LOCAL COHOMOLOGY OF DU BOIS SINGULARITIES AND
APPLICATIONS TO FAMILIES

arXiv:1605.02755v3 [math.AG] 14 Apr 2017

LINQUAN MA, KARL SCHWEDE, AND KAZUMA SHIMOMOTO

Abstract. In this paper we study the local cohomology modules of Du Bois singularities.
i
i
Let (R, m) be a local ring, we prove that if Rred is Du Bois, then Hm
(R) −→ Hm
(Rred ) is
surjective for every i. We find many applications of this result. For example we answer a
question of Kovács and the second author [KS16b] on the Cohen-Macaulay property of Du
Bois singularities. We obtain results on the injectivity of Ext that provide substantial partial answers of questions in [EMS00] in characteristic 0. These results can also be viewed as
generalizations of the Kodaira vanishing theorem for Cohen-Macaulay Du Bois varieties. We
prove results on the set-theoretic Cohen-Macaulayness of the defining ideal of Du Bois singularities, which are characteristic 0 analogs and generalizations of results of Singh-Walther
and answer some of their questions in [SW05]. We extend results of Hochster-Roberts on
the relation between Koszul cohomology and local cohomology for F -injective and Du Bois
singularities [HR76]. We also prove that singularities of dense F -injective type deform.

1. Introduction
The notion of Du Bois singularities was introduced by Steenbrink based on work of Du Bois
[DB81] which itself was an attempt to localize Deligne’s Hodge theory of singular varieties
[Del74]. Steenbrink studied them initially because families with Du Bois fibers have remarkable base-change properties [Ste81]. On the other hand, Du Bois singularities have recently
found new connections across several areas of mathematics [KK10, KS16a, Lee09, CHWW11,
HJ14]. 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 (S, m) is a local ring essentially of finite type over C
√
and that Sred = S/ 0 has Du Bois singularities. Then
is surjective for every i.

Hmi (S) −
→ Hmi (Sred )

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 ⊆ X be a Cartier divisor.
If H is Du Bois and X\H is Cohen-Macaulay, then X is Cohen-Macaulay and hence so is
H.
The first named author was supported in part by the NSF Grant DMS #1600198 and NSF CAREER
Grant DMS #1252860/1501102.
The second named author was supported in part by the NSF FRG Grant DMS #1265261/1501115 and
NSF CAREER Grant DMS #1252860/1501102.
The third named author is partially supported by Grant-in-Aid for Young Scientists (B) # 25800028.
1

This first consequence answers a question of Kovács and the second author and should be
viewed as a generalization of several results in [KK10, Section 7]. In particular, we do not
need a projective family. This result holds if one replaces Cohen-Macaulay with Sk and also
generalizes to the context of Du Bois pairs as in [KS16b]. Theorem A should also be viewed as
a characteristic 0 analog of [FW89, Proposition 2.13] and [HMS14, Corollary A.4]. Theorem
A also implies that if X \ H has rational singularities and H has Du Bois singularities, then
X has rational singularities (see Corollary 3.10), generalizing [KS16b, Theorem E].
Our next consequence of the key lemma is the following.
Theorem B (Proposition 4.1). Let (R, m) be a Gorenstein local ring essentially of finite
type over C, and let I ⊆ R be an ideal such that R/I has Du Bois singularities. Then the
natural map ExtjR (R/I, R) −
→ HIj (R) is injective for every j.
This gives a partial characteristic 0 answer to [EMS00, Question 6.2], asking when such
maps are injective. A special case of the analogous characteristic p > 0 result was obtained
by Singh-Walther [SW07]. This result leads to an answer of [DN16, Questioin 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, m) be a reduced Noetherian N-graded (R0 = C)-algebra
with m the unique homogeneous maximal ideal. Suppose RP is Du Bois for all P 6= m.
Write R = A/I where A = C[x1 , . . . , xn ] is a polynomial ring with deg xi = di > 0 and I is a
homogeneous ideal. Then the natural degree-preserving map ExtjA (R, A) −
→ HIj (A) induces
an injection




ExtjA (R, A) ≥−d ֒→ HIj (A) ≥−d
P
for every j, where d = di .

Theorem C leads to new vanishing for local cohomology for N-graded isolated non-Du
Bois singularities Theorem 4.8, a generalization of the Kodaira vanishing theorem for CohenMacaulay 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 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 ≫ 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 C [BST13]. On the other hand, several recent papers have
studied how Du Bois and F -injective singularities deform [KS16a, HMS14]. We know that if
a Cartier divisor H ⊆ 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 [Fed83]. In fact F -injective singularities were introduced because it
was observed that Cohen-Macaulay F -injective deform in this way but F -pure singularities1
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.
1We

now know that F -pure singularities are analogs of log canonical singularities [HW02].
2

Theorem D (Theorem 5.3). Let (R, m) be a local ring essentially of finite type over C and
let x be a nonzerodivisor on R. Suppose R/xR has dense F -injective type. Then for infinitely
many p > 0, the Frobenius action xp−1 F on Hmi p (Rp ) is injective for every i, where (Rp , mp )
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 [HR76],
and we can give a characteristic p > 0 generalization as well.
Theorem E (Corollary 4.13). Let R be a Noetherian N-graded k-algebra, with m the unique
homogeneous maximal ideal. Suppose R is equidimensional and Cohen-Macaulay on Spec R−
{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 (x, R) 0 ∼
= Hmr (R) for every r < n = dim R and every homogeneous system of
parameters x = x1 , . . . , xn , where H r (x, R) denotes the r-th Koszul cohomology of 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 Theorem 4.5, Theorem 4.8.
Acknowledgements: The authors are thankful to Sándor Kovács, Shunsuke Takagi and the
referee for valuable discussions and comments on a previous draft of the paper. We would
like to thank Pham Hung Quy for discussions which motivate the proof of Theorem 3.8. We
also thank Zsolt Patakfalvi, Anurag K. Singh and Kevin Tucker for valuable discussions.
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 C. Of course, nearly all of our results also hold over all other
fields of characteristic zero 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 [KS11] and [Kol13]. Frequently we will work in the setting
of pairs in the Du Bois sense.
Definition 2.1. Suppose X and Z are schemes of essentially finite type over 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 C. Associated to X is an object
b
∈ Dcoh
(X) with a map OX −
→ Ω0X functorial in the following way. If f : Y −
→ X is a
map of schemes then we have a commutative square

Ω0X

OX
/

Ω0X




Rf∗ OY
/
3

Rf∗ Ω0Y

→X
If X is non-reduced, then Ω0X = Ω0Xred by definition. To define Ω0X in general let π : X q −
0
be a hyperresolution and set ΩX = Rπ∗ OX q (alternatively see [Sch07]).
If Z ⊆ X is a closed subscheme, then we define Ω0X,Z as the object completing the following
distinguished triangle
+1

→ Ω0Z −→ .
→ Ω0X −
Ω0X,Z −

We observe that there is an induced map IZ⊆X −
→ Ω0X,Z . We also note that by this definition
and the fact Ω0X = Ω0Xred , we have Ω0X,Z = Ω0Xred ,Zred .
Definition 2.2 (Du Bois singularities). We say that X has Du Bois singularities if the map
OX −
→ Ω0X is a quasi-isomorphism. If Z ⊆ X is a closed subscheme we say that (X, Z) has
Du Bois singularities if the map IZ⊆X −
→ Ω0X,Z is a quasi-isomorphism.
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 X or Z is reduced. However, if a pair (X, Z) is
Du Bois and X is reduced, then so is Z [KS16b, Lemma 2.9].
2.2. Cyclic covers of non-reduced schemes. In this paper we will need to take cyclic
covers of non-reduced 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 [KM98, Section 2.4] or [dFEM, Subsection 2.1.1]). Note also that the reason we are able
to work with non-reduced schemes is because our L is actually locally free and not just a
reflexive rank-1 sheaf.
Setting 2.4. Suppose X is a scheme of finite type over C (typically projective). Suppose
also that L is a line bundle (typically semi-ample). Choose a (typically general) global
section s ∈ H 0 (X, L n ) for some n > 0 (typically n ≫ 0) and form the sheaf of rings:
R(X, L , s) = OX ⊕ L −1 ⊕ · · · ⊕ L −n+1

where for i, j < n and i + j > n, the multiplication L −i ⊗ L −j −
→ L −i−j+n is performed
by the formula a ⊗ b 7→ abs. We define ν : Y = YL ,s = SpecR(X, L , s) −
→ 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 L , where U = Spec R ⊆ X with
corresponding ν −1 (U) = Spec S. Fix an isomorphism L |U ∼
= OU and write
S = R ⊕ Rt ⊕ Rt2 ⊕ · · · ⊕ Rtn−1

where t is a dummy variable used to help keep track of the degree. The choice of the section
s|U ∈ H 0 (X, L n ) is the same as the choice of map OX −
→ L n (send 1 to the section s). If
n
s is chosen to be general and L is very ample then this map is an inclusion, but it is not
injective in general. Working locally where we have fixed L |U = OU , we also have implicitly
chosen L n = OU and hence we have just specified that tn = s|U ∈ Γ(U, L n ) = Γ(U, OU ). In
other words
S = R[t]/htn − si.
In particular, it follows that ν is a finite map.

Lemma 2.5. The map ν is functorial with respect to taking closed subschemes of X. In
particular if Z ⊆ X is a closed subscheme and L and s are as above, then we have a
4

commutative diagram
YL ,s

SpecR(X, L , s)

WL |Z ,s|Z

SpecR(Z, L ⊗ OZ , s|Z )

νX

/

O
?

XO
?

νZ

/

Z.

Furthermore, WL |Z ,s|Z = π −1 (Z) is the scheme theoretic preimage of Z.
Proof. The first statement follows since R(X, L , s) = OX ⊕L −1 ⊕. . .⊕L −n+1 surjects onto
−n+1
R(Z, L |Z , s|Z ) = OZ ⊕ L |−1
in the obvious way. For the second statement
Z ⊕ . . . ⊕ L |Z
we simply notice that
R(X, L , s) ⊗OX OZ = R(Z, L |Z , s|Z ).



Lemma 2.6. Suppose we are working in Setting 2.4 and that X is projective and also reduced
(respectively normal), L n is globally generated and s is chosen to be general. Then Y = YL ,s
is also reduced (respectively normal).
Proof. To show that Y is reduced we will show it is S1 and R0 . We work locally on an open
affine chart U = Spec R ⊆ X.
To show it is R0 , it suffices to consider the case 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]/htn − si is a product of fields. But this is obvious since we
are working in characteristic zero.
Next we show it is S1 . Indeed if (R, m) is a local ring of depth ≥ 1 then obviously
S = R[t]/htn − si has depth ≥ 1 as well since it is a free R-module. Finally, the depth
condition is preserved after localizing at the maximal ideals of the semi-local ring of S.
This proves the reduced case. The normal case is well known but we sketch it. The R1
condition follows from [KM98, Lemma 2.51] and the fact that s is chosen generally, utilizing
Bertini’s theorem. The S2 condition follows in the same way that S1 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 [Sch09], [BST13]. In this short subsection we collect some definitions about singularities in positive characteristic. Our main
focus are F -pure and F -injective singularities.
A local ring (R, m) is called F -pure if the Frobenius endomorphism F : R → R is pure.2
Under mild conditions, for example when R is F -finite, which means the Frobenius map
F
R−
→ R is a finite map, R being F -pure is equivalent to the condition that the Frobenius
F
endomorphism R −
→ R is split [HR76, Corollary 5.3]. The Frobenius endomorphism on R
induces a natural Frobenius action on each local cohomology module Hmi (R) and we say a
local ring is F -injective if this natural Frobenius action on Hmi (R) is injective for every i
[Fed83]. This holds if R is F -pure [HR76, Lemma 2.2]. For some other basic properties of
F -pure and F -injective singularities, see [HR76, Fed83, EH08].
2A

map of R-modules N → N ′ is pure if for every R-module M the map N ⊗R M → N ′ ⊗R M is injective.
This implies that N → N ′ is injective, and is weaker than the condition that 0 → N → N ′ be split.
5

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 [Kol13, 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 principals but we believe the path
we take below is quicker.
Lemma 3.1. Let X be a projective scheme over C and let Z ⊆ X be a closed subscheme
(X, Z are not necessarily reduced). Then the natural map
H i (X, IZ ) −
→ Hi (Xred , Ω0Xred ,Zred ) ∼
= Hi (X, Ω0X,Z )
is surjective for every i ∈ Z.
Proof. Note that if the result holds for C, it certainly also holds for other fields of characteristic zero. 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\Z.
Note that the rows are not exact.
Hci (Ured , C)
/

O

H i (Xred , IZred )
O

/

Hi (Xred , Ω0Xred ,Zred )
O
∼
=

∼
=

Hci (U, C)
/

H i (X, IZ )
/

Hi (X, Ω0X,Z )

The composite map in the top horizontal line is a surjection by [KS16b, Lemma 2.17] or
[Kov11, Corollary 4.2]. The vertical isomorphism on the left holds because the constant
sheaf C does not see the non-reduced structure. Likewise for the vertical isomorphism on
the right where Ω0X,Z does not see the non-reduced structure. The diagram then shows that
H i (X, IZ ) −
→ Hi (Xred , Ω0Xred ,Zred ) is a surjection.

Next we prove the key injectivity lemma for possibly non-reduced pairs. The proof is
essentially the same as in [KS16b] or [KS16a]. We reproduce it here carefully because we
need it in the non-reduced setting.
Lemma 3.2. Let X be a scheme of essentially finite type over C and Z ⊆ X a closed
subscheme (X and Z are not necessarily reduced). Then the natural map
hj (ω X,Z ) ֒→ hj (R H omOX (IZ , ωX ))
q

q

is injective for every j ∈ Z, where ω X,Z = R H omOX (Ω0X,Z , ωX ).
q

q

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 L . Let s ∈ H 0 (X, L n ) be a general
n−1
global section for some n ≫ 0 and let η: Y = Spec ⊕i=0
L −i −
→ X be the n-th cyclic cover
−1
corresponding to s. Set W = η (Z). Then for n ≫ 0 and s general, the restriction of η to
n−1
W is the cyclic cover W = Spec ⊕i=0
L |−i
→ Z by Lemma 2.5. Likewise η also induces
Z −
the corresponding cyclic covers of the closed subschemes Yred −
→ Xred and Wred −
→ Zred by
n−1
Lemma 2.5 and Lemma 2.6. We have η∗ IW = ⊕i=0
IZ ⊗ L −i and
∼
⊗ L −i ∼
η∗ Ω0 ∼
= η∗ Ω0
= ⊕n−1 Ω0
= ⊕n−1 Ω0 ⊗ L −i
Y,W

Yred ,Wred

i=0

Xred ,Zred

6

i=0

X,Z

where the second isomorphism is [KS16b, Lemma 3.1] (see also [KS16a, Lemma 3.1]). Since
Lemma 3.1 implies that H j (Y, IW ) ։ Hj (Y, Ω0Y,W ) is surjective for every j ∈ Z, we know
that H j (X, IZ ⊗ L −i ) ։ Hj (X, Ω0X,Z ⊗ L −i ) is surjective for every i ≥ 0 and j ∈ Z.
Applying Grothendieck-Serre duality we obtain an injection
Hj (X, ω X,Z ⊗ L i ) ֒→ Hj (X, R H omOX (IZ , ωX ) ⊗ L i )
q

q

for all i ≥ 0 and j ∈ Z. Since L is ample, for i ≫ 0 the spectral sequence computing the
above hypercohomology degenerates. Hence for i ≫ 0 we get
H 0 (X, hj (ω X,Z ) ⊗ L i ) ֒→ H 0 (X, hj (R H omOX (IZ , ωX )) ⊗ L i ).
q

q

But again since L is ample, the above injection for i ≫ 0 implies the injection hj (ω X,Z ) ֒→
q
hj (R H omOX (IZ , ωX )).

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 (Spec S, Spec S/J).
Lemma 3.3. Let (S, m) be a local ring essentially of finite type over C and let J ⊆ S be an
ideal. Suppose further that S ′ = S/N where N ⊆ S is an ideal contained in the nilradical
(for instance, N could be the nilradical and then S ′ = Sred ). Suppose (S ′ , S ′ /JS ′ ) is a Du
Bois pair. Then the natural map Hmi (J) −
→ Hmi (JS ′ ) is surjective for every i. In particular,
i
i
if Sred is Du Bois, then Hm (S) −
→ Hm (Sred ) is surjective for every i.
Proof. We consider the following commutative diagram:
Hmi (J)
/

Hmi (JS ′ )
∼
=



Him (Ω0S,S/J )



∼
=

/

Him (Ω0S ′ ,S ′/JS ′ )

Here the left vertical map is surjective by the Matlis dual of Lemma 3.2 applied to X =
Spec S and Z = Spec S/J, the right vertical map is an isomorphism because (S ′ , S ′ /JS ′ ) is
′
a Du Bois pair, and the bottom map is an isomorphism because Sred = Sred
. Chasing the
i
i
′
diagram shows that Hm (J) −
→ Hm (JS ) is surjective. The last sentence is the case that J is
the unit ideal (i.e., the non-pair version).

Remark 3.4. The characteristic p > 0 analog of the above lemma (in the case J = S, i.e.,
the non-pair version) holds if Sred is F -pure. We give a short argument here. Without
loss
√
of generality we may assume S and Sred are complete, so S = A/I and
√ Sered = A/ I where
A is a complete regular local ring. We may pick e ≫ 0 such that ( I)[p ] ⊆ I. We have a
composite map
√ e
√
φ
Hmi (A/( I)[p ] ) −
→ Hmi (A/I) ∼
→ Hmi (A/ I) ∼
= Hmi (S) −
= Hmi (Sred ).

We know from [Lyu06, Lemma 2.2] that the image of the composite map is equal to
spanA hf e (Hmi (Sred ))i, where f e denotes the natural e-th Frobenius action on Hmi (Sred ). In
particular, the Frobenius action on Hmi (Sred )/ im φ is nilpotent. However, since im φ is an
F -stable submodule of Hmi (Sred ) and Sred is F -pure, [Ma14, Theorem 3.7] shows that Frobenius acts injectively on Hmi (Sred )/ im φ. Hence we must have Hmi (Sred ) = im φ, that is,
Hmi (S) −
→ Hmi (Sred ) is surjective.
7

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 [EH08, 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) as in [EH08, Example 2.16]. Now let R=K + (x, y)L[[x, y]]
with m = (x, y)L[[x, y]]. Then it is easy to see that (R, m) is a local ring of dimension 2 and
we have a short exact sequence:
0−
→R−
→ L[[x, y]] −
→ L/K −
→ 0.
The long exact sequence of local cohomology gives
Hm1 (R) ∼
= L/K,

Hm2 (R) ∼
= Hm2 (L[[x, y]]).

Moreover, one can check that the Frobenius action on Hm1 (R) is exactly the Frobenius action
on L/K. The Frobenius action on L/K is injective since Lp ∩ K = K p , and the Frobenius
action on Hm2 (L[[x, y]]) is injective because L[[x, y]] is regular. Hence the Frobenius action
on both Hm1 (R) and Hm2 (R) are injective. This proves that R is F -injective.
Write R = A/I for a regular local ring (A, m). One checks that the Frobenius action F :
1
Hm (R) −
→ Hm1 (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 6= Lp [K] by our choice of K and L (see [EH08,
Example 2.16] for a detailed computation on this). But now [Lyu06, Lemma 2.2] shows that
Hm1 (A/I [p]) −
→ Hm1 (A/I) is not surjective. Therefore we can take S = A/I [p] with Sred = R
that is F -injective, but Hm1 (S) −
→ Hm1 (Sred ) 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, m) is a local ring such that Rred has these more
restrictive F -injective singularities, then it should follow that Hmi (R) −
→ Hmi (Rred ) surjects
for all i.
Theorem 3.6. Suppose that (X, Z) is a reduced pair and that H ⊆ X is a Cartier divisor
such that H does not contain any irreducible components of either X or Z. If (H, Z ∩ H) is
a Du Bois pair, then for all i and all points η ∈ H ⊆ X, the following segment of the long
exact sequence
→0
0−
→ Hηi (IZ,η · OH,η ) ֒→ Hηi+1 (IZ,η · OX,η (−H)) ։ Hηi+1 (IZ,η ) −
is exact for all i. Dually, in the special case that IZ = OX , we can also phrase this as saying
that
q
q
q
0−
→ h−i (ωX ) −
→ h−i (ωX (H)) −
→ h−i+1 (ωH ) −
→0
is exact for all i.

Proof. Localizing at η, we may assume that X = Spec R, Z = Spec R/I, H = div(x) with
(R, m) a local ring and η = {m}. Moreover, the hypotheses imply that x is a nonzerodivisor
on both R and R/I. It is enough to show that the segment of the long exact sequence
·x

(3.6.1)
·x

→0
→ Hmi+1 (I) −
→ Hmi+1 (I) −
0−
→ Hmi (I/xI) −

induced by 0 −
→I−
→I −
→ I/xI −
→ 0 is exact.
8

Consider S = R/xn R and J = I(R/xn R). The pair (S ′ = R/xR, S ′ /IS ′ ) = (H, Z ∩ H) is
Du Bois by hypothesis (note we do not assume that H is reduced). Thus Lemma 3.3 implies
that
Hmi (I(R/xn R)) ։ Hmi (I(R/xR))

(3.6.2)

is surjective for every i, n. Since x is a nonzerodivisor on R/I, xn I = I ∩ xn R. Thus
I(R/xn R) =
Hence (3.6.2) shows that

I + xn R ∼
I
= I/xn I.
=
n
n
x R
I ∩x R

Hmi (I/xn I) ։ Hmi (I/xI)
is surjective for every i, n. The long exact sequence of local cohomology induced by 0 −
→
·x
I/xn−1 I −
→ I/xn I −
→ I/xI −
→ 0 tells us that
·x

Hmi (I/xn−1 I) −
→ Hmi (I/xn I)
is injective for every i, n. Hence after taking a direct limit we obtain that:
φi : Hmi (I/xI) −
→ lim Hmi (I/xn I) ∼
(I/xn I)) ∼
= Hmi+1 (I)
= Hmi (lim
= Hmi (Hx1 (I)) ∼
−→
−→
n

n

is injective for every i. Here the last two isomorphism come from the fact that x is a
nonzerodivisor on I (since it is a nonzerodivisor on R) and a simple computation using the
local cohomology spectral sequence.
Claim 3.7. The φi ’s are exactly the connection maps in the long exact sequence of local
·x
cohomology induced by 0 −
→I−
→I−
→ I/xI −
→ 0:
φi

·x

··· −
→ Hmi (I) −
→ Hmi (I/xI) −
→ Hmi+1 (I) −
→ Hmi+1 (I) −
→ ··· .
This claim immediately produces (3.6.1) because we proved that φi is injective for every
i. Thus it remains to prove the Claim.
Proof of claim. Observe that by definition, φi is the natural map in the long exact sequence
of local cohomology
φi

·x

··· −
→ Hmi (I/xI) −
→ Hmi (Ix /I) −
→ Hmi (Ix /I) −
→ ···
·x

which is induced by 0 −
→ I/xI −
→ Ix /I −
→ Ix /I −
→ 0 (note that x is a nonzerodivisor on I
·x
1
∼
→
and Hx (I) = Ix /I). However, it is easy to see that the multiplication by x map Hmi (Ix /I) −
·x
Hmi (Ix /I) can be identified with the multiplication by x map Hmi+1 (I) −
→ Hmi+1 (I) because
we have a natural identification Hmi (Ix /I) ∼
= Hmi+1 (I). This finishes the proof
= Hmi (Hx1 (I)) ∼
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 [KS16b, Question 5.7] and is a generalization to pairs of a characteristic zero
analog of [HMS14, Corollary A.5].
9

Theorem 3.8. Suppose that (X, Z) is a reduced pair and that H ⊆ X is a Cartier divisor
such that H does not contain any irreducible components of either X or Z. If (H, Z ∩ H) is
a Du Bois pair and IZ |X\H is Sk , then IZ is Sk .
Proof. Suppose IZ is not Sk , choose an irreducible component Y of the non-Sk locus of IZ .
Since IZ |X\H is Sk , we know that Y ⊆ H. Let η be the generic point of Y . Theorem 3.6
tells us that
Hηi (IZ · OX (−H)) ։ Hηi (IZ )
is surjective for every i. Note that if we localize at η and use the same notation as in the
·x
proof of Theorem 3.6, this surjection is the multiplication by x map Hmi (I) −
→ Hmi (I).
However, after we localize at η, I is Sk on the punctured spectrum, Hmi (I) has finite length
for every i < k. In particular, for h ≫ 0, xh annihilates Hmi (I) for every i < k. Therefore for
·x
i < k, the multiplication by x map Hmi (I) −
→ Hmi (I) cannot be surjective unless Hmi (I) = 0,
which means I is Sk as an R-module. But this contradicts our choice of Y (because we pick
Y an irreducible component of the non-Sk locus of IZ ). Therefore IZ is Sk .

Corollary 3.9. Let X be a reduced scheme and let H ⊆ X be a Cartier divisor. If H is Du
Bois and X\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
[KK10, Corollary 1.3, Corollary 7.13] and [KS16b, Theorem 5.5]. In those results, one
considered a flat projective family X −
→ B with Du Bois fibers such that the general fiber is
Sk . They then show that the special fiber is Sk . Let us explain this result in a simple case.
Suppose that B is smooth curve and Z = 0, the special fiber is a Cartier divisor H and
X \ 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 [KS16b, 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 dlt singularities, see [Kol13]), the interested reader can look it up, or simply assume that D = 0.
Corollary 3.10. Suppose (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 ∩ H) is Du Bois and such that (X \ H, D \ 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 the [KS16b, Proof of Theorem 7.1]. It goes through word for word once one
observes that OX (−D) is Cohen-Macaulay which follows immediately from Theorem 3.8. 
This corollary is also an analog of [FW89, Proposition 2.13] and [HMS14, 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 [HR76, SW05, SW07, Var13, DN16]
as well as answer their questions. Moreover, our results can give a generalization of the
classical Kodaira vanishing theorem.
10

4.1. Injectivity of Ext and a generalization of the Kodaira vanishing theorem.
Our first application gives a solution to [EMS00, Question 6.2] on the injectivity of Ext in
characteristic 0, which parallels to its characteristic p > 0 analog [SW07, Theorem 1.3].
Proposition 4.1. Let (R, m) be a Gorenstein local ring essentially of finite type over C,
and let I ⊆ R be an ideal such that R/I has Du Bois singularities. Then the natural map
ExtjR (R/I, R) −
→ HIj (R) is injective for every j.
Proof. By Lemma 3.3 applied to S = R/I t and applying local duality, we know that
ExtjR (R/I, R) −
→ ExtjR (R/I t , R)
is injective for every j and every t > 0. Now taking a direct limit we obtain the desired
injectivity ExtjR (R/I, R) −
→ HIj (R).

Remark 4.2. [SW07, Theorem 1.3] proves the same injectivity result in characteristic p > 0
when R/I is F -pure (and when R is regular). Perhaps it is worth to 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 S = A/I is F -injective but Hm1 (A/I [p]) −
→ Hm1 (A/I) is
not surjective. Hence applying local duality shows that
A−1
A−1
Extdim
(A/I, A) −
→ Extdim
(A/I [p], A)
A
A

dim A−1
is not injective, so neither is ExtA
(A/I, A) −
→ HIdim A−1 (A).

An immediate corollary of Proposition 4.1 is the following, which is a characteristic 0
analog of [DN16, Theorem 7.3] (this also answers [DN16, Question 7.5], which is motivated
by Stillman’s conjecture).
Corollary 4.3. Let R be a regular ring of essentially finite type over C. Let I ⊆ R be an
ideal such that S = R/I has Du Bois singularities. Then pdR S ≤ ν(I) where ν(I) denotes
the minimal number of generators of I.
j
Proof. For every j > ν(I) and every maximal ideal m of R, we have HIR
(Rm ) = 0. Therefore
m
j
by Proposition 4.1, we have ExtRm (Sm , Rm ) = 0 for every j > ν(I). Since pdRm Sm < ∞, we
have
pdRm Sm = sup{j| ExtjRm (Sm , Rm ) 6= 0} ≤ ν(I)3.

Because this is true for every maximal ideal m, we have pdR S ≤ ν(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 N-graded rings. This result
and related results should be well-known to experts (or at least well-believed by experts):
for example [Ma15, Theorem 4.4], [BST13, Lemma 2.14], [KS16b, Lemma 2.12] or [GK14].
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 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.
3This

follows from applying the functor HomRm (−, Rm ) to a minimal free resolution of Sm and observing
that the matrix defining the maps have elements contained in m. See also [DGI06, 2.4]
11

Proposition 4.4. Let (R, m) be a reduced Noetherian N-graded (R0 = k)-algebra with m the
unique homogeneous maximal ideal. Suppose RP is Du Bois for all P 6= m. Then we have


hi (Ω0 ) ∼
= H i+1(R)
R

>0

m

for every i > 0 and



h0 (Ω0R )/R ∼
= Hm1 (R) >0 .


In particular, R is Du Bois if and only if Hmi (R) >0 = 0 for every i > 0.

Proof. Let R♮ denote the Rees algebra of R with respect to the natural filtration R≥t . That
is, R♮ = R ⊕ R≥1 ⊕ R≥2 ⊕ · · · . Let Y = Proj R♮ . We first claim that Y is Du Bois: Y is
covered by D+ (f ) for homogeneous
elements f ∈ R≥t and t ∈ N. 
 
♮ ∼
If deg f > t, then Rf 0 = Rf is Du Bois. If deg f = t, then Rf♮ 0 ∼
= Rf ≥0 is also Du
Bois4 (see [KSSW09, Lemma 5.4] for an analogous analysis on rational singularities).
Since R and thus R♮ are Noetherian, there exists n such that R≥nt = (R≥n )t for every
t ≥ 1 [Bou98, Chapter III, Section 3, Proposition 3]. Let I = (R≥n ), then we immediately see
that Y ∼
= Proj R ⊕ I ⊕ I 2 ⊕ . . . is the blow up of
 Spec R at I. We define the exceptional
2
2
3
∼
divisor to be E = Proj R/I ⊕ I/I ⊕ I /I ⊕ · · · . We next claim that
(4.4.1)

(R/I ⊕ I/I 2 ⊕ I 2 /I 3 ⊕ · · · )red ∼
= R0 ⊕ Rn ⊕ R2n ⊕ · · · .

The point is that, for x ∈ I t /I t+1 = R≥nt /R≥n(t+1) , if x ∈ Rnt+a for some a > 0, then we can
pick b > 0 such that ba > n, we then have
xb ∈ Rbnt+ba ⊆ R≥n(bt+1) .

But this means xb = 0 in I bt /I bt+1 and thus x is nilpotent in R/I ⊕ I/I 2 ⊕ I 2 /I 3 ⊕ · · · . This
proves (4.4.1).
 
By (4.4.1) we have Ered ∼
= Proj R is Du Bois (because Rf 0
= Proj R0 ⊕ Rn ⊕ R2n ⊕ · · · ∼
is Du Bois for every homogeneous f ∈ R). We consider the following commutative diagram:
E
/

Y
π

π




Spec R/I
/

Spec R

Since Y , Ered and (Spec R/I)red ∼
= Spec k are all Du Bois by the above discussion, the exact
+1
0
0
0
triangle ΩR −
→ Rπ∗ ΩY ⊕ ΩR/I −
→ Rπ∗ Ω0E −→ reduces to
(4.4.2)

+1

Ω0R −
→ Rπ∗ OY ⊕ k −
→ Rπ∗ OEred −→ .

Next we study the map Rπ∗ OY −
→ Rπ∗ OEred using the Čech complex. We pick x1 , . . . , xm ∈
2
I = (R≥n ) in R ⊕ I ⊕ I ⊕ · · · such that
(a) The internal degree, deg xi = n for each xi (in other words, xi ∈ Rn ⊆ R≥n = I);
(b) The images x1 , . . . , xm in (R/I ⊕ I/I 2 ⊕ I 2 /I 3 ⊕ · · · )red = R0 ⊕ Rn ⊕ R2n ⊕ · · · are
algebra generators of R0 ⊕ Rn ⊕ R2n ⊕ · · · over R0 = k.
general, if S is a Z-graded ring, then S≥0 ∼
= S[z]0 where deg z = −1. Hence if S is Du Bois, then S≥0 ,
being a summand of S[z], is also Du Bois [Kov99].
4In

12

Note that conditions (a) and (b) together imply that the radical of (x1 , . . . , xm ) in R ⊕ I ⊕
I 2 ⊕ · · · is the irrelevant ideal I ⊕ I 2 ⊕ · · · . In particular, {D+ (xi )}1≤i≤m forms an affine
open cover of Y . The point is that for any y ∈ I t = R≥tn , y n ∈ I tn = R≥tn2 as an element
in R ⊕ I ⊕ I 2 ⊕ · · · is always contained in the ideal (x1 , . . . , xm ), this is because the internal
degree of y n is divisible by n, so it can be written as a sum of monomials in xi by (b).
The natural map OY → OEred induces a map between the s-th spot of the Čech complexes
of OY and OEred with respect to the affine cover {D+ (xi )}1≤i≤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 ≤ i1 < · · · < is ≤ m):
L
L
/
OY (D+ (xi1 xi2 · · · xis ))
OEred (D+ (xi1 xi2 · · · xis ))


L

(

y
(xi1 xi2 ···xis )h

∼
=



h > 0, y ∈ I sh = R≥nsh

)
/

∼
=



L
Rxi1 xi2 ···xis ≥0

L

(

y
(xi1 xi2 ···xis )h

∼
=

h > 0, y ∈ Rnsh

)

φ

∼
=

/



L
Rxi1 xi2 ···xis 0

y
y
to
n
(xi1 xi2 · · · xis )
(xi1 xi2 · · · xis )n
where y denotes the image of y in Rnsh . Hence the same map φ on the third line is exactly
“taking the degree 0 part”. Therefore we have


Ri π∗ OY ∼
= H i (Y, OY ) = H i+1(R)
The induced map on the second line takes the element

m

while

≥0



Ri π∗ OEred ∼
= H i (Ered , OEred ) ∼
= Hmi+1 (R) 0

for every i ≥ 1, and the map Ri π∗ OY −
→ Ri π∗ OEred is taking degree 0 part. Therefore taking
cohomology of (4.4.2), we have


for every i ≥ 2.
(4.4.3)
hi (Ω0 ) ∼
= H i+1 (R)
R

>0

m

Moreover, for i = 0, 1, the cohomology of (4.4.2) gives



φ
(4.4.4) 0 −
→ h0 (Ω0R ) −
→ H 0 (Y, OY ) ⊕ k −
→ H 0 (Ered , OEred ) −
→ h1 (Ω0R ) → Hm2 (R) >0 −
→ 0.

A similar Čech complex computation as above shows that
  

 
H 0 (Y, OY ) = ker ⊕ Rxi ≥0 −
→ ⊕ Rxi xj ≥0
  


while H 0 (Ered , OEred ) = ker ⊕ Rxi 0 −
→ ⊕ Rxi xj 0 . Therefore φ is surjective, which
implies


(4.4.5)
h1 (Ω0 ) ∼
= H 2 (R) .
R

m

>0

Taking degree > 0 part of (4.4.4) we get an exact sequence
 


→ ⊕ Rxi >0 −
→ ⊕ Rxi xj >0 .
0−
→ h0 (Ω0R )>0 −
13

This implies that h0 (Ω0R )>0 ∼
= (Γ(Spec R \ m, OSpec R ))>0 and so


(4.4.6)
h0 (Ω0 )>0 /R>0 ∼
= H 1 (R) .
R

>0

m

∼
Finally we notice that h
= Rsn is the seminormalization of R [Sai00, 5.2]. We know
sn
sn
sn
R0 ⊆ R is reduced. But we can also view R0sn as the quotient Rsn /R>0
, in particular the
sn
prime ideals of R0sn correspond to prime ideals of Rsn that contain R>0
, so they all contract
to m in R. Since seminormlization induces a bijection on spectrum, R0sn has a unique prime
ideal. Thus R0sn , being a reduced Artinian local ring, must be a field. Since seminormalization
also induces isomorphism on residue fields, R0sn = k and thus h0 (Ω0R )0 ∼
= R0sn = R0 . Hence
(4.4.6) tells us that


(4.4.7)
h0 (Ω0 )/R ∼
= H 1 (R) .
0

(Ω0R )

R

m

>0

Now (4.4.3), (4.4.5) and (4.4.7) together finish the proof.



Now we prove our result on injectivity of Ext. Later we will see that this theorem can be
viewed as a generalization of the Kodaira vanishing theorem.
Theorem 4.5. Let (R, m) be a reduced Noetherian N-graded (R0 = C)-algebra with m the
unique homogeneous maximal ideal. Suppose RP is Du Bois for all P 6= m. Write R = A/I
where A = C[x1 , . . . , xn ] is a polynomial ring with deg xi = di > 0 and I is a homogeneous
ideal. Then the natural degree-preserving map ExtjA (R, A) −
→ HIj (A) induces an injection




ExtjA (R, A) ≥−d ֒→ HIj (A) ≥−d
P
for every j, where d = di .
Proof. We have the hypercohomology spectral sequence

0
Hmp (hq (Ω0R )) ⇒ Hp+q
m (ΩR ).

Since R is Du Bois away from V (m), hq (Ω0R ) has finite length when q ≥ 1. Thus we know
Hmp (hq (Ω0R )) = 0 unless p = 0 or q = 0. We also have that Hm0 (hi (Ω0R )) ∼
= hi (Ω0R ) for i ≥ 1.
Hence the above spectral sequence carries the data of a long exact sequence:
0 −
→ Hm1 (h0 (Ω0R )) −
→ H1m (Ω0R ) −
→ h1 (Ω0R ) −
→
(4.5.1)

−
→ Hm2 (h0 (Ω0R )) −
→ H2m (Ω0R ) −
→ h2 (Ω0R ) −
→
···

···

···

→ Him (Ω0R ) −
→ hi (Ω0R ) −
→ ···
−
→ Hmi (h0 (Ω0R )) −


We also have short exact sequence 0 −
→R−
→ h0 (Ω0R ) → Hm1 (R) >0 −
→ 0 by Proposition 4.4.


Therefore the long exact sequence of local cohomology and the observation that Hm1 (R) >0
has finite length tells us that


H 1 (h0 (Ω0 )) ∼
= H 1 (R)/ H 1 (R)
m

R

m

and

m

>0

Hmi (h0 (Ω0R )) ∼
= Hmi (R) for every i ≥ 2.
Now taking the degree ≤ 0 part of (4.5.1) and again using Proposition 4.4 yields:



 i
(4.5.2)
Hm (R) ≤0 ∼
= Him (Ω0R ) ≤0 for every i.
14

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 J = I, and √
thus Sred = A/I = R), we always have a surjection
0
i
i
Hm (A/J) ։ Hm (ΩR ) for every i and J = I. Taking the degree ≤ 0 part and applying
(4.5.2), we thus get:
 i



(4.5.3)
Hm (A/J) ≤0 ։ Hmi (R) ≤0
√
is surjective for every i and J = I. Now taking J = I t and applying graded local duality (we
refer to [BS98] 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:




ExtjA (R, A(−d)) ≥0 ֒→ ExtjA (A/I t , A(−d)) ≥0

is injective for every j and t. So after taking a direct limit and a degree shift, we have




ExtjA (R, A) ≥−d ֒→ HIj (A) ≥−d

is injective for every j. This finishes the proof.


 i

 i

Remark 4.6. The dual form of Theorem 4.5 says that Hm (A/J) t ։ Hm (R) t is surjective
√
for every i, every t ≤ 0 and every J = I, see (4.5.3). When R = A/I has isolated
singularity, A is standard graded and t = 0, this was proved in [Var13, Proposition 3.8].
Therefore our Theorem 4.5 greatly generalized this result.




In general, we cannot expect ExtjA (R, A) <−d ֒→ HIj (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. Example 3.5 in [SW07]). Let R = C[s4 , s3 t, st3 , t4 ]. Then we can write
R = A/I where A = 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


Hm1 (R) = Hm1 (R) >0 6= 0


(in particular depth R = 1). By graded local duality, we have Ext3A (R, A) <−4 6= 0. On
the other hand, using standardvanishing theorems

in 3[HL90,
 Theorem 2.9] we know that
3
3
HI (A) = 0. Therefore the map ExtA (R, A) <−4 → HI (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, m) be a Noetherian N-graded (R0 = C)-algebra with m the unique
homogeneous maximal ideal.
RP is Du Bois for all P 6= m and Hmi (R) has finite
 i Suppose

length for some i. Then Hm (R) <0 = 0.
In particular,
 if R is equidimensional and is Cohen-Macaulay Du Bois on Spec R − {m},
i
then Hm (R) <0 = 0 for every i < dim R.

Proof. Let R = A/I where A = C[x1 , . . . , xn ] is a (not necessarily standard
P graded) polynomial ring and I is a homogeneous ideal. Set deg xi = di > 0 and d =
di . The graded
canonical module of A is A(−d). By graded local duality,
Extn−i (R, A)(−d) ∼
= H i (R)∗ .
A

m

15





n−i
Therefore if Hmi (R) −j 6= 0 for some j > 0, then ExtA
(R, A) j−d 6= 0 for some j > 0. By
Theorem 4.5, we have an injection:




n−i
ExtA
(R, A) j−d ֒→ HIn−i (A) j−d .

n−i
Since Hmi (R) has finite length, ExtA
(R, A) also has finite length. Hence the natural degreen−i
n−i
preserving map ExtA (R, A) −
→ HI (A) factors through
n−i
ExtA
(R, A) −
→ Hm0 HIn−i (A) −
→ HIn−i (A).

Taking the degree j − d part, we thus get an injection:




n−i
ExtA
(R, A) j−d → Hm0 HIn−i (A) j−d.


It follows that Hm0 HIn−i (A) j−d 6= 0 for some j > 0. However, Hm0 HIn−i (A) is an Eulerian graded D-module supported only at m. Thus
by [MZ14, Theorem 1.2],5 the socle of

Hm0 HIn−i (A) is concentrated in degree −d so that Hm0 HIn−i (A) >−d = 0, a contradiction. 

If R is the section ring of a normal Cohen-Macaulay
and Du Bois projective variety X with
 i

respect to an ample line bundle L , then Hm (R) <0 = 0 is exactly the Kodaira vanishing for
X (which is well-known, for example see [Ma15] or [Pat15]). But Theorem 4.8 can handle
more general R, i.e., R need not be a section ring of an ample line bundle. If R is normal,
then any graded ring is the section ring of some Q-divisor [Dem88] and in that case, our
results yield variants and consequences of Kawamata-Viehweg vanishing (also see [Wat81,
Lemma 2.1 and Proposition 2.2]). But for general graded rings we do not know how to view
them as section rings. Thus our results Theorem 4.5 and Theorem 4.8 should be viewed as
generalizations of the Kodaira vanishing theorem for Cohen-Macaulay Du Bois projective
varieties. It would also be natural to try to generalize the results of Proposition 4.4 through
Theorem 4.8 to the context of Du Bois pairs. One particular obstruction is the use of the
Eulerian graded D-module at the end of the proof of Theorem 4.8.

4.2. Set-theoretic Cohen-Macaulayness. Our next application is a characteristic 0 analog of [SW05, Lemma 3.1] on set-theoretic Cohen-Macaulayness. Recall that an ideal I in
a√ regular
√ ring R is set theoretically Cohen-Macaulay if there exists an ideal J such that
I = J and R/J is Cohen-Macaulay.
Proposition 4.9. Let (R, m) be a regular local ring essentially of finite type over C, and
let I ⊆ R be an ideal. If R/I is Du Bois but not Cohen-Macaulay, then the ideal I is not
set-theoretically Cohen-Macaulay.
√
Proof. Suppose I = J for some J such that R/J is Cohen-Macaulay. Applying Lemma 3.3
to S = R/J, we find that for every i < dim R/I,
0 = Hmi (R/J) ։ Hmi (R/I)
is surjective and thus R/I is Cohen-Macaulay, a contradiction.



In the graded characteristic 0 case, we have a stronger criterion for set-theoretic CohenMacaulayness.
5[MZ14,

P
Theorem 1.2] assumes A = C[x1 , . . . , xn ] has standard grading, i.e., di = 1 and hence d =
di = n.
However,
the
same
proof
can
be
adapted
to
the
general
case:
one
only
needs
to
replace
the
Euler
operator
P
P
xi ∂i by
di xi ∂i . The reader is referred to [Put15, Section 2] for a discussion on this.
16

Corollary 4.10. Let R = C[x1 , . . . , xn ] be a polynomial ring with possibly non-standard
grading. Let I be a homogeneous ideal of R such that R/I is Du Bois on Spec R − {m} (e.g.,
R/I has an isolated singularity
at {m}).

Suppose Hmi (R/I) ≤0 6= 0 for some i < dim R/I (e.g., R/I is not Cohen-Macaulay on
Spec R − {m}, or H i (X, OX ) 6= 0 for X = Proj R/I). Then I is not set-theoretically CohenMacaulay.
√
Proof. Suppose I = J for some J such that R/J is Cohen-Macaulay. Applying the dual
form of Theorem 4.5 (see Remark 4.6), we get




0 = Hmi (R/J) ≤0 ։ Hmi (R/I) ≤0
is surjective for every i < dim R/I. This clearly contradict our hypothesis.



We point out the following example as an application.
Example 4.11. Let k be a field and let Ek ⊆ P2k be a smooth elliptic curve over k. We want
to study the defining ideal of the Segre embedding Ek ×P1k ⊆ P5k . We let k[x0 , . . . , x5 ]/I = A/I
be this homogeneous coordinate ring. It is well known that A/I is not Cohen-Macaulay.
(a) k has characteristic p > 0 and Ek is an ordinary elliptic curve. In this case it is well
known that A/I is F -pure, so [SW05, Lemma 3.1] shows I is not set-theoretically
Cohen-Macaulay.
(b) k has characteristic p > 0 and Ek is supersingular. We want to point out that, at least
when k is F -finite, I is still not set-theoretically Cohen-Macaulay.
√ This answers a
question in [SW05, Remark 3.4]. Suppose there exists J such that J = I and A/J is
e
Cohen-Macaulay. Let e ≫ 0 such that I [p ] ⊆ J. The composite of the Frobenius map
e
F
on A/I with the natural surjection A/I −
→ A/I [p ] ։ A/J makes A/J a small CohenMacaulay algebra over A/I (note that k, and hence A, is F -finite). However, by a
result of Bhatt [Bha14, Example 3.11], A/I does not have any small Cohen-Macaulay
algebra, a contradiction.
(c) k has characteristic 0. In this case, it is easy to check using Proposition 4.4 that
A/I is Du Bois. Hence Proposition 4.9 immediately shows I is not set-theoretically
Cohen-Macaulay. This example was originally obtained in [SW05, Theorem 3.3] using
reduction to characteristic p > 0. Thus our Proposition 4.9 can be viewed as a vast
generalization of their result.
It is worth√to point out that in Example 4.11, we know Hmi (A/J) ։ Hmi (A/I) for every
i and every J = I in characteristic 0 since A/I√ is Du Bois. In characteristic p > 0, we
have Hmi (A/J) ։ Hmi (A/I) for every i and every J = I when Ek is ordinary, however it is
straightforward to check that Hm2 (A/I [p] ) −
→ Hm2 (A/I) is not surjective (it is the zero map)
when Ek is supersingular. Therefore the surjective property proved in Lemma 3.3 does not
pass to reduction mod p ≫ 0.
4.3. Koszul cohomology versus local cohomology. Our last application in this section
is a strengthening of the main result of [HR76]. We start by proving a general result which
is characteristic-free. The proof is inspired by [Sch82].
Theorem 4.12. Let R be a Noetherian N-graded k-algebra where k = R0 is an
 arbitrary
field. Let m be the unique homogenous maximal ideal. If Hmr (R) = Hmr (R) 0 for every
17



r < n = dim R, then H r (x, R) 0 ∼
= Hmr (R) for every r < n and every homogeneous system
of parameters x = x1 , . . . , xn , where H r (x, R) denotes the r-th Koszul cohomology of x. In
other words, it is not necessary to take a direct limit when computing the local cohomology.
Proof. We fix x = x1 , . . . , xn a homogeneous system of parameters. Let deg xi = di > 0.
Consider the graded Koszul complex:
Kq : 0−
→ R(−d1 − d2 − · · · − dn ) −
→ ··· −
→ ⊕R(−di ) −
→R−
→ 0.
After we apply HomR (−, R) we obtain the graded Koszul cocomplex:
K :0−
→R−
→ ⊕R(di ) −
→ ··· −
→ R(d1 + d2 + · · · + dn ) −
→ 0.
q

We note that K lives in cohomology degree 0, 1, . . . , n.
q
q
Let ωR be the graded normalized dualizing complex of R, thus ωR = h−n ωR is the graded
canonical module of R. Let (−)∗ = HomR (−, ∗ E) where ∗ E is the graded injective hull of k.
We have a triangle
q

+1

ωR [n] −
→ ωR −
→ τ>−n ωR −→
q

q

Applying R HomR (K , −) we get:
q

+1

R HomR (K , ωR [n]) −
→ R HomR (K , ωR ) −
→ R HomR (K , τ>−n ωR ) −→
q

q

q

q

q

Applying HomR (−, ∗ E) and using graded local duality, we obtain:

+1

R HomR (K , τ>−n ωR )∗ −
→K −
→ R HomR (K , ωR [n])∗ −→
q

q

q

q

Note that R HomR (K , ωR [n])∗ lives in cohomological degree n, n+1, . . . , 2n, hence we obtain
a graded isomorphism in the derived category:
q
q
q
τ <n K ∼
= τ <n R HomR (K , τ>−n ω )∗ .
q

R

Therefore for every r < n, we have:
q
q
q
hr (K ) ∼
= hr (R HomR (K , τ>−n ωR )∗ ).


At this point, notice that Hmr (R) = Hmr (R) 0 for every r < n implies R is Buchsbaum
q
by [Sch82, Theorem 3.1]. This means τ>−n ωR is quasi-isomorphic to a complex of graded kvector spaces.
Moreover,
we know that all these graded vector spaces have degree 0 because


Hmi (R) = Hmi (R) 0 for every i < n. In other words, we have:
q
τ>−n ω ∼
→ k sn−1 −
→ k sn−2 −
→ ··· −
→ k s1 −
→ k s0 −
→0
=0−
R

where the complex on the right hand side has zero differentials,
k si has internal degree 0 and

sits in cohomology degree −i, with si = dimk Hmi (R) 0 .
Recall that di > 0, hence by keeping track of the internal degrees we see that

q
q 
q
R HomR (K , τ>−n ω ) ∼
= τ>−n ω .
R

0

R

Now by graded local duality, we have:
 r




q 
q
q
q
H (x, R) 0 = hr (K ) 0 ∼
= Hmr (R)
= hr (R HomR (K , τ>−n ωR )∗ ) 0 ∼
= hr ((τ>n ωR )∗ ) ∼

for every r < n. This finishes the proof.

Now we can prove the following extension of the main result of [HR76].
18



Corollary 4.13. Let R be a Noetherian N-graded (R0 = k)-algebra with m the unique
homogeneous maximal ideal. Suppose R is equidimensional and Cohen-Macaulay on Spec R−
{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 (x, R) 0 ∼
= Hmr (R) for every r < n = dim R and every homogeneous system of
parameters x = x1 , . . . , xn , where H r (x, R) denotes the r-th Koszul cohomology of x. In
other words, it is not necessary to take a direct limit when computing the local cohomology.
Proof. Since R is equidimensional and Cohen-Macaulay on the punctured spectrum,
 r we know
r
r
that Hm (R) has finite length for every r < n = dim R. We will show Hm (R) = Hm (R) 0 for
every r < n in situation (a) or (b).
 r This will finish the proof by rTheorem 4.12.
r
In situation (a), Hm (R) = Hm (R) 0 is obvious because Hm (R) has finite length and


Frobenius acts injectively on it. In situation (b), notice that Hmr (R) <0 = 0 by Theorem 4.8




while Hmr (R) >0 = 0 by Proposition 4.4, hence Hmr (R) = Hmr (R) 0 .



Remark 4.14. In situation (b) above, if R is normal standard graded, then Hmr (R) = Hmr (R) 0
also follows from [Ma15, Theorem 4.5]. However, in the above proof we don’t need any normal
or standard graded hypothesis thanks to Proposition 4.4 and Theorem 4.8.
Remark 4.15. Corollary 4.13 was proved when R is F -pure and k is perfect in [HR76, Theorem 1.1], and by a technical reduction to p > 0 technique, it was also proved when R is of
F -pure type [HR76, Theorem 4.8]. Since F -pure certainly implies F -injective and F -injective
type implies Du Bois (see [Sch09]), our theorem gives a generalization of Hochster-Roberts’s
result, and our proof is quite different from that of [HR76].
We end this section by pointing out an example showing that in Theorem 4.12 or Corollary 4.13,
it is possible that H r (x, R) 6= Hmr (R), i.e., we must take the degree 0 piece of the Koszul
cohomology. This is a variant of Example 4.11.
Example 4.16. Let R = x3k[x,y,z]
#k[a, b, c] be the Segre product of x3k[x,y,z]
and k[a, b, c]
+y 3 +z 3
+y 3 +z 3
where the characteristic of k is either 0 or congruent to 1 mod 3. Therefore R is the
homogeneous coordinate ring of the Segre embedding of X = E × P2 to P8 , where E =
Proj x3k[x,y,z]
is an elliptic curve. Notice that dim X = 3 and dim R = 4. Since R has an
+y 3 +z 3
isolated singularity it is Cohen-Macaulay on the punctured spectrum. It is easy to check
that R is Du Bois in characteristic 0, and F -pure (and thus F -injective) in characteristic
p > 0 since p ≡ 1 mod 3. So we know that Hmi (R) = [Hmi (R)]0 for every i < 4 where m is
the unique homogeneous maximal ideal of R. Now we compute:
Hm2 (R) = [Hm2 (R)]0 = H 1 (X, OX ) = ⊕i+j=1 H i (E, OE ) ⊗k H j (P2 , OP2 ) ∼
=k
Hm3 (R) = [Hm3 (R)]0 = H 2 (X, OX ) = ⊕i+j=2 H i (E, OE ) ⊗k H j (P2 , OP2 ) = 0.
The first line shows that R is not Cohen-Macaulay (actually we have depth R = 2), so for
any homogeneous system of parameters x of R, we have H 3 (x, R) ∼
= H1 (x, R) 6= 0. Hence
3
3
H (x, R) 6= Hm (R).
19

5. Deformation of dense F -injective type
In this section we use results in section 3 to prove that singularities of (dense) F -injective
type in characteristic 0 deform. In fact, we prove a slightly stronger result. Our motivation for studying this question is a recent result of Kovács-Schwede [KS16a] that Du Bois
singularities in characteristic 0 deform.
By the main result of [Sch09], singularities of dense F -injective type are always Du Bois.
It is conjectured [BST13, Conjecture 4.1] that Du Bois singularities should be equivalent to
singularities of dense F -injective type. This conjecture is equivalent to the weak ordinarity
conjecture of Mustaţă-Srinivas [MS11]. If this conjecture is true, then singularities of dense
F -injective type deform because Du Bois singularities deform [KS16a]. However, the weak
ordinarity conjecture is wide open.
Setup 5.1 (Reduction to characteristic p > 0). We recall briefly reduction to characteristic
p > 0. For more details in our setting, see [HH06, Section 2.1] and [Sch09, Section 6].
Suppose (R, m) is essentially of finite type over C (or another field of characteristic
zero, which we will also call C) so that (R, m) is a homomorphic image of TP where
T = C[x1 , . . . , xt ] and P ⊆ T is a prime ideal so that R = (T /J)P . Given a finite collection of finitely generated R-modules Mi (and finitely many maps between them), we may
assume that each Mi = (Mi′ )P for a finitely generated T -module Mi′ annihilated by J. Suppose E −
→ Spec T /J is the reduced preimage of T /J in a log resolution of (Spec T, Spec T /J).
We also keep track of E −
→ Spec T /J in the reduction to characteristic p > 0 process, as well
i
0
as the modules h (ΩT /J ) = Ri π∗ OE . Assume x is the image of h(x1 , . . . , xt ) ∈ T (note that
if x has a problematic denominator, we can replace x by another element that generates the
same ideal and does not have a denominator).
We pick a finitely generated regular Z-algebra A ⊆ C such that the coefficients of the generators of P , the coefficients of h, the coefficients of J, and the coefficients of a presentation
of the Mi′ are contained in A. Form TA = A[x1 , . . . , xt ] and let PA = P ∩ TA , JA = J ∩ TA
and observe that PA ⊗A C = P, JA ⊗A C = J by construction. Note that by generic flatness,
we can replace A by A[b−1 ] and so shrink Spec A if necessary to assume that RA is a flat
A-module and x is a nonzerodivisor on RA . Likewise form (Mi′ )A with the same presentation
matrix of Mi′ (and likewise with maps between the modules) and form EA −
→ Spec TA /JA so
′
′
i
i
that (Mi )A ⊗A C = Mi and that (R π∗ OEA ) ⊗A C = (R π∗ OEA ). Shrinking Spec A yet again
if necessary, we can assume all these modules are flat over A (and that any relevant kernels
and cokernels of the finitely many maps between them are also flat).
We now mod out by a maximal ideal n of A with κ = A/n. In particular, we use RA =
(TA /JA )PA , etc. and Rκ , Tκ , Eκ etc. to denote the corresponding rings, schemes and modules
over A and κ, where κ = A/n for n a maximal ideal in A. Since all relevant kernels and
cokernels of the maps are flat by shrinking Spec A, x is a nonzerodivisor on Rκ for every κ.
We also need a slightly different version of the main result of [Sch09]. In that paper, it was
assumed that if R is finite type over a field of characteristic zero, and of dense F -injective
type, then R has Du Bois singularities. We need a version of this result in which R is local.
The reason is, that, in the notation of Theorem 5.3, if R is of finite type, and Rp /xp Rp is F injective, then for p sufficiently divisible, we obtain that Rp is F -injective in a neighborhood
of V (xp ). The problem is that we do not know to control these neighborhoods as p varies.
20

Thus we need the following preliminary result. In particular, we do not know whether having
dense F -injective type is an open property in characteristic zero.
Theorem 5.2. Let (R, m) be a local ring essentially of finite type over C and suppose that
R has dense F -injective type. Then R is Du Bois.
The strategy is the same as in [Sch09], indeed, the proof only differs in how carefully we
keep track of a minimal prime of the non-Du Bois locus.
Sketch of the proof. It is easy to see that R is seminormal so we need to show that hi (Ω0R ) = 0
for all i > 0. We use the notation of Setup 5.1. Let QC ⊆ P ⊆ T /J correspond to a minimal
prime of the non-Du Bois locus of (R, m), so that hi (Ω0(T /J)Q ) = hi (Rπ∗ OE )QC has finite
C
length for i > 0. Since (R, m) has dense F -injective type, it is easy to see that so does
(RQC , QC RQC ) so we may assume that m = QC . Now using Setup 5.1, reduce to some model
so that (Rκ , mκ ) is F -injective. The proof now follows exactly that of [Sch09, Proof 6.1]
where we obtain that
(Rκ ).
hi (R(πκ )∗ OEκ )Qκ ֒→ HQi+1
κ
But the left side is annihilated by a power of Frobenius by [Sch09, Theorem 7.1] and Frobenius
acts injectively on the right by hypothesis. The result follows.

Theorem 5.3. Let (R, m) be a local ring essentially of finite type over C and let x be a
nonzerodivisor on R. Suppose R/xR has dense F -injective type. Then for infinitely many
p > 0, the Frobenius action xp−1 F on Hmi p (Rp ) is injective for every i, where (Rp , mp ) is the
reduction mod p of R. In particular, R has dense F -injective type.
Proof. By Theorem 5.2, R/xR has Du Bois singularities. By Theorem 3.6 (taking Z = ∅),
·x
·x
Hmi (R) −
→ Hmi (R) surjects for all i. By Matlis duality, ExtiT (R, T ) −
→ Exti (R, T ) injects
for all i. Spreading this out to A, and possibly inverting an element of A, we see that
·x
→ ExtiTA (RA , TA ) injects for all i (note there are only finitely many Ext to
ExtiTA (RA , TA ) −
consider). Inverting another element of A if necessary, we deduce that
·x

→ ExtiTκ (Rκ , Tκ )
ExtiTκ (Rκ , Tκ ) −

injects for every i and each maximal ideal n ⊆ A, setting κ = A/n. We abuse notation
and let x denote the image of x in Rκ . Applying Matis duality again and considering the
Frobenius on the local cohomology modules, we have the following collection of short exact
sequences for each i.
0
/

Hmi κ (Rκ /xRκ )

(Rκ )
Hmi+1
κ
/

/

Hmi κ (Rκ /xRκ )

(Rκ )
Hmi+1
κ
/

xp−1 F

F



0

·x

/



(Rκ )
Hmi+1
κ

/

0

/

0

F
·x

/



(Rκ )
Hmi+1
κ

where p is the characteristic of κ and F denotes the natural Frobenius action on Hmi κ (Rκ /xRκ )
(Rκ ).
and Hmi+1
κ
At this point recall that R/xR has dense F -injective type. It follows that for infinitely
many p > 0, if the residue field κ of A has characteristic p > 0 then the natural Frobenius
action on Hmi κ (Rκ /xRκ ) is injective for every i. Now chasing the above diagram, if xp−1 F is
(Rκ )) ∩ ker(xp−1 F ). Since y is in the socle
not injective, then we can pick 0 6= y ∈ socle(Hmi+1
κ
21

(Rκ ), it maps to zero under multiplication by x. But then 0 6= y ∈ Hmi κ (Rκ /xRκ ),
of Hmi+1
κ
and chasing the diagram we find that xp−1 F (y) 6= 0, which is a contradiction.
We have established that for infinitely many p, after we do reduction to p, the Frobenius
action xp−1 F on Hmi+1
(Rκ ) is injective for every i. This certainly implies the natural Frobenius
κ
(R

action F on Hmi+1
κ ) is injective for every i. Hence R has dense F -injective type.
κ
Remark 5.4. It is still unknown whether F -injective singularities in characteristic p > 0
deform (and this has been open since [Fed83]). Our theorem is in support of this conjecture:
it shows this is true “in characteristic p ≫ 0”. For the most recent progress on deformation of
F -injectivity in characteristic p > 0, we refer to [HMS14]. On the other hand, Theorem 5.3
provides evidence for the weak ordinarity conjecture [MS11] because of the relation between
the weak ordinarity conjecture and the conjecture that Du Bois singularities have dense
F -injective type [BST13].
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Department of Mathematics, University of Utah, Salt Lake City, UT 84112
E-mail address: lquanma@math.utah.edu
Department of Mathematics, University of Utah, Salt Lake City, UT 84112
E-mail address: schwede@math.utah.edu
Department of Mathematics College of Humanities and Sciences, Nihon University, Setagayaku, Tokyo 156-8550, Japan
E-mail address: shimomotokazuma@gmail.com

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