0\rightarrow I\xrightarrow[{}]{\cdot x}I\rightarrow I/xI\rightarrow 0$ :
This claim immediately produces (3.6.1) because we proved that $\unicode[STIX]{x1D719}_{i}$ is injective for every $i$ . Thus, it remains to prove the claim.
Proof of claim.
Observe that, by definition, $\unicode[STIX]{x1D719}_{i}$ is the natural map in the long exact sequence of local cohomology
which is induced by $0\rightarrow I/xI\rightarrow I_{x}/I\xrightarrow[{}]{\cdot x}I_{x}/I\rightarrow 0$ (note that $x$ is a nonzero divisor on $I$ and $H_{x}^{1}(I)\cong I_{x}/I$ ). However, it is easy to see that the multiplication by $x$ map $H_{\mathfrak{m}}^{i}(I_{x}/I)\xrightarrow[{}]{\cdot x}H_{\mathfrak{m}}^{i}(I_{x}/I)$ can be identified with the multiplication by $x$ map $H_{\mathfrak{m}}^{i+1}(I)\xrightarrow[{}]{\cdot x}H_{\mathfrak{m}}^{i+1}(I)$ because we have a natural identification $H_{\mathfrak{m}}^{i}(I_{x}/I)\cong H_{\mathfrak{m}}^{i}(H_{x}^{1}(I))\cong H_{\mathfrak{m}}^{i+1}(I)$ . This finishes the proof of the claim and thus also the local cohomology statement.◻
The global statement can be checked locally, where it is simply a special case of the local dual of the local cohomology statement. ◻
The following is the main theorem of the section; it answers a question of Kovács and the second author [Reference Kovács and SchwedeKS16b, Question 5.7] and is a generalization to pairs of a characteristic $0$ analog of [Reference Horiuchi, Miller and ShimomotoHMS14, Corollary A.5].
Theorem 3.8. Suppose that $(X,Z)$ is a reduced pair and that $H\subseteq X$ is a Cartier divisor such that $H$ does not contain any irreducible components of either $X$ or $Z$ . If $(H,Z\cap H)$ is a Du Bois pair and $\mathscr{I}_{Z}_{X\backslash H}$ is $\text{S}_{k}$ , then $\mathscr{I}_{Z}$ is $\text{S}_{k}$ .
Proof. Suppose that $\mathscr{I}_{Z}$ is not $\text{S}_{k}$ ; choose an irreducible component $Y$ of the non $\text{S}_{k}$ locus of $\mathscr{I}_{Z}$ . Since $\mathscr{I}_{Z}_{X\backslash H}$ is $\text{S}_{k}$ , we know that $Y\subseteq H$ . Let $\unicode[STIX]{x1D702}$ be the generic point of $Y$ . Theorem 3.6 tells us that
is surjective for every $i$ . Note that if we localize at $\unicode[STIX]{x1D702}$ and use the same notation as in the proof of Theorem 3.6, this surjection is the multiplication by $x$ map $H_{\mathfrak{m}}^{i}(I)\xrightarrow[{}]{\cdot x}H_{\mathfrak{m}}^{i}(I)$ .
However, after we localize at $\unicode[STIX]{x1D702}$ , $I$ is $\text{S}_{k}$ on the punctured spectrum and $H_{\mathfrak{m}}^{i}(I)$ has finite length for every $i<k$ . In particular, for $h\gg 0$ , $x^{h}$ annihilates $H_{\mathfrak{m}}^{i}(I)$ for every $i<k$ . Therefore, for $i<k$ , the multiplication by $x$ map $H_{\mathfrak{m}}^{i}(I)\xrightarrow[{}]{\cdot x}H_{\mathfrak{m}}^{i}(I)$ cannot be surjective unless $H_{\mathfrak{m}}^{i}(I)=0$ , which means that $I$ is $\text{S}_{k}$ as an $R$ module. But this contradicts our choice of $Y$ (because we pick $Y$ as an irreducible component of the non $\text{S}_{k}$ locus of $\mathscr{I}_{Z}$ ). Therefore, $\mathscr{I}_{Z}$ is $\text{S}_{k}$ .◻
Corollary 3.9. Let $X$ be a reduced scheme and let $H\subseteq X$ be a Cartier divisor. If $H$ is Du Bois and $X\backslash H$ is Cohen–Macaulay, then $X$ is Cohen–Macaulay and hence so is $H$ .
As mentioned in the Introduction, these results should be viewed as a generalization of [Reference Kollár and KovácsKK10, Corollaries 1.3 and 7.13] and [Reference Kovács and SchwedeKS16b, Theorem 5.5]. In those results, one considered a flat projective family $X\rightarrow B$ with Du Bois fibers such that the general fiber is $\text{S}_{k}$ . It was then shown that the special fiber is $S_{k}$ . Let us explain this result in a simple case. Suppose that $B$ is a smooth curve, $Z=0$ , the special fiber is a Cartier divisor $H$ and $X\setminus H$ is Cohen–Macaulay. Then we are trying to show that $H$ is Cohen–Macaulay, which is of course the same as showing that $X$ is Cohen–Macaulay, which is exactly what our result proves. The point is that we had to make no projectivity hypothesis for our family.
This result also implies [Reference Kovács and SchwedeKS16b, Conjecture 7.9], which we state next. We leave out the definition of a rational pair here (this is a rational pair in the sense of Kollár and Kovács: a generalization of rational singularities analogous to divisorially log terminal singularities; see [Reference KollárKol13]); the interested reader can look it up, or simply assume that $D=0$ .
Corollary 3.10. Suppose that $(X,D)$ is a pair with $D$ a reduced Weil divisor. Further suppose that $H$ is a Cartier divisor on $X$ not having any components in common with $D$ such that $(H,D\cap H)$ is Du Bois and such that $(X\setminus H,D\setminus H)$ is a rational pair in the sense of Kollár and Kovács. Then $(X,D)$ is a rational pair in the sense of Kollár and Kovács.
Proof. We follow [Reference Kovács and SchwedeKS16b, Proof of Theorem 7.1]. It goes through word for word once one observes that ${\mathcal{O}}_{X}(D)$ is Cohen–Macaulay, which follows immediately from Theorem 3.8.◻
This corollary is also an analog of [Reference Fedder and WatanabeFW89, Proposition 2.13] and [Reference Horiuchi, Miller and ShimomotoHMS14, Corollary A.4].
4 Applications to local cohomology
In this section we give many applications of Lemma 3.3 to local cohomology; several of these applications are quite easy to prove. However, they provide strong and surprising characteristic $0$ analogs and generalizations of results in [Reference Hochster and RobertsHR76, Reference Singh and WaltherSW05, Reference Singh and WaltherSW07, Reference VarbaroVar13, Reference De Stefani and NúñezBetancourtDN16] as well as answer their questions. Moreover, our results can give a generalization of the classical Kodaira vanishing theorem.
4.1 Injectivity of Ext and a generalization of the Kodaira vanishing theorem
Our first application gives a solution to [Reference Eisenbud, Mustaţă and StillmanEMS00, Question 6.2] on the injectivity of $\operatorname{Ext}$ in characteristic $0$ , which parallels its characteristic $p>0$ analog [Reference Singh and WaltherSW07, Theorem 1.3].
Proposition 4.1. Let $(R,\mathfrak{m})$ be a Gorenstein local ring essentially of finite type over $\mathbb{C}$ , and let $I\subseteq R$ be an ideal such that $R/I$ has Du Bois singularities. Then the natural map $\operatorname{Ext}_{R}^{j}(R/I,R)\rightarrow H_{I}^{j}(R)$ is injective for every $j$ .
Proof. By Lemma 3.3 applied to $S=R/I^{t}$ and applying local duality, we know that
is injective for every $j$ and every $t>0$ . Now taking a direct limit, we obtain the desired injectivity $\operatorname{Ext}_{R}^{j}(R/I,R)\rightarrow H_{I}^{j}(R)$ . ◻
Remark 4.2. Theorem 1.3 of [Reference Singh and WaltherSW07] proves the same injectivity result in characteristic $p>0$ when $R/I$ is $F$ pure (and when $R$ is regular). Perhaps it is worth pointing out that Proposition 4.1 fails in general if we replace Du Bois by $F$ injective: Example 3.5 is a counterexample, because there we have that $S=A/I$ is $F$ injective but $H_{\mathfrak{m}}^{1}(A/I^{[p]})\rightarrow H_{\mathfrak{m}}^{1}(A/I)$ is not surjective. Hence, applying local duality shows that
is not injective, so neither is $\operatorname{Ext}_{A}^{\dim A1}(A/I,A)\rightarrow H_{I}^{\dim A1}(A)$ .
An immediate corollary of Proposition 4.1 is the following, which is a characteristic $0$ analog of [Reference De Stefani and NúñezBetancourtDN16, Theorem 7.3] (this also answers [Reference De Stefani and NúñezBetancourtDN16, Question 7.5], which is motivated by Stillman’s conjecture).
Corollary 4.3. Let $R$ be a regular ring of essentially finite type over $\mathbb{C}$ . Let $I\subseteq R$ be an ideal such that $S=R/I$ has Du Bois singularities. Then $\operatorname{pd}_{R}S\leqslant \unicode[STIX]{x1D708}(I)$ , where $\unicode[STIX]{x1D708}(I)$ denotes the minimal number of generators of $I$ .
Proof. For every $j>\unicode[STIX]{x1D708}(I)$ and every maximal ideal $\mathfrak{m}$ of $R$ , we have $H_{IR_{\mathfrak{m}}}^{j}(R_{\mathfrak{m}})=0$ . Therefore, by Proposition 4.1, we have $\operatorname{Ext}_{R_{\mathfrak{m}}}^{j}(S_{\mathfrak{m}},R_{\mathfrak{m}})=0$ for every $j>\unicode[STIX]{x1D708}(I)$ . Since $\operatorname{pd}_{R_{\mathfrak{m}}}S_{\mathfrak{m}}<\infty$ , we have
Because this is true for every maximal ideal $\mathfrak{m}$ , we have $\operatorname{pd}_{R}S\leqslant \unicode[STIX]{x1D708}(I)$ .◻
Next, we want to prove a stronger form of Proposition 4.1 in the graded case. We will first need the following criterion of Du Bois singularities for $\mathbb{N}$ graded rings. This result and related results should be well known to experts (or at least well believed by experts): for example [Reference MaMa15, Theorem 4.4], [Reference Bhatt, Schwede and TakagiBST13, Lemma 2.14], [Reference Kovács and SchwedeKS16b, Lemma 2.12] or [Reference Graf and KovácsGK14]. However, all these references only deal with the case that $R$ is the section ring of a projective variety with respect to a certain ample line bundle, while here we allow arbitrary $\mathbb{N}$ graded rings which are not even normal. We could not find a reference that handles the generality that we will need and hence we write down a careful argument.
Proposition 4.4. Let $(R,\mathfrak{m})$ be a reduced Noetherian $\mathbb{N}$ graded $(R_{0}=k)$ algebra with $\mathfrak{m}$ the unique homogeneous maximal ideal. Suppose that $R_{P}$ is Du Bois for all $P\neq \mathfrak{m}$ . Then we have
for every $i>0$ and
In particular, $R$ is Du Bois if and only if $[H_{\mathfrak{m}}^{i}(R)]_{{>}0}=0$ for every $i>0$ .
Proof. Let $R^{\natural }$ denote the Rees algebra of $R$ with respect to the natural filtration $R_{{\geqslant}t}$ . That is, $R^{\natural }=R\oplus R_{{\geqslant}1}\oplus R_{{\geqslant}2}\oplus \cdots \,.$ Let $Y=\operatorname{Proj}R^{\natural }$ . We first claim that $Y$ is Du Bois: $Y$ is covered by $D_{+}(f)$ for homogeneous elements $f\in R_{{\geqslant}t}$ and $t\in \mathbb{N}$ .
If $\deg f>t$ , then $[R_{f}^{\natural }]_{0}\cong R_{f}$ is Du Bois. If $\deg f=t$ , then $[R_{f}^{\natural }]_{0}\cong [R_{f}]_{{\geqslant}0}$ is also Du BoisFootnote ^{4} (see [Reference Kurano, Sato, Singh and WatanabeKSSW09, Lemma 5.4] for an analogous analysis on rational singularities).
Since $R$ and thus $R^{\natural }$ are Noetherian, there exists $n$ such that $R_{{\geqslant}nt}=(R_{{\geqslant}n})^{t}$ for every $t\geqslant 1$ [Reference BourbakiBou98, ch. III, §3, Proposition 3]. Let $I=(R_{{\geqslant}n})$ ; then we immediately see that $Y\cong \operatorname{Proj}(R\oplus I\oplus I^{2}\oplus \cdots \,)$ is the blow up of $\operatorname{Spec}R$ at $I$ . We define the exceptional divisor to be $E\cong \operatorname{Proj}(R/I\oplus I/I^{2}\oplus I^{2}/I^{3}\oplus \cdots \,)$ . We next claim that
The point is that, for $\overline{x}\in I^{t}/I^{t+1}=R_{{\geqslant}nt}/R_{{\geqslant}n(t+1)}$ , if $x\in R_{nt+a}$ for some $a>0$ , then we can pick $b>0$ such that $ba>n$ ; we then have
But this means that $\overline{x}^{b}=0$ in $I^{bt}/I^{bt+1}$ and thus $\overline{x}$ is nilpotent in $R/I\oplus I/I^{2}\oplus I^{2}/I^{3}\oplus \cdots \,$ . This proves (4.4.1).
By (4.4.1), we have $E_{\text{red}}\cong \operatorname{Proj}R_{0}\oplus R_{n}\oplus R_{2n}\oplus \cdots \cong \operatorname{Proj}R$ is Du Bois (because $[R_{f}]_{0}$ is Du Bois for every homogeneous $f\in R$ ). We consider the following commutative diagram.
Since $Y$ , $E_{\text{red}}$ and $(\operatorname{Spec}R/I)_{\text{red}}\cong \operatorname{Spec}k$ are all Du Bois by the above discussion, the exact triangle $\text{}\underline{\unicode[STIX]{x1D6FA}}_{R}^{0}\rightarrow \mathbf{R}\unicode[STIX]{x1D70B}_{\ast }\text{}\underline{\unicode[STIX]{x1D6FA}}_{Y}^{0}\oplus \text{}\underline{\unicode[STIX]{x1D6FA}}_{R/I}^{0}\rightarrow \mathbf{R}\unicode[STIX]{x1D70B}_{\ast }\text{}\underline{\unicode[STIX]{x1D6FA}}_{E}^{0}\xrightarrow[{}]{+1}$ reduces to
Next, we study the map $\mathbf{R}\unicode[STIX]{x1D70B}_{\ast }{\mathcal{O}}_{Y}\rightarrow \mathbf{R}\unicode[STIX]{x1D70B}_{\ast }{\mathcal{O}}_{E_{\text{red}}}$ using the Čech complex. We pick $x_{1},\ldots ,x_{m}\in I=(R_{{\geqslant}n})$ in $R\oplus I\oplus I^{2}\oplus \cdots \,$ such that:

(a) the internal degree $\deg x_{i}=n$ for each $x_{i}$ (in other words, $x_{i}\in R_{n}\subseteq R_{{\geqslant}n}=I$ );

(b) the images $\overline{x}_{1},\ldots ,\overline{x}_{m}$ in $(R/I\oplus I/I^{2}\oplus I^{2}/I^{3}\oplus \cdots \,)_{\text{red}}=R_{0}\oplus R_{n}\oplus R_{2n}\oplus \cdots \,$ are algebra generators of $R_{0}\oplus R_{n}\oplus R_{2n}\oplus \cdots \,$ over $R_{0}=k$ .
Note that conditions (a) and (b) together imply that the radical of $(x_{1},\ldots ,x_{m})$ in $R\oplus I\oplus I^{2}\oplus \cdots \,$ is the irrelevant ideal $I\oplus I^{2}\oplus \cdots \,$ . In particular, $\{D_{+}(x_{i})\}_{1\leqslant i\leqslant m}$ forms an affine open cover of $Y$ . The point is that for any $y\in I^{t}=R_{{\geqslant}tn}$ , $y^{n}\in I^{tn}=R_{{\geqslant}tn^{2}}$ as an element in $R\oplus I\oplus I^{2}\oplus \cdots \,$ is always contained in the ideal $(x_{1},\ldots ,x_{m})$ ; this is because the internal degree of $y^{n}$ is divisible by $n$ , so it can be written as a sum of monomials in $x_{i}$ by (b).
The natural map ${\mathcal{O}}_{Y}\rightarrow {\mathcal{O}}_{E_{\text{red}}}$ induces a map between the $s$ th spot of the Čech complexes of ${\mathcal{O}}_{Y}$ and ${\mathcal{O}}_{E_{\text{red}}}$ with respect to the affine cover $\{D_{+}(x_{i})\}_{1\leqslant i\leqslant m}$ . The induced map on Čech complexes can be explicitly described as follows (all the direct sums in the following diagram are taken over all $s$ tuples $1\leqslant i_{1}<\cdots <i_{s}\leqslant m$ ):
The induced map on the second line takes the element $y/(x_{i_{1}}x_{i_{2}}\cdots x_{i_{s}})^{n}$ to $\overline{y}/(\overline{x}_{i_{1}}\overline{x}_{i_{2}}\cdots \overline{x}_{i_{s}})^{n}$ , where $\overline{y}$ denotes the image of $y$ in $R_{nsh}$ . Hence, the same map $\unicode[STIX]{x1D719}$ on the third line is exactly ‘taking the degree $0$ part’. Therefore, we have
while
for every $i\geqslant 1$ , and the map $\mathbf{R}^{i}\unicode[STIX]{x1D70B}_{\ast }{\mathcal{O}}_{Y}\rightarrow \mathbf{R}^{i}\unicode[STIX]{x1D70B}_{\ast }{\mathcal{O}}_{E_{\text{red}}}$ is taking the degree $0$ part. Therefore, taking cohomology of (4.4.2), we have
Moreover, for $i=0,1$ , the cohomology of (4.4.2) gives
A similar Čech complex computation as above shows that
while $H^{0}(E_{\text{red}},{\mathcal{O}}_{E_{\text{red}}})=\ker (\oplus [R_{x_{i}}]_{0}\rightarrow \oplus [R_{x_{i}x_{j}}]_{0})$ . Therefore, $\unicode[STIX]{x1D719}$ is surjective, which implies that
Taking the degree ${>}0$ part of (4.4.4), we get an exact sequence
This implies that $h^{0}(\text{}\underline{\unicode[STIX]{x1D6FA}}_{R}^{0})_{{>}0}\cong (\unicode[STIX]{x1D6E4}(\operatorname{Spec}R\setminus \mathfrak{m},{\mathcal{O}}_{\operatorname{Spec}R}))_{{>}0}$ and so
Finally, we notice that $h^{0}(\text{}\underline{\unicode[STIX]{x1D6FA}}_{R}^{0})\cong R^{\text{sn}}$ is the seminormalization of $R$ [Reference SaitoSai00, 5.2]. We know that $R_{0}^{\text{sn}}\subseteq R^{\text{sn}}$ is reduced. But we can also view $R_{0}^{\text{sn}}$ as the quotient $R^{\text{sn}}/R_{{>}0}^{\text{sn}}$ ; in particular, the prime ideals of $R_{0}^{\text{sn}}$ correspond to prime ideals of $R^{\text{sn}}$ that contain $R_{{>}0}^{\text{sn}}$ , so they all contract to $\mathfrak{m}$ in $R$ . Since seminormalization induces a bijection on spectrum, $R_{0}^{\text{sn}}$ has a unique prime ideal. Thus, $R_{0}^{\text{sn}}$ , being a reduced Artinian local ring, must be a field. Since seminormalization also induces isomorphism on residue fields, $R_{0}^{\text{sn}}=k$ and thus $h^{0}(\text{}\underline{\unicode[STIX]{x1D6FA}}_{R}^{0})_{0}\cong R_{0}^{\text{sn}}=R_{0}$ . Hence, (4.4.6) tells us that
Now (4.4.3), (4.4.5) and (4.4.7) together finish the proof. ◻
Now we prove our result on injectivity of $\operatorname{Ext}$ . Later we will see that this theorem can be viewed as a generalization of the Kodaira vanishing theorem.
Theorem 4.5. Let $(R,\mathfrak{m})$ be a reduced Noetherian $\mathbb{N}$ graded $(R_{0}=\mathbb{C})$ algebra with $\mathfrak{m}$ the unique homogeneous maximal ideal. Suppose that $R_{P}$ is Du Bois for all $P\neq \mathfrak{m}$ . Write $R=A/I$ , where $A=\mathbb{C}[x_{1},\ldots ,x_{n}]$ is a polynomial ring with $\deg x_{i}=d_{i}>0$ and $I$ is a homogeneous ideal. Then the natural degreepreserving map $\operatorname{Ext}_{A}^{j}(R,A)\rightarrow H_{I}^{j}(A)$ induces an injection
for every $j$ , where $d=\sum d_{i}$ .
Proof. We have the hypercohomology spectral sequence
Since $R$ is Du Bois away from $V(\mathfrak{m})$ , $h^{q}(\text{}\underline{\unicode[STIX]{x1D6FA}}_{R}^{0})$ has finite length when $q\geqslant 1$ . Thus, we know that $H_{\mathfrak{m}}^{p}(h^{q}(\text{}\underline{\unicode[STIX]{x1D6FA}}_{R}^{0}))=0$ unless $p=0$ or $q=0$ . We also have that $H_{\mathfrak{m}}^{0}(h^{i}(\text{}\underline{\unicode[STIX]{x1D6FA}}_{R}^{0}))\cong h^{i}(\text{}\underline{\unicode[STIX]{x1D6FA}}_{R}^{0})$ for $i\geqslant 1$ . Hence, the above spectral sequence carries the data of a long exact sequence
We also have a short exact sequence $0\rightarrow R\rightarrow h^{0}(\text{}\underline{\unicode[STIX]{x1D6FA}}_{R}^{0})\rightarrow [H_{\mathfrak{m}}^{1}(R)]_{{>}0}\rightarrow 0$ by Proposition 4.4. Therefore, the long exact sequence of local cohomology and the observation that $[H_{\mathfrak{m}}^{1}(R)]_{{>}0}$ has finite length tell us that
and
Now taking the degree ${\leqslant}0$ part of (4.5.1) and again using Proposition 4.4 yields
At this point, note that by the Matlis dual of Lemma 3.2 (see the proof of Lemma 3.3 applied to $S=A/J$ for $\sqrt{J}=I$ , and thus $S_{\text{red}}=A/I=R$ ), we always have a surjection $H_{\mathfrak{m}}^{i}(A/J){\twoheadrightarrow}\mathbb{H}_{\mathfrak{m}}^{i}(\text{}\underline{\unicode[STIX]{x1D6FA}}_{R}^{0})$ for every $i$ and $\sqrt{J}=I$ . Taking the degree ${\leqslant}0$ part and applying (4.5.2), we thus get that
is surjective for every $i$ and $\sqrt{J}=I$ . Now taking $J=I^{t}$ and applying graded local duality (we refer to [Reference Brodmann and SharpBS98] for definitions and standard properties of graded canonical modules, graded injective hulls and graded local duality, but we emphasize here that the graded canonical module of $A$ is $A(d)$ ), we have that
is injective for every $j$ and $t$ . So, after taking a direct limit and a degree shift, we have
is injective for every $j$ . This finishes the proof.◻
Remark 4.6. The dual form of Theorem 4.5 says that $[H_{\mathfrak{m}}^{i}(A/J)]_{t}{\twoheadrightarrow}[H_{\mathfrak{m}}^{i}(R)]_{t}$ is surjective for every $i$ , every $t\leqslant 0$ and every $\sqrt{J}=I$ ; see (4.5.3). When $R=A/I$ has an isolated singularity, $A$ is standard graded and $t=0$ , this was proved in [Reference VarbaroVar13, Proposition 3.8]. Therefore, our Theorem 4.5 greatly generalizes this result.
In general, we cannot expect that $[\operatorname{Ext}_{A}^{j}(R,A)]_{{<}d}{\hookrightarrow}[H_{I}^{j}(A)]_{{<}d}$ is injective under the hypothesis of Theorem 4.5 (even if $R$ is an isolated singularity). Consider the following example.
Example 4.7 (Cf. [Reference Singh and WaltherSW07, Example 3.5]).
Let $R=\mathbb{C}[s^{4},s^{3}t,st^{3},t^{4}]$ . Then we can write $R=A/I$ , where $A=\mathbb{C}[x,y.z,w]$ with standard grading (i.e., $x,y,z,w$ all have degree one). It is straightforward to check that $R$ is an isolated singularity with
(in particular, $\operatorname{depth}R=1$ ). By graded local duality, we have $[\operatorname{Ext}_{A}^{3}(R,A)]_{{<}4}\neq 0$ . On the other hand, using standard vanishing theorems in [Reference Huneke and LyubeznikHL90, Theorem 2.9], we know that $H_{I}^{3}(A)=0$ . Therefore, the map $[\operatorname{Ext}_{A}^{3}(R,A)]_{{<}4}\rightarrow [H_{I}^{3}(A)]_{{<}4}$ is not injective.
An important consequence of Theorem 4.5 (in fact, we only need the injectivity in degree ${>}d$ ) is the following vanishing result.
Theorem 4.8. Let $(R,\mathfrak{m})$ be a Noetherian $\mathbb{N}$ graded $(R_{0}=\mathbb{C})$ algebra with $\mathfrak{m}$ the unique homogeneous maximal ideal. Suppose that $R_{P}$ is Du Bois for all $P\neq \mathfrak{m}$ and $H_{\mathfrak{m}}^{i}(R)$ has finite length for some $i$ . Then $[H_{\mathfrak{m}}^{i}(R)]_{{<}0}=0$ .
In particular, if $R$ is equidimensional and is Cohen–Macaulay Du Bois on $\operatorname{Spec}R\{\mathfrak{m}\}$ , then $[H_{\mathfrak{m}}^{i}(R)]_{{<}0}=0$ for every $i<\dim R$ .
Proof. Let $R=A/I$ , where $A=\mathbb{C}[x_{1},\ldots ,x_{n}]$ is a (not necessarily standard graded) polynomial ring and $I$ is a homogeneous ideal. Set $\deg x_{i}=d_{i}>0$ and $d=\sum d_{i}$ . The graded canonical module of $A$ is $A(d)$ . By graded local duality,
Therefore, if $[H_{\mathfrak{m}}^{i}(R)]_{j}\neq 0$ for some $j>0$ , then $[\operatorname{Ext}_{A}^{ni}(R,A)]_{jd}\neq 0$ for some $j>0$ . By Theorem 4.5, we have an injection