F -SINGULARITIES VIA ALTERATIONS

arXiv:1107.3807v4 [math.AG] 5 May 2014

MANUEL BLICKLE, KARL SCHWEDE, KEVIN TUCKER

Abstract. We give characterizations of test ideals and F -rational singularities via (regular) alterations. Formally, the descriptions are analogous to standard characterizations of
multiplier ideals and rational singularities in characteristic zero via log resolutions. Lastly,
we establish Nadel-type vanishing theorems (up to finite maps) for test ideals, and further
demonstrate how these vanishing theorems may be used to extend sections.

1. Introduction
In this paper we define an ideal, in arbitrary equal characteristic, which coincides with
the multiplier ideal over C, and coincides with the test ideal in characteristic p > 0. This
justifies the maxim:
The test ideal and multiplier ideal are morally equivalent.
We state our main theorem.
Main Theorem (Theorem 4.6, Corollary 4.8, Theorem 8.2). Suppose that X is a normal
algebraic variety over a perfect field and ∆ is a Q-divisor on X such that KX + ∆ is
Q-Cartier. Consider the ideal


\
Tr
J :=
Image π∗ OY (⌈KY − π ∗ (KX + ∆)⌉) −−−π→ K(X) .
π: Y −
→X
Here the intersection runs over all generically finite proper separable maps π : Y −→ X
where Y is regular (or equivalently just normal), and the map to the function field K(X)
is induced by the Grothendieck trace map Trπ : π∗ ωY −→ ωX (if KY = π ∗ KX + Ramπ over
the locus where π is finite, then Trπ : π∗ OY (KY ) −→ OX (KX ) is induced by the field trace
TrK(Y )/K(X) : K(Y ) −→ K(X)). We obtain the following:
(a) If X is of equal characteristic zero, then J = J (X; ∆), the multiplier ideal of (X, ∆).
(b) If X is of equal characteristic p > 0, then J = τ (X; ∆), the test ideal of (X, ∆).
Furthermore, in either case, the intersection defining J stabilizes: in other words, there
is always a generically finite separable proper map π : Y −→ X with Y regular such that

Tr
J = Image π∗ OY (KY − π ∗ (KX + ∆)) −−−π→ K(X) .

In fact, we prove a number of variants on the above theorem in further generality, i.e.
for various schemes other than varieties over a perfect field.
2010 Mathematics Subject Classification. 14F18, 13A35, 14F17, 14B05, 14E15.
Key words and phrases. Test ideal, multiplier ideal, alteration, rational singularities, F -rational singularities, Nadel vanishing.
The first author was partially supported by a DFG Heisenberg Fellowship and the DFG research grant
SFB/TRR45.
The second author was partially supported by an NSF Postdoctoral Fellowship #0703505, the NSF grant
#1064485, the NSF FRG grant #1265261, the NSF CAREER grant #1252860 and a Sloan Fellowship.
The third author was partially supported by an NSF Postdoctoral fellowship #1004344 and NSF grant
#1303077.
1

F -SINGULARITIES VIA ALTERATIONS

2

Of course, there are two different statements here. In characteristic zero, this statement
can be viewed as a generalization of the transformation rule for multiplier ideals under generically finite proper dominant maps, see [Laz04, Theorem 9.5.42] or [Ein97, Proposition 2.8].
In positive characteristic, a basic case of the theorem is the following characterization of
F -rational singularities – which is interesting in its own right. Recall that an alteration is
a proper and generically finite map π : Y −
→ X, it is called a regular alteration if Y is a
regular scheme [dJ96].
Corollary (Corollary 3.6, cf. [HL07, HY11]). An F -finite ring R of characteristic p > 0 is
F -rational if and only if it is Cohen-Macaulay and for every alteration (equivalently every
regular separable alteration if R is of finite type over a perfect field, equivalently every finite
dominant map with Y normal) f : Y −→ Spec R, the map f∗ ωY −→ ωR is surjective.
The proof of this special case is in fact the key step in the proof of the main theorem in
positive characteristic. The central ingredients in its proof are the argument of K. Smith
[Smi97b] that F -rational singularities are pseudo-rational, and the work of C. Huneke and
G. Lyubeznik on annihilating local cohomology using finite covers [HL07] (cf. [HH92, HY11,
SS12]). The proof of the Main Theorem additionally utilizes transformation rules for test
ideals under finite morphisms [ST10].
In this paper, we also give a transformation rule for test ideals under proper dominant
(and in particular proper birational) maps between varieties of the same dimension. More
precisely, for any normal (but not necessarily proper) variety Y and Q-divisor Γ, we define
a canonical submodule T 0 (Y, Γ) ⊆ H 0 (Y, OY (⌈KY + Γ⌉)). We use this submodule to obtain
a transformation rule for test ideals under alterations.
Theorem 1 (Theorem 6.8). Suppose that π : Y −→ X = Spec R is a proper dominant generically finite map of normal varieties over a perfect (or even F -finite) field of characteristic
p > 0. Further suppose that ∆ is a Q-divisor on X such that KX + ∆ is Q-Cartier.
Consider the canonically determined submodule (see Definition 6.1)
T 0 (Y, −π ∗ (KX + ∆)) ⊆ H 0 (Y, OY (⌈KY − π ∗ (KX + ∆))⌉))
of sections which are in the image of the trace map for any alteration of Y . Then the global
sections of τ (X; ∆) coincide with the image of T 0 (Y, −π ∗ (KX + ∆)) under the map
Tr

H 0 (Y, OY (⌈KY − π ∗ (KX + ∆)⌉)) −−−π→ K(X)
which is induced by the trace Trπ : π∗ ωY −→ ωX .
We also prove a related transformation rule for multiplier ideals under arbitrary proper
dominant maps in Theorem 8.3.
Perhaps the most sorely missed tools in positive characteristic birational algebraic geometry (in comparison to characteristic zero) are vanishing theorems for cohomology. Indeed,
Kodaira vanishing fails in positive characteristic [Ray78]. However, if X is projective in characteristic p > 0 and L is a “positive” line-bundle, cohomology classes z ∈ H i (X, L −1 ) can
often be killed by considering their images in H i (Y, f ∗ L −1 ) for finite covers f : Y −
→ X.
For example, if i ≥ 0 and L big and semi-ample, it was shown in [Bha12, Bha10] that
there exists such a cover killing any cohomology class η ∈ H i (X, L −1 ) for i < dim X (cf.
[HH92, Smi97c, Smi97a]). When we combine our main result with results from [Bha10], we
obtain the following variant of a Nadel-type vanishing theorem in characteristic p > 0 (and
a relative version). Notably, we need not require a W2 lifting hypothesis.
Theorem 2 (Theorem 5.5). Let X be a normal proper algebraic variety of finite type over
a perfect (or even F -finite) field of characteristic p > 0, L a Cartier divisor, and ∆ ≥ 0

F -SINGULARITIES VIA ALTERATIONS

3

a Q-divisor such that KX + ∆ is Q-Cartier. Suppose that L − (KX + ∆) is a big and
semi-ample Q-divisor. Then there exists a finite surjective map f : Y −→ X such that:
(a) The natural map f∗ OY (⌈KY + f ∗ (L − KX − ∆)⌉) −→ OX (L), induced by the trace
map, has image τ (X; ∆) ⊗ OX (L).
(b) The induced map on cohomology
H i (Y, OY (⌈KY + f ∗ (L − KX − ∆)⌉)) −→ H i (X, τ (X; ∆) ⊗ OX (L))
is zero for all i > 0.
Applying the vanishing theorem above, we obtain the following extension result.
Theorem 3 (Theorem 7.6). Let X be a normal algebraic variety which is proper over
a perfect (or even F -finite) field of characteristic p > 0 and D is a Cartier divisor on
X. Suppose that ∆ is a Q-divisor having no components in common with D and such
that KX + ∆ is Q-Cartier. Further suppose that L is a Cartier divisor on X such that
L − (KX + D + ∆) is big and semi-ample. Consider the natural restriction map
γ : H 0 (X, OX (⌈L − ∆⌉) −→ H 0 (D, OD (L|D − ⌊∆⌋|D ) .
Then




T 0 D, L|D − (KD + ∆|D ) ⊆ γ T 0 X, D + L − (KX + D + ∆)

noting that T 0 (D, L|D − (KD + ∆|D )) ⊆ H 0 (D, OD (⌈L|D − ⌊∆⌋|D )⌉). In particular, if
T 0 (D, L|D − (KD + ∆|D )) 6= 0, then H 0 (X, OX (⌈L − ∆⌉) 6= 0.
Finally, let us remark that many of the results contained herein can be extended to
excellent (but not necessarily F -finite) local rings with dualizing complexes; in fact, this is
the setting of C. Huneke and G. Lyubeznik in [HL07]. However, moving beyond the local
case is then difficult essentially because we do not know the existence of test elements. For
this reason, and also because our inspiration comes largely from (projective) geometry, we
restrict ourselves to the F -finite setting throughout (note that any scheme of finite type
over a perfect field is automatically F -finite).
Acknowledgements: The authors would like to thank Bhargav Bhatt, Christopher Hacon,
Mircea Mustaţă and Karen Smith for valuable conversations. The authors would also like
to thank all the referees for many very useful comments. Finally, the authors worked on
this paper while visiting the Johannes Gutenberg-Universität Mainz during the summers
of 2010 and 2011. These visits were funded by the SFB/TRR45 Periods, moduli, and the
arithmetic of algebraic varieties.
2. The trace map, multiplier ideals and test ideals
Multiplier ideals and test ideals are prominent tools in the study of singularities of algebraic varieties. Later in this section, we will briefly review their constructions together with
those of certain variants – the multiplier and test module, respectively – related to various
notions of rational singularities. In doing so, we emphasize a viewpoint that relies heavily
on the use of the trace map of Grothendieck-Serre duality. In fact, the whole paper (particularly Sections 5, 6, and 7) relies on some of the more subtle properties of this theory.
First however we give a brief introduction to this theory necessary to understand the main
results of this paper that does not rely on any of these more subtle aspects.

F -SINGULARITIES VIA ALTERATIONS

4

2.1. Maps derived from the trace map. This section is designed to be a friendly and
brief introduction to the trace map at the level we will apply it for our main theorem.
Therefore, in this subsection we only deal algebraic varieties of finite type over a perfect
field (although everything can be immediately generalized to F -finite integral schemes).
More general versions will be discussed later in Section 2.3. We will assume that the reader
is familiar with canonical and dualizing modules at the level of [KM98, Section 5.5] and
[Har77, Chapter III, Section 7].
Suppose that π : Y −
→ X is a proper generically finite map of varieties of finite type over
a field k. A key tool in this paper is a trace map
Trπ : π∗ ωY −
→ ωX .
Here ωY and ωX denote suitable canonical modules on Y and X (which we assume exist).
We will explain the origin of this map explicitly. Since any generically finite map can be
factored into a composition of a finite and proper birational map, it suffices to deal with
these cases separately:
Example 2.1 (Trace for proper birational morphism). Suppose that π : Y −
→ X is a proper
birational map between normal varieties. In this case, the trace map Trπ : π∗ ωY −
→ ωX
can be described in the following manner. Fix a canonical divisor KY on Y and set KX =
π∗ KY (in other words, recall by definition that ωY ∼
= OX (KY ) and requiring that π∗ KY =
KX simply means that KX is the divisor on X that agrees with KY wherever π is an
isomorphism). Then π∗ OY (KY ) is a torsion-free sheaf whose reflexification is just OX (KX ),
since π is an isomorphism outside a codimension 2 set of X. The trace map is simply the
natural (reflexification) map π∗ OY (KY ) ֒→ OX (KX ).
Example 2.2 (Trace for finite morphism). Suppose that π : Y −
→ X is a finite surjective map
of varieties. The trace map Trπ : π∗ ωY −
→ ωX is then identified with the evaluation-at-1
map, π∗ ωY := H omOX (π∗ OY , ωX ) −
→ ωX (the neophyte reader should take on faith that
π∗ ω Y ∼
= H omOX (π∗ OY , ωX ), or see Section 2.3 and Remark 2.17 for additional discussion).
Assuming additionally that π : Y −
→ X is a finite separable map of normal varieties with
ramification divisor Ramπ , we fix a canonical divisor KX on X and set KY = π ∗ KX +Ramπ .
Then the field-trace map
TrK(Y )/K(X) : K(Y ) −
→ K(X)
restricts to a map π∗ OY (KY ) −
→ OX (KX ) which can be identified with the Grothendieck
trace map (cf. [ST10]).
Below in subsection 2.3 we will explain that this construction of a trace map is just an
instance of a much more general theory contained in Grothendieck-Serre duality. We do
not need this generality for our main theorem however.
We now mention two key properties that we will use repeatedly in this basic context.
Lemma 2.3 (Compatibilities of the trace map). Suppose that π : Y −→ X is a proper
generically finite dominant morphism between varieties (or integral schemes). Fix Trπ :
π∗ ωY −→ ωX to be the trace map as above.
(a) If additionally, ρ : Z −→ Y is another proper generically finite dominant morphism
and Trρ : ρ∗ ωZ −→ ωY is the associated trace map, then Trπ ◦(π∗ Trρ ) = Trπ◦ρ .
(b) Additionally, if U ⊆ X is open and W = π −1 (U ), then Trπ|W = Trπ |U (here Trπ :
π∗ ωY −→ ωX is a map of sheaves on X and so can be restricted to an open set). In
other words, the trace map is compatible with open immersions.
Proof. These properties follow directly from the definition given.



F -SINGULARITIES VIA ALTERATIONS

5

Because much of our paper is devoted to studying singularities defined by Frobenius,
utilizing Lemma 2.3 we specialize Example 2.2 to the case where π is the Frobenius.
Example 2.4 (Trace of Frobenius). Suppose that X is a variety of finite type over a perfect
field of characteristic p > 0. Then consider the absolute Frobenius map F : X −
→ X, this
map is not a map of varieties over k, but it is still a map of schemes. Using the fact (cf.
Example 2.15) that H omOX (F∗ OX , ωX ) ∼
= F∗ ωX , and applying Example 2.2, we obtain the
evaluation-at-1 trace map,
TrF : F∗ ωX −
→ ωX .
Because of the importance of this map in what follows, we will use the notation ΦX to
denote TrF . As an endomorphism of X one can compose the Frobenius with itself and
obtain the e-iterated Frobenius F e . It follows from Lemma 2.3(a) that TrF e then coincides
with the composition of TrF with itself e-times (appropriately pushed forward). Because of
this, we use ΦeX to denote TrF e .
Now we come to a compatibility statement for images of trace maps that will be absolutely
crucial later in the paper. This is essentially the dual statement to a key observation
from [Smi97b]. We will generalize this later in Proposition 2.21 and also in the proof of
Proposition 4.2.
Proposition 2.5. If π : Y −→ X is a proper dominant generically finite map of varieties,
then the image of the trace map
Jπ := Trπ (π∗ ωY ) ⊆ ωX
satisfies ΦX (F∗ Jπ ) ⊆ Jπ .
Proof. Consider the commutative diagram:
Y

F

/Y
π

π





X

F

/X

where the horizontal maps are the Frobenius on X and Y respectively. It follows from
Lemma 2.3(a) that there is a commutative diagram
F∗ π∗ ωY

π ∗ ΦY

/ π∗ ω Y

F∗ Trπ

Trπ



F∗ ωX
The claimed result follows immediately.



ΦX

/ ωX .



2.2. Pairs. The next step is to extend the trace map to incorporate divisors. Suppose that
X is a normal integral scheme. A Q-divisor Γ on XPis a formal linear combination of prime
Weil divisors with coefficients
in Q. Writing P
Γ=
bi Bi where the Bi are distinct prime
P
divisors, we use ⌈Γ⌉ = ⌈bi ⌉Bi and ⌊Γ⌋ = ⌊bi ⌋Bi to denote the round up and round
down of Γ, respectively. We say Γ is Q-Cartier if there exists an integer n > 0 such that
nΓ is an integral (i.e. having integer coefficients) Cartier divisor, and the smallest such n
is called the index of Γ. An integral divisor KX with OX (KX ) ∼
= ωX is called a canonical
divisor.

F -SINGULARITIES VIA ALTERATIONS

6

Definition 2.6. A pair (X, ∆) is the combined data of a normal integral scheme X together
with a Q-divisor ∆ on X. The pair (X, ∆) is called log-Q-Gorenstein if KX +∆ is Q-Cartier.
We emphasize that log-Q-Gorenstein pairs need not be Cohen-Macaulay.
Convention 2.7. For X normal and integral let ∆ be a Q-divisor on X. The choice of a
rational section s ∈ ωX gives a canonical divisor KX = KX,s = div s and also a map ωX ⊆
s7→1

s7→1

ωX ⊗ K(X) −−−−→ K(X). Then the image of the inclusion ωX (−KX,s − ⌊∆⌋) ֒−−−−→ K(X)
is the subsheaf OX (−⌊∆⌋) ⊆ K(X). Note the image is independent of the choice of s but
the inclusion maps for different sections may differ by multiplication with a unit of OX .
Hence, every OX -submodule of ωX (−⌊KX +∆⌋) corresponds uniquely to an OX -submodule
of OX (−⌊∆⌋) (or even ⊆ OX when ∆ is effective). As such, we have chosen to accept certain abuses of notation in order to identify such submodules. For example, we may write
ωX (−⌊KX + ∆⌋) ⊆ K(X) (or ⊆ OX when ∆ is effective); however, while it is canonical as
a subset (and equals OX (−⌊∆⌋)), the actual inclusion map involves the choice of a section
(and is well defined only up to a multiplication by a unit of OX ).
We now state a result incorporating divisors into the trace map.
Proposition 2.8. Suppose that π : Y −→ X is a proper dominant generically finite morphism between normal varieties, and let ∆ be a Q-divisor on X such that KX + ∆ is
Q-Cartier. Then the trace map of π induces a non-zero map
Trπ : π∗ ωY (−⌊π ∗ (KX + ∆)⌋) −→ ωX (−⌊KX + ∆⌋) .
Proof. This result is simple based upon Examples 2.1 and 2.2 and so we leave it to the reader
to verify. We carefully prove a more general result in Propositions 2.13 2.18 below.

2.3. Duality and the trace map. This section restates the results of the previous section
in the more general language of dualizing complexes. While these results will be important
for generalizations of our main theorem and for some of the Kodaira-type vanishing theorems, they are not needed for the main result stated in the introduction. Therefore, we
invite the reader to skip the next section and instead read ahead to Section 2.4.
From now on we assume that all schemes X are Noetherian, excellent, separated and
q
possess a dualizing complex ωX . This is a relatively mild condition, since, for example,
all Noetherian schemes that are of finite type over a local Gorenstein ring of finite Krull
dimension have a dualizing complex [Har66, Chapter V, Section 10]. By definition, [Har66,
b (X) which has fiChapter V, Section 2], a dualizing complex on X is an object in Dcoh
q
q
nite injective dimension and such that the canonical map OX −
→ RH omX (ωX , ωX ) is an
b (X).
isomorphism in Dcoh
Since dualizing complexes are defined by properties in the derived category, they are only
q
unique up to quasi-isomorphism. But even worse, if ω is a dualizing complex, then so is
q
ω ⊗ L [n] for any integer n and line-bundle L . But this is all the ambiguity there is for a
q
connected scheme: if Ω is another dualizing complex then there is a unique line-bundle L
q
q
and a unique shift n such that Ω is quasi-isomorphic to ω ⊗ L [n], see [Har66, Theorem
V.3.1]. The ambiguity with respect to shift is the least serious. For this, we say that a
dualizing complex on an integral scheme (or a local scheme) is said to be normalized if the
q
first non-zero cohomology of ωX lies in degree (− dim X).
A canonical module ωX on a reduced and connected scheme X is a coherent OX -module
q
that agrees with the first non-zero cohomology of a dualizing complex ωX . In particular,
q
q
for a normalized dualizing complex ωX , its (− dim X)-th cohomology ωX := h− dim X ωX is
a canonical module; since it is the first non-zero cohomology there is a natural inclusion

F -SINGULARITIES VIA ALTERATIONS

7

ωX [dim X] ֒−
→ ωX . If X is S1 , i.e. satisfies Serre’s first condition, then ωX is S2 by [Har07,
Lemma 1.3]. Also see [KM98, Corollary 5.69] where it is shown that any (quasi-)projective
scheme over a field has an S2 canonical module. We also note that if X is integral, then
ωX can be taken to be the S2 -module agreeing with the dualizing complex on the CohenMacaulay locus of X.
We shall make extensive use of the trace map from Grothendieck-Serre duality, see
[Con00, Har66]. For S a base scheme, Noetherian, excellent, and separated, we consider the
category SchS of S-schemes (essentially) of finite type over S, with S-morphisms between
q
them. We assume as before that S has a dualizing complex ωS . Then Grothendieck duality
theory provides us with a functor f ! for every S-morphism f : Y −
→ X with the properties
!
(a) ( ) is compatible with composition, i.e. if g : Z −
→ Y is a further S-morphism,
then there is a natural isomorphism of functors (f ◦ g)! ∼
= g! ◦ f ! which is compatible
with triple composition.
q
q
(b) If f is of finite type, and ωX is a dualizing complex on X then f ! ωX is a dualizing
complex on Y . If additionally f is a dominant morphism of integral schemes and
q
q
ωX is normalized, then f ! ωX is also normalized.
q
(c) If f is a finite map, then f ! ( ) = RH omOX (f∗ OY , ) viewed as a complex of
OY -modules. Note that the right hand side is defined for any finite morphism, not
just for an S-morphism.
Therefore we may define for each S-scheme X with structural map πX : X −
→ S the dualq
! ω q . After this choice of dualizing complexes on Sch , the compatizing complex ωX := πX
S
S
ibility with composition now immediately implies that for any S-morphism f : Y −
→ X we
q ∼ q
!
have a canonical isomorphism f ωX = ωY .
q

Remark 2.9. In the remainder of the paper, the base scheme S will typically either be a
field, or it will be the scheme X we are interested in. Note that in either case the absolute
Frobenius map F : X −
→ X is not a map of S-schemes with the obvious choice of (the
same) structural maps. However, using the composition F : X −
→X −
→ S, we do obtain a
new S-scheme structure for X and so we view F : X −
→ X as a map of different S-schemes.
→ X proper there is a natural
A key point in the construction of ( )! is that for f : Y −
transformation of functors
Rf∗ f ! −
→ idX
called the trace map which induces a natural isomorphism of functors in the derived category
Rf∗ RH omY (M , f ! N ) −
→ RH omY (Rf∗ M , N )
q

q

q

q

for any bounded above complex of quasi-coherent OY -modules M and bounded below
q
complex of coherent OX -modules N . This statement, which expresses that f ! is right
adjoint to Rf∗ for proper f , is the duality theorem in its general form.
q
Applying trace map to the dualizing complex ωX we obtain
q
q
q
→ω
(2.10)
Trf q : Rf∗ ω ∼
= Rf∗ f ! ω −
q

Y

X

X

which we also refer to as the trace map and denote by Trf q . Equivalently, by the duality
theorem, the trace map is Grothendieck-Serre dual to the corresponding map of structure
sheaves f : OX −
→ Rf∗ OY . The key properties of the trace relevant for us are
(a) Compatibility with composition, i.e. if g : Z −
→ Y is another proper S-morphism
then Tr(f ◦g) q = Trf q ◦Rf∗ Trg q .
(b) The trace map is compatible with certain base changes. In general this is a difficult
and subtle issue (see [Con00]); however, we will only need this for open inclusions
U ⊆ X and localization at a point, where it is not problematic.

F -SINGULARITIES VIA ALTERATIONS

8

(c) In the case that f is finite, Trf q is locally given by evaluation at 1,
→ ωX
Trf q : Rf∗ f ! ωX = Rf∗ RH omOX (f∗ OY , ωX ) −
q

q

q

q

For a proper dominant morphism π : Y −
→ X of integral schemes, the trace gives rise to
maps on canonical modules as well (not just dualizing complexes). Taking the (− dim X)-th
q
q
cohomology of Trπ q : Rπ∗ ωY −
→ ωX gives
(2.11)

Trπ q : h− dim X (Rπ∗ ωY ) −
→ h− dim X ωX =: ωX
q

q

which we will also denote by Trπ q . Further composing with the inclusion ωY [dim Y ] −
→ ωY
then gives
q

(2.12)

Tr

Trπ : hdim Y −dim X Rπ∗ ωY −
→ h− dim X Rπ∗ ωY −−−π−→ ωX
q

q

which we also refer to as the trace map and now denote by Trπ . Note that the above
construction remains compatible with localization on the base scheme, which we make use
of below in showing under mild assumptions that this trace map is non-zero.
Proposition 2.13. If π : Y −→ X is a proper dominant morphism of integral schemes, then
the trace map
Trπ : hdim Y −dim X Rπ∗ ωY −→ ωX
is non-zero.
Proof. To show the statement we may assume that X = Spec K for K a field. The
q
q
q
map H0 (Y, ωY ) = h0 Rπ∗ ωY −
→ ωK ∼
= K is non-zero as it is Grothendieck-Serre dual
to the natural inclusion K −
→ H 0 (Y, OY ). Hence it is sufficient to show that the map
q
hdim Y (Rπ∗ ωY ) −
→ h0 (Rπ∗ ωY ) is surjective. If Y is Cohen-Macaulay, i.e. ωY [dim Y ] =
q
ωY , we are done. More generally consider the hypercohomology spectral sequence E2 =
q
q
q
q
H i (Y, hj ωY ) ⇒ Hi+j (Y, ωY ). The only terms contributing to H0 (Y, ωY ) are H i (Y, hj ωY )
q
with i + j = 0. In the next lemma, it is shown that dim supp hj ωY < −j for j > − dim Y ,
which hence implies that H dim Y (Y, ωY ) = hdim Y (Rπ∗ ωY ) is the only non-vanishing term
q

among them and thus surjects onto h0 (Rπ∗ ωY ) as claimed.
Lemma 2.14. Let (R, m) be an equidimensional local S1 ring of dimension n with normalq
ized dualizing complex ωR . Then
dim supp h−j ωR < j
q

for j < n.
Proof. By local duality h0 ωR is Matlis dual to Γm (R) which by the S1 condition is zero.
This shows the lemma for j = 0, and hence in particular for n = 1, so that we may proceed
by induction on n.
q
Assuming that 1 ≤ j < n, if dim supp h−j ωR = 0 we are done. Otherwise, we have
q
c = dim supp h−j ωR > 0 and can take p 6= m to be a minimal prime in the support
q
q
of h−j ωR with dim R/p = c. Since dim Rp = n − c < n, we have that (ωR )p [−c] is a
normalized dualizing complex for Rp . Thus, by the induction hypothesis it follows
q

0 = dim supp h−j (ωR )p = dim supp h−j+c (ωR )p [−c] < j − c
q

so that c = dim supp h−j ωR < j as desired.
q

q



We address now a particularly subtle issue surrounding the upper shriek functor and
Frobenius.

F -SINGULARITIES VIA ALTERATIONS

9

Example 2.15 (Trace of Frobenius and behavior of dualizing complexes). A particularly
important setting in this paper is that of a scheme X essentially of finite type over an F finite base scheme S of positive characteristic p (e.g. a perfect field of characteristic p > 0).
This means simply that the (absolute) Frobenius or p-th power map F : X −
→ X is a finite
morphism, and hence proper. Thus we have the “evaluation at 1” trace map
q ∼
q
q
TrF q : F∗ F ! ω =
F∗ RH omX (F∗ OX , ω ) −
→ω .
X

X

X

However, note that since the Frobenius F is generally not an S-morphism we do not have,
q
q
a priori, that F ! ωX ∼
= ωX . This needs an additional assumption, namely that this property
q
q
holds for the base scheme S. Using a fixed isomorphism F ! ωS ∼
= ωS , the compatibility of
( )! with composition in the commutative diagram
X

F

/X
π

π





S
shows that indeed

F

/S

q
q
q
q
q
F ! ωX ∼
= ωX .
= π ! ωS ∼
= π ! F ! ωS ∼
= F ! π ! ωS ∼

Convention 2.16. For simplicity, we will always assume that all our base schemes S of
q
q
positive characteristic p are F -finite and satisfy F ! ωS ∼
= ωS for the given choice of dualizing
q
complex ωS . This is automatic if S is the spectrum of a local ring (e.g. a field).
q ∼ q
Remark 2.17. The assumption F ! ωS =
ωS is convenient but could nonetheless be avoided.
q
q
!
Notice that regardless, F ωS is a dualizing complex, and so it already agrees with ωS up to
a shift and up to tensoring with an invertible sheaf. It is easy to see that the shift is zero
q
q
since F is a finite map, thus we have F ! ωS ∼
= ωS ⊗OS L for some invertible sheaf L . One
option would be to carefully keep track of L throughout all constructions and arguments
– this we chose to avoid. In any case, if one is willing to work locally over the base S (as is
q
q
the case with most of our main theorems) one may always assume that F ! ωS = ωS simply
by restricting to charts where L is trivial.

Proposition 2.8 and Proposition 2.13 can be combined as follows.
Proposition 2.18. Suppose that π : Y −→ X is a proper dominant morphism between
normal integral schemes, and let ∆ be a Q-divisor on X such that KX + ∆ is Q-Cartier.
Then the trace map of π induces a non-zero map
Trπ : hdim Y −dim X Rπ∗ ωY (−⌊π ∗ (KX + ∆)⌋) −→ ωX (−⌊KX + ∆⌋) .
Similarly, if additionally π ∗ (KX + ∆) is a Cartier divisor, then we have another non-zero
map
q
Trπ q : h− dim X Rπ∗ ωY (−⌊π ∗ (KX + ∆)⌋) −→ ωX (−⌊KX + ∆⌋) .
Proof. The difficulty here lies in that ⌊KX + ∆⌋ need not be Q-Cartier. To overcome this,
i
let U ֒−→ X be the regular locus of X, then we have the trace map
Tr

hdim Y −dim X Rπ∗ ωπ−1 (U ) −−−π−→ ωU
q

which is just the restriction of (2.12) to U . Tensoring this map by the invertible sheaf
OU (−⌊(KX + ∆)⌋), we have by the projection formula
hdim Y −dim X Rπ∗ ωπ−1 (U ) (−π ∗ ⌊(KX + ∆)⌋) −
→ ωU (−⌊(KX + ∆)⌋) .

F -SINGULARITIES VIA ALTERATIONS

10

Since −π ∗ ⌊KX + ∆⌋ ≥ −⌊π ∗ (KX + ∆)⌋ on U (where again ⌊KX + ∆⌋ is Cartier), we have
an induced map
(2.19)

hdim Y −dim X Rπ∗ ωπ−1 (U ) (−⌊π ∗ (KX + ∆)⌋) −
→ ωU (−⌊(KX + ∆)⌋) .

Applying i∗ ( ) to (2.19) and composing with the restriction map
hdim Y −dim X Rπ∗ ωY (−⌊π ∗ (KX + ∆)⌋) −
→ i∗ hdim Y −dim X Rπ∗ ωπ−1 (U ) (−⌊π ∗ (KX + ∆)⌋)
now gives the desired first map
Trπ : h− dim X Rπ∗ ωY (−⌊π ∗ (KX + ∆)⌋) −
→ ωX (−⌊KX + ∆⌋)
by noting that i∗ ωU (−⌊(KX + ∆)⌋) = ωX (−⌊(KX + ∆)⌋). To see that Trπ is non-zero,
localize to the generic point of X (where KX + ∆ is trivial) and apply Proposition 2.13.
The construction of the second map Trπ q is similar, rather starting from (2.11) in place
of (2.12). Notice that we require that π ∗ (KX + ∆) to be Cartier so that we have a means
q
of interpreting ωY (−⌊π ∗ (KX + ∆)⌋). Since Trπ q agrees with Trπ generically by the proof
of Proposition 2.13, this map is again non-zero.

In the previous Proposition, we studied trace maps twisted by Q-divisors. In the next
Lemma, we study a special case of this situation which demonstrates that sometimes this
trace map can be re-interpreted as generating a certain module of homomorphisms.
Lemma 2.20. Let X be a normal integral F -finite scheme. Suppose that ∆ is a Q-divisor
such that (pe − 1)(KX + ∆) = div c for some e > 0 and 0 6= c ∈ K(X). If ΦeX : F e ωX −→ ωX
is the trace of the e-iterated Frobenius, then the homomorphism φ( ) = ΦeX (F∗e c · )
generates HomOX (F∗e OX (⌈(pe − 1)∆⌉), OX ) as an F∗e OX -module.
Proof. Essentially by construction (and the definition of ωX ), we have that ΦeX generates
HomOX (F∗e ωX , ωX ) as an F∗e OX -module. Using the identification ωX = OX (KX ) we may
consider ΦeX to generate
q

HomOX (F∗e OX ((1 − pe )KX ), OX ) = HomOX (F∗e ωX , ωX ) .
·c

But then multiplication by c induces an isomorphism OX ((pe − 1)∆) −−→ OX ((1 − pe )KX )
(note (pe − 1)∆ is integral), so that ΦX (F∗e c · ) generates HomOX (F∗e OX (⌈(pe − 1)∆⌉), OX )

as an F∗e OX -module as well.
A main technique in this paper is the observation that the images of the various trace
maps Trπ are preserved under the trace of the Frobenius. We will show this now for
Trπ : hdim Y −dim X Rπ∗ ωY −
→ ωX . We will obtain a partial generalization involving Qdivisors within the proof of Proposition 4.2.
Proposition 2.21. If π : Y −→ X is a proper dominant map of integral schemes, the image
of the trace map
Jπ := Trπ (hdim Y −dim X Rπ∗ ωY ) ⊆ ωX
satisfies ΦX (F∗ Jπ ) ⊆ Jπ .
Proof. Since Frobenius commutes with any map, we get the following diagram for which we
consider the corresponding commutative diagram of structure sheaves
YO

π

F

/X
O
F

Y

π

/X

Rπ∗ OY o

π

OX

Rπ∗ F

F



F∗ Rπ∗ OY o

F∗ π



F∗ OX

.

F -SINGULARITIES VIA ALTERATIONS

11

Applying duality now gives the following commutative diagram of trace maps
(2.22)

Trπ

Rπ∗ ωY
q

q
/ωq
X
O

O

Rπ∗ TrF

TrF

q

F∗ Rπ∗ ωY
q

F∗ Trπ q

.
q

/ F∗ ω q
X

Taking the (− dim X)-th cohomology and composing with the inclusion ωY [dim Y ] −
→ ωY
on the left, we get a diagram
q

(2.23)

hdim Y −dimO X Rπ∗ ωY

/ h− dim X Rπ∗ ω q
Y
O

Trπ

/ h− dim X F∗ Rπ∗ ω q
Y

F∗ Trπ q

q

hdim Y −dim X Rπ∗ ΦY

hdim Y −dim X F∗ Rπ∗ ωY

/ ωX
O
ΦX

/ F∗ ωX

where the left vertical map exists since F is finite and hence F∗ is exact. The horizontal
composition on the top is Trπ and the image Trπ (hdim Y −dim X Rπ∗ ωY ) is Jπ ⊆ ωX . By the
exactness of F∗ , the horizontal composition on the bottom is F∗ Trπ and we get that F∗ Jπ
is the image, and the result now follows.

2.4. Multiplier ideals and pseudo-rationality.
Definition 2.24. [LT81] We say that a reduced connected scheme X is pseudo-rational if
(a) X is Cohen-Macaulay, and
(b) π∗ ωY = ωX for every proper birational map π : Y −
→ X.
Furthermore, if there exists a resolution of singularities π : Y −
→ X, then it is sufficient to
check (b) for this one map π.
If X is of characteristic zero, this coincides via Grauert-Riemenschneider vanishing [GR70]
with the classical definition of rational singularities, meaning there exists a resolution of
singularities π : Y −
→ X such that OX ∼
= π∗ OY and hi Rπ∗ OY = 0 for all i. In positive or
mixed characteristic, it is a distinct notion.
Remark 2.25. It was remarked in [GR70] that if π : Y −
→ X is a resolution of singularities
in characteristic zero, then the subsheaf π∗ ωY ⊆ ωX is independent of the choice of resolution of singularities. This subsheaf should be viewed as an early version of the multiplier
ideal. Compare with the definition of the multiplier module below and the parameter test
submodule in Definition 2.33.
Going back to ideas in K. Smith’s thesis and [Smi95], the natural object to deal with
rational singularities of pairs is the multiplier module, cf. [Bli04, ST08].
Definition 2.26. Given a pair (X, Γ) with Γ a Q-Cartier Q-divisor, then the multiplier module
is defined as
\
J (ωX ; Γ) :=
π∗ ωY (⌈−π ∗ Γ⌉)
π: Y −
→X
where π ranges over all proper birational maps with normal Y .
Note that from this definition it is not clear that J (ωX ; Γ) is even quasi-coherent, as the
infinite intersection of coherent subsheaves need not be quasi-coherent in general. However, if there is a theory of resolution of singularities available (for example over a field
of characteristic zero [Hir64], or in dimension ≤ 2 [Lip78]), it is straightforward to check
coherence by showing that the above intersection stabilizes. Recall that a log resolution of

F -SINGULARITIES VIA ALTERATIONS

12

the pair (X, Γ) is a proper birational map π : Y −
→ X with Y regular and exceptional set
E of pure codimension one such that Supp(E) ∪ π −1 (Supp(Γ)) is a simple normal crossings
divisor. Assuming every normal proper birational modification can be dominated by a log
resolution, one can in fact show
J (ωX ; Γ) = π∗ ωY (⌈−π ∗ Γ⌉)
for any single log resolution π : Y −
→ X, which is in particular coherent. Note that, for
effective Γ it is a subsheaf of ωX via the natural inclusion π∗ ωY ⊆ ωX as in Example 2.1.
Immediately from this definition it follows that X is pseudo-rational if and only if X is
Cohen-Macaulay and J (ωX ) := J (ωX ; 0) = ωX . Hence one defines:
Definition 2.27 ([ST08]). A pair (X, Γ) with Γ ≥ 0 a Q-Cartier Q-divisor is called pseudorational if X is Cohen-Macaulay and J (ωX ; Γ) = ωX . Note that this implies that ⌊Γ⌋ = 0.
The classical notion is of course that of multiplier ideals, which have been defined primarily in characteristic zero. See [Laz04] for a complete treatment in this setting. Historically,
while multiplier ideals first appeared in more analytic contexts and were originally defined
using integrability conditions, one facet of their pre-history was defined for any normal
integral scheme – J. Lipman’s adjoint ideals [Lip94]. The definition we give here (which
makes sense in arbitrary characteristic) is a slight generalization of Lipman’s definition to
the modern setting of pairs.
Definition 2.28 ([Laz04, Lip94]). Given a log-Q-Gorenstein pair (X, ∆) then the multiplier
ideal is defined as
\
J (X; ∆) :=
π∗ OY (⌈KY − π ∗ (KX + ∆)⌉)
π:Y −
→X
where π ranges over all proper birational maps with normal Y and, for each individual π,
we have that KX and KY agree wherever π is an isomorphism.
As above, in a non-local setting, for J (X; ∆) to be quasi-coherent one needs a good
theory of resolution of singularities. In this situation,
J (X; ∆) = π∗ OY (⌈KY − π ∗ (KX + ∆)⌉)
for any log resolution π : Y −
→ X of the pair (X, ∆). In general, J (X; ∆) depends heavily on ∆ and not simply the corresponding linear or Q-linear equivalence class; a similar
observation holds for the multiplier module as well.
If (X, ∆) is a pair, strictly speaking the object J (ωX ; KX + ∆) is ambiguous as KX is
not uniquely determined (and represents a linear equivalence class of divisors). Nonetheless,
for each choice of KX we have that J (ωX ; KX + ∆) ⊆ ωX (−⌊KX + ∆⌋), and is thereby
identified with a submodule of OX (−⌊∆⌋) using Convention 2.7. This construction is in fact
independent of the choice of KX , and allows one to relate multiplier ideals and multiplier
modules in general.
Lemma 2.29. If (X, ∆) is a log-Q-Gorenstein pair, then J (ωX ; (KX + ∆)) = J (X; ∆).
Proof. Suppose π : Y −
→ X is a proper birational map, Y is normal, and KY and KX agree
wherever π is an isomorphism. Making full use of Convention 2.7, we have
=

OX (−⌊∆⌋)
⊆

⊆

ωX (−⌊KX + ∆⌋)

π∗ ωY (−⌊π ∗ (KX + ∆)⌋) = π∗ OY (⌈KY − π ∗ (KX + ∆)⌉)
and the desired conclusion now follows immediately from the definitions.



F -SINGULARITIES VIA ALTERATIONS

13

Definition 2.30. A log-Q-Gorenstein pair (X, ∆) with ∆ ≥ 0 effective is called Kawamata
log terminal if J (X; ∆) = OX .
2.5. The parameter test submodule and F -rationality. We now turn to the characteristic p > 0 notion of F -rationality, extensively studied in [FW89] and [Smi97b], which is
central to our investigations.
Definition 2.31. Suppose that X is reduced, connected, and F -finite (and satisfies Convention 2.17). We say that X is F -rational if
(a) X is Cohen-Macaulay.
(b) There is no proper submodule M ⊆ ωX , non-zero on every irreducible component of
X where ωX is non-zero, such that ΦX (F∗ M ) ⊆ M where ΦX : F∗ ωX −
→ ωX is the
trace of Frobenius as in Example 2.4.
Remark 2.32. While the preceding characterization of F -rationality differs from the definition used historically throughout the literature, it is nonetheless readily seen to be equivalent. Indeed, when X = Spec R for a local ring R with maximal ideal m, the characterization of the tight closure of zero in Hmdim R (R) found in [Smi97b] implies that condition (b)
is Matlis dual to the statement 0∗H dim R (R) = 0.
m

Definition 2.33 ([Smi95, Bli04, ST08]). Suppose that X is reduced, connected, and F -finite
(and satisfies Convention 2.17). The parameter test submodule τ (ωX ) is the unique smallest
subsheaf M of ωX , non-zero on every irreducible component of X where ωX is nonzero,
such that ΦX (F∗ M ) ⊆ M where ΦX : F∗ ωX −
→ ωX is trace of Frobenius.
For X normal and integral, the parameter test submodule τ (ωX ; Γ) of a pair (X, Γ)
with Γ ≥ 0 is the unique smallest non-zero subsheaf M of ωX such that φ(F∗e M ) ⊆ M
for all local sections φ ∈ H omOX (F∗e ωX (⌈(pe − 1)Γ⌉), ωX ) and all e > 0, noting that
ωX ⊆ ωX (⌈(pe − 1)Γ⌉). The further observation φ(F∗ ωX (⌈−Γ⌉)) ⊆ ωX (⌈−Γ⌉) gives that
τ (ωX ; Γ) ⊆ ωX (⌈−Γ⌉).
Once again, the preceding definition is separate from but equivalent to that which is
commonly used throughout the literature. Moreover, standard arguments on the existence
of test elements are required to show the (non-obvious) fact that τ (ωX ) and τ (ωX ; Γ) (as
above) are well-defined. See, for example, [Sch11, Proposition 3.21] (cf. [Sch10, Lemma
2.17]), or more generally [BB11, Bli13].
Lemma 2.34. Suppose that X is reduced, connected, and F -finite. Then X is F -rational
if and only if it is Cohen-Macaulay and τ (ωX ) = ωX . Furthermore, any F -rational scheme
is normal.
Proof. The first statement is an immediate consequence of the definitions, so we need only
show the second. Without loss of generality, we may assume that X = Spec R where R is
an F -rational local ring. If RN is the normalization of R, we will show the inclusion map
i:R−
→ RN is an isomorphism. As R is already assumed Cohen-Macaulay it is S2 , and so
by Serre’s criterion for normality we simply need to check that R is regular in codimension
1. Thus, by localizing we may assume that R is one dimensional (and thus so is RN , which
now must be regular). Consider the following commutative diagram of rings together with
its corresponding Grothendieck-Serre dual (all rings in question are Cohen-Macaulay)
RO N

F

F∗ i

i

R

/ F∗ (RN )
O

F

/ F∗ R

ωRN o

ΦR N

F∗ ωRN
F∗ (i∨ )

i∨


ωR o



ΦR

F∗ ωR

.

F -SINGULARITIES VIA ALTERATIONS

14

As in Example 2.2, i∨ is identified with the evaluation-at-1 map HomR (RN , ωR ) −
→ ωR .
Since i is birational, it is easy to see i∨ is injective. In particular, i∨ is non-zero and
thus also surjective by the definition of F -rationality. It follows that i∨ and hence i are
isomorphisms, whence R is normal as desired.

In order to consider pairs (X, ∆) with ∆ not necessarily effective, we need to recall the
following lemma.
Lemma 2.35. [ST08, Proposition 7.10(3)][ST10, Lemma 6.12] Suppose that (X, ∆) is a pair
with ∆ ≥ 0, and that D ≥ 0 an integral Cartier divisor on X = Spec R, then τ (ωX ; ∆+D) =
τ (ωX ; ∆) ⊗ OX (−D).
Definition 2.36. Suppose that (X, Γ) is a pair. Fix a Cartier divisor D on X such that
Γ + D is effective (these always exist on affine charts). Then the parameter test module of
(X, Γ) is defined as
τ (ωX ; Γ) := τ (ωX ; Γ + D) ⊗ OX (D) .
It is also easy to verify that this definition is independent of the choice of D, hence our local
definition globalizes. It is straightforward to check that τ (ωX ; Γ) ⊆ ωX (⌈−Γ⌉).
In defining the test ideal of a pair below, we handle the non-effective case analogously.
Definition 2.37. Suppose that X is reduced and F -finite. The test ideal τ (X) is the
unique smallest ideal J of OX , non-zero on every irreducible component of X, such that
φ(F∗e J) ⊆ J for every local section φ ∈ H omOX (F∗e OX , OX ) and all e > 0.
If (X, ∆) is a pair with ∆ ≥ 0, the test ideal τ (X; ∆) is the unique smallest non-zero ideal
J of OX such that φ(F∗e J) ⊆ J for any local section φ ∈ H omOX (F∗e OX (⌈(pe − 1)∆⌉), OX )
and all e > 0, noting that J ⊆ OX ⊆ OX (⌈(pe − 1)∆⌉).
As with the parameter test module, one has for any effective integral Cartier Divisor D
the equality τ (X; ∆ + D) = τ (X; ∆) ⊗ OX (−D), which allows one to extend the definition
to the non-effective case as above (see [ST08] for further details). Furthermore, the same
subtle albeit well-known arguments are again required to show these ideals exist [Sch11,
Proposition 3.21] (see also [ST12a]).
Remark 2.38. As before, the preceding definition is non-standard; rather, what we have
just defined is an alternative yet equivalent characterization of the big or non-finitistic
test ideal, commonly denoted in the literature by τb (X, ∆) or τ̃ (X, ∆). However, in many
situations (and conjecturally in general) the big test ideal agrees with the classically defined
or finitistic test ideal. Indeed, these two notions are known to coincide whenever KX + ∆
is Q-Cartier [Tak04, BSTZ10] – the only setting considered in this paper. For this reason,
as well as our belief that the big test ideal is the correct object of study in general, we will
drop the adjective big from the terminology throughout.
Strictly speaking the object τ (ωX ; KX + ∆) is ambiguous as KX is not uniquely determined. Indeed supposing ∆ ≥ 0, as seen from Lemma 2.35, different choices of KX give rise
to different submodules τ (ωX ; KX + ∆) of ωX . However, we in fact have τ (ωX ; KX + ∆) ⊆
ωX (−⌊KX + ∆⌋), so that τ (ωX ; KX + ∆) is identified with a submodule of OX (−⌊∆⌋)
using Convention 2.7. As was the case with the multiplier module, this construction is
independent of the choice of KX .
Lemma 2.39. If X is an F -finite and (X, ∆) is a pair, then τ (ωX ; KX + ∆) = τ (X; ∆).
Proof. Choose a rational section s ∈ ωX determining a canonical divisor KX = KX,s = div s
s7→1

and the embedding ωX ⊆ ωX ⊗ K(X) −−−−→ K(X). We will write ωX = OX (KX ) for the

F -SINGULARITIES VIA ALTERATIONS

15

duration of the proof without further remark. Working locally, it is harmless to assume KX
and ∆ are both effective by Definition 2.36, so that τ (ωX ; KX + ∆) ⊆ ωX (−⌊KX + ∆⌋) =
OX (−⌊∆⌋) gives in particular τ (ωX ; KX + ∆) ⊆ OX ⊆ ωX .
Next, observe for all e > 0 that
H omOX (F∗e ωX (⌈(pe − 1)(KX + ∆)⌉), ωX ) = H omOX (F∗e OX (⌈(pe − 1)∆⌉), OX )
in a very precise sense; they are equal after mapping to the corresponding stalks at the
generic point of X, which are explicitly identified with one another. Said another way, the
image of F∗e OX (⌈(pe −1)∆⌉) ⊆ F∗e ωX (⌈(pe −1)(KX +∆)⌉) under every local homomorphism
φ ∈ H omOX (F∗e ωX (⌈(pe − 1)(KX + ∆)⌉), ωX ) satisfies φ(F∗e OX (⌈(pe − 1)∆⌉)) ⊆ OX , giving
rise to a commutative diagram

F∗e OX

ωX
/

⊆

F∗e OX (⌈(pe − 1)∆⌉)

⊆

φ

⊆

⊆ F∗e ωX (⌈(pe − 1)(KX + ∆)⌉)

⊆

F∗e ωX

φ

/

OX

and uniquely accounting for every local homomorphism in H omOX (F∗e OX (⌈(pe −1)∆⌉), OX ).
The desired conclusion is now an exercise in manipulating definitions. As τ (X; ∆) ⊆ ωX
is preserved under the local homomorphisms in H omOX (F∗e ωX (⌈(pe − 1)(KX + ∆)⌉), ωX ),
we must have τ (ωX ; KX + ∆) ⊆ τ (X; ∆) by the definition of τ (ωX ; KX + ∆). Conversely,
as τ (ωX ; KX + ∆) ⊆ OX is preserved under H omOX (F∗e OX (⌈(pe − 1)∆⌉), OX ), we must
have τ (X; ∆) ⊆ τ (ωX ; KX + ∆) and the statement follows.

2.6. Transformation behavior of test ideals and multiplier ideals. One of the contributions of this paper is a further clarification of the transformation behavior of test and
multiplier ideals. Let us summarize what is known so far for alterations, which can always
be viewed as compositions of proper birational maps and finite dominant maps.
Let us first consider the classical (and transparent) case of the multiplier ideal in characteristic zero [Laz04, 9.5.42]. Essentially by definition, the multiplier ideal of a logQ-Gorenstein pair (X, ∆X ) is well-behaved under a proper birational morphism π : Y −
→X
and satisfies
(2.40)

π∗ J (Y, ∆Y ) = J (X, ∆X ) with ∆Y := π ∗ (KX + ∆X ) − KY

where we have arranged that π∗ KY = KX as usual. If rather π : Y −
→ X is a finite dominant
∗
map, one sets KY = π KX + Ramπ where Ramπ is the ramification divisor, and has the
transformation rule
(2.41)
π∗ J (Y, ∆Y ) ∩ K(X) = J (X, ∆X ) with ∆Y := π ∗ (KX + ∆X ) − KY = π ∗ ∆X − Ramπ .
In Section 8, we further generalize this rule to incorporate the trace map Example 2.2
(2.42)

Trπ (π∗ J (Y, ∆Y )) = J (X, ∆X )

in the process of showing our main theorem in characteristic zero.
In characteristic p > 0, the transformation rule (2.40) for the multiplier ideal under proper
birational maps once again follows immediately from the definition. However, the behavior
of the multiplier ideal for finite maps is more complicated and not fully understood in general
– even for finite separable maps. In Section 8, we will show that both (2.41) and (2.42) hold
for separable finite maps of degree prime to p (and more generally when Trπ (π∗ OY ) = OX ).
However, Examples 3.10, 6.13, and 7.12 in [ST10] together show that neither formula is
valid for arbitrary separable finite maps in general.

F -SINGULARITIES VIA ALTERATIONS

16

In contrast, the last two authors in [ST10] have completely described the behavior of
the test ideal of a pair (X, ∆X ) under arbitrary finite dominant maps π : Y −
→ X. In the
separable case, one again has
Trπ (π∗ τ (Y, ∆Y )) = τ (X, ∆X )

with ∆Y := π ∗ (KX + ∆X ) − KY = π ∗ ∆X − Ramπ .

More generally, when π is not necessarily separable, a similar description holds after reinterpretation of the ramification divisor (via the Grothendieck trace). However, a formula
as simple as (2.40) relating test ideals under a birational map cannot hold. Indeed, if
π: Y −
→ X is a log resolution of (X, ∆X ), then the multiplier and test ideal of (Y, ∆Y )
will agree while those of (X, ∆X ) may not (cf. [Tak04, Theorems 2.13, 3.2]). Nonetheless
in Theorem 6.8 we will give a transformation rule – albeit far more complex – for the test
ideal under alterations in general, so in particular for birational morphisms.
In summary we observe that the test ideal behaves well under finite maps (and may
be computed using either finite maps or alterations by Theorem 4.6), whereas its behavior
under birational maps is much more subtle. On the other hand, the multiplier ideal in
characteristic zero behaves well under finite and birational maps, however finite maps will
not suffice for its computation. The multiplier ideal in positive characteristic is still well
behaved under birational transformations, but its behavior under finite maps is more subtle.
3. F -rationality via alterations
In this section, we will characterize F -rationality and, more generally, the parameter test
submodule in terms of alterations. This is – at the same time – a special case of our Main
Theorem as well as a key ingredient in its proof. The full proof of our Main Theorem in
positive characteristic consists of a reduction to the cases treated here and will follow in the
next section.
The crux of the argument to follow is based on a version of the equational lemma of [HH92]
in the form that is found in [HL07]. In fact, we require a variant with an even stronger
conclusion; namely, that the guaranteed finite extension may be assumed separable. This
generalization follows from a recent result of A. Sannai and A. Singh [SS12].
Lemma 3.1 (equational lemma). Consider a domain R with characteristic p > 0. Let K
be the fraction field of R, K̄ an algebraic closure of K, and I an ideal of R. Suppose i ≥ 0
2
and α ∈ HIi (R) is such that α, αp , αp , . . . belong to a finitely generated R-submodule of
HIi (R). Then there exists a separable R-subalgebra R′ of K̄ that is a finite R-module such
that the induced map HIi (R) −→ HIi (R′ ) sends α to zero.
Proof. The statement of the equational lemma in [HL07, Lemma 2.2] is the same as above
without the desired separability. Applying this weaker version we therefore have a finite
extension R ⊆ R′ such that HIi (R) −
→ HIi (R′ ) maps α to zero. Let Rs be the separable
′
closure of R in R , that is all elements of R′ that are separable over R. The extension
Rs ⊆ R′ is then purely inseparable, i.e. some power of the Frobenius has the property that
F i (R′ ) ⊆ Rs . Applying the functor HIi ( ) yields the diagram
HIi (R)

/ H i (Rs )
I

/ H i (R′ )
I
tt
t
t
F i ttt
 ztt F i

HIi (Rs )

Denoting the image of α in HIi (Rs ) by a, this element is mapped to zero HIi (R′ ), hence,
under the Frobenius F i it is mapped also to zero in HIi (Rs ). But this shows that F i (a) = 0

F -SINGULARITIES VIA ALTERATIONS

17

in HIi (Rs ) by the above diagram. Now [SS12, Theorem 1.3(1)] states that there is a module
finite separable (even Galois with solvable Galois group) extension Rs ⊆ R′′ such that a is
mapped to zero in HIi (R′′ ). But this means that the finite separable extension R ⊆ R′′ is
as required: the image of α ∈ HIi (R) in HIi (R′′ ) is zero.

The main result of this section, immediately below, is a straightforward application of
the method of C. Huneke and G. Lyubeznik in [HL07]. Since making this observation, we
have been informed that a Matlis dual version of the theorem below has also been obtained
by M. Hochster and Y. Yao (in a non-public preprint [HY11]).
Theorem 3.2. Suppose X is an integral F -finite scheme.
(a) For all proper dominant maps π : Y −→ X with Y integral, the image of the trace
map (2.12) contains the parameter test module, i.e.
Tr

τ (ωX ) ⊆ Image(hdim Y −dim X Rπ∗ ωY −−−π→ ωX ) .
(b) There exists a finite separable map π : Y −→ X with Y integral such that the image
of the trace map equals the parameter test module, i.e.
Tr

τ (ωX ) = Image(hdim Y −dim X Rπ∗ ωY −−−π→ ωX ) .
Remark 3.3. In the above result, it is possible to work with equidimensional reduced (rather
than integral) schemes of finite type over a field at the expense of removing “separable”
from the conclusion in (b).
An immediate Corollary of this statement is a characterization of the test submodule. Recall that an alteration is a generically finite proper dominant morphism of integral schemes.
Corollary 3.4. Suppose X is an integral F -finite scheme. Then
\
Tr
τ (ωX ) =
Image(hdim Y −dim X Rπ∗ ωY −−−π→ ωX ).
π: Y −
→X
where π ranges over all of the maps from an integral scheme Y to X that are either:
◦ (separable) finite dominant maps, or
◦ (separable) alterations, or
◦ (separable) proper dominant maps.
Additionally, if X is a variety over a perfect field, then we may allow π to range over
◦ all regular separable alterations to X, or
◦ all separable proper dominant maps with Y regular.
Furthermore, in this case, there always exists a separable regular alteration π : Y −→ X such
Tr
that Image(π∗ ωY −−−π→ ωX ) is equal to the parameter test submodule of X.
Proof. This follows immediately from Theorem 3.2 and from the existence of regular alterations [dJ96].

Proof of Theorem 3.2 (a). The statement follows immediately from Propositions 2.21 and
2.13 as well as the definition of τ (ωX ).

The proof of (b) follows closely the strategy of [HL07]; note that a local version of the
statement is also related to the result of K. Smith that “plus closure equals tight closure
for parameter ideals” [Smi94]. The proof comes down to Noetherian induction once we
establish the following lemma.

F -SINGULARITIES VIA ALTERATIONS

18

Lemma 3.5. Let R ⊆ S be a module finite inclusion of domains and denote the image of
the trace map by JS = Image(ωS −→ ωR ). If τ (ωR ) ( JS , then there is a separable finite
extension of domains S ⊆ S ′ such that the support of JS ′ /τ (ωR ) is strictly contained in the
support of JS /τ (ωR ), where JS ′ = Image(ωS ′ −→ ωR ).
Proof. Choose η ∈ Spec R to be the generic point of a component of the support of JS /τ (ωR )
and set d = dim Rη . By construction (JS /τ (ωR ))η = JSη /τ (ωRη ) has finite length, and hence
so also must its Matlis dual
(JS /τ (ωR ))∨
η = Hom(JSη /τ (ωRη ), E(Rη /η)) .
By Matlis duality applied to the sequence ωSη −
→
→(JSη /τ (ωRη )) ֒−
→ ωRη /τ (ωRη ) one obd (S ) is identified with the image of the tight closure of zero
serves that (JS /τ (ωR ))∨
⊆
H
η
η
η
0∗H d (Rη ) = (ωRη /τ (ωRη ))∨ in Hηd (Sη ). By Proposition 2.21 we have for ΦR : F∗ ωR −
→ ωR
η

that ΦR (JS ) ⊆ JS . Hence the dual of JS /τ (ωR ) is stable under the action of the Frobenius
∨
on Hηd (Sη ). i.e. F ((JS /τ (ωR ))∨
η ) ⊆ (JS /τ (ωR ))η (phrased differently: the tight closure
of zero is Frobenius stable, hence so is its image). This implies that for any element
p
p2
∨
α ∈ (JS /τ (ωR ))∨
η the powers α, α , α , . . . must also lie in the finite length (JSη /τ (ωRη )) .
∨
Applying Lemma 3.1 to α ∈ (JSη /τ (ωRη )) repeatedly (e.g. for a finite set of generators) we obtain a separable integral extension Sη ⊆ T such that Hηd (Sη ) −
→ Hηd (T ) maps
∨
′
(JSη /τ (ωRη )) to zero. By taking S to be the normalization of S in the total field of fractions of T we see that T = Sη′ and that the finite extension R ⊆ S ′ is separable. Translating
this back via Matlis duality this means that the map ωSη′ −
→ ωRη sends JSη′ into τ (ωRη ). In

particular η 6∈ Supp(JS ′ /τ (ωR )).
Proof of Theorem 3.2 (b). Without loss of generality, we may assume that X = Spec R is
affine. Starting with the identity R = S0 we successively produce, using Lemma 3.5, a
sequence of separable finite extensions R = S0 ⊆ S1 ⊆ S2 ⊆ S3 . . . such that the support
of JSi+1 /τ (ωR ) is strictly smaller than the support of JSi /τ (ωR ) until JSi = τ (ωR ) by
Noetherian induction.

The following important corollary (which can also be proven directly from the equational
lemma without reference to the above results) should be viewed in the context of the
definition of pseudo-rationality (see Section 2.4), as well as the Kempf-criterion for rational
singularity [KKMSD73, p. 50] in characteristic zero.
Corollary 3.6. For an F -finite Cohen-Macaulay domain R the following are equivalent.
(a) R is F -rational, i.e. there is no non-trivial submodule M ⊆ ωR which is stable under
ΦR : F∗ ωR −→ ωR .
(b) For all finite extensions R −→ S (which may be taken to be separable if desired) the
induced map ωS −→ ωR is surjective.
(c) For all alterations π : Y −→ X = Spec R (π may be taken to be separable and or
regular if R is of finite type over a perfect field) the induced map π∗ ωY −→ ωX is
surjective.
In fact, utilizing local duality and [HL07] once again to obtain a further finite cover which
annihilates the local cohomology modules below the dimension, we obtain the following
characterization of F -rationality without the Cohen-Macaulay hypothesis.
Corollary 3.7. For an F -finite domain R, the following are equivalent:
(a) R is F -rational.

F -SINGULARITIES VIA ALTERATIONS

19

(b) For each integer i ∈ Z, and every (separable) finite extension of rings R −→ S, the
q
q
induced map hi ωS −→ hi ωR is surjective.
(c) For each integer i ∈ Z and every (separable regular, if R is of finite type over a
q
q
perfect field) alteration π : Y −→ X = Spec R , the induced map hi Rπ∗ ωY −→ hi ωX
is surjective.
Remark 3.8. The main result of the paper [HL07] which inspired our proof is that for a local
ring (R, m) of dimension d that is a homomorphic image of a Gorenstein ring, there is a
module finite extension R ⊆ S such that the induced map Hmi (R) −
→ Hmi (S) is zero for i < d.
q
q
A local dual statement to this is that the induced map on dualizing complexes ωS −
→ ωR
q
q
is zero on cohomology hi (ωS ) −
→ hi (ωR ) for i 6= − dim R. What we accomplish here is
that we clarify the case i = − dim R. With d = dim R we just showed that one can also
q
q
achieve that the image h−d (ωS ) −
→ h−d (ωR ) ∼
= ωR is the parameter test submodule τ (ωR ).
Of course, the dual statement thereof is: the tight closure of zero 0∗H d (R) is mapped to zero
m

→ Hmd (S). It is exactly this statement, and further generalizations,
under the map Hmd (R) −
which are first and independently by M. Hochster and Y. Yao in [HY11].
4. Test ideals via alterations
In this section, we explore the behavior of test ideals under proper dominant maps and
prove our main theorem in characteristic p > 0 in full generality. First we show that various
images are compatible with the Φ from Example 2.4, and so they contain the test ideal.
Proposition 4.1. Suppose that f : Y −→ X is a proper dominant generically finite map of
normal F -finite varieties and that (X, ∆) is a log-Q-Gorenstein pair. Then the test ideal is
contained in the image of the trace map, i.e.


Trf
∗
τ (X; ∆) ⊆ Image f∗ ωY (−⌊f (KX + ∆)⌋) −−−→ ωX (−⌊KX + ∆⌋) ⊆ K(X)
where Trf is the map induced by trace as in Proposition 2.8.
g

h

Proof. This follows easily by Stein factorization. Factor f as Y −
→ Z −
→ X where g is
birational and h is finite. Then it follows from [ST10, Theorem 6.25] (cf. [ST12b, Lemma
4.4(a)]) that
Trh (h∗ τ (Z; −KZ + h∗ (KX + ∆))) = τ (X; ∆).
On the other hand, it follows from the argument that the test ideal is contained in the
multiplier ideal (since the test ideal is the unique smallest ideal satisfying a certain property),
that
Trg (g∗ ωY (−⌊f ∗ (KX + ∆)⌋)) ⊇ τ (Z; −KZ + h∗ (KX + ∆)).
See [Tak04] or see [ST12a, Theorem 4.17] for a sketch of a simpler version of this argument
(or see immediately below for a generalization).
This completes the proof since Trh ◦h∗ Trg = Trf .

We also prove a more general statement whose proof also partially generalizes Proposition 2.21,
and which uses heavily the material from the preliminary section on Duality, Section 2.3.
The reader who has so far avoided that section, can skip Proposition 4.2 and rely on
Proposition 4.1 instead.
Proposition 4.2. Suppose that f : Y −→ X is a proper dominant map of normal integral
F -finite schemes and that (X, ∆) is a log-Q-Gorenstein pair. Then the test ideal is contained

F -SINGULARITIES VIA ALTERATIONS

20

in the image of the trace map, i.e.


Trf
dim Y −dim X
∗
τ (X; ∆) ⊆ Image h
Rf∗ ωY (−⌊f (KX + ∆)⌋) −−−→ ωX (−⌊KX + ∆⌋) ⊆ K(X)
where Trf is the map induced by trace as in Proposition 2.18. Similarly, if additionally
f ∗ (KX + ∆) is a Cartier divisor, then


Trf q
q ∗
− dim X
τ (X; ∆) ⊆ Image h
Rf∗ ωY (f (KX + ∆)) −−−−→ ωX (−⌊KX + ∆⌋) ⊆ K(X)
where again Trf q is the map induced by trace as in Proposition 2.18.

Proof. The statement is local (if it fails to hold, then it fails to hold locally), so we assume
that X is the spectrum of a local ring. Without loss of generality, as in Definition 2.37, we
may assume that ∆ ≥ 0. Let n = dim Y − dim X.
→ OX such that ∆φ ≥ ∆, where ∆φ is defined as in
Fix an OX -linear map φ : F∗e OX −
[Sch09, Theorem 3.11, 3.13] or [ST12a, Subsection 4.4]. Recall that (pe − 1)(KX + ∆φ ) ∼ 0,
so in particular we may write (pe − 1)(KX + ∆) = div c for some c ∈ K(X). For the first
statement, it is sufficient to show that


φ F∗e Trf (hn Rf∗ ωY (−⌊f ∗ (KX + ∆)⌋)) ⊆ Trf hn Rf∗ ωY (−⌊f ∗ (KX + ∆)⌋)

and by Lemma 2.20, we may assume φ( ) = ΦeX (F∗e c ·
φ(F∗e

Trf ( )) =

ΦeX

((F∗e c)

· F∗e Trf (

)) =

ΦeX (F∗e Trf (c

). Now we have

·

)) = Trf (hn Rf∗ ΦeY (F∗e c ·

))

where we have used that Trf is OX -linear, and that ΦeX (F∗e Trf ( )) = Trf (hn Rf∗ ΦeY (F∗e ))
as shown in (2.23). Thus, it is enough to show

ΦeY F∗e (c · ωY (−⌊f ∗ (KX + ∆)⌋)) ⊆ ωY (−⌊f ∗ (KX + ∆)⌋) .

Since

−⌊f ∗ (KX + ∆)⌋ + (1 − pe )f ∗ (KX + ∆φ ) = −⌊f ∗ (pe KX + (pe − 1)∆φ + ∆)⌋
and ∆φ ≥ ∆, we have
c · ωY (−⌊f ∗ (KX + ∆)⌋) ⊆ ωY (−⌊pe f ∗ (KX + ∆)⌋) = ωY (−⌊(F e )∗ f ∗ (KX + ∆)⌋) .
But then, according to Proposition 2.18, we have
ΦeY (F∗e ωY (−⌊(F e )∗ f ∗ (KX + ∆)⌋)) ⊆ ωY (−⌊f ∗ (KX + ∆)⌋)
which completes the proof of the first statement. For the second statement, simply notice
that Image(Trf q ) ⊇ Image(Trf ).

Lemma 4.3. Suppose that (X = Spec R, Γ) is a pair such that Γ = t div(g) for some g ∈ R
and t ∈ Q≥0 . Fix c ∈ R such that Rc is regular and that Supp(Γ) = V (g) ⊆ V (c). Then
there exists a power of cN of c such that
X
e
τ (ωX ; Γ) =
Φe (F∗e cN g ⌈t(p −1)⌉ ωR ).
e≥0

where

ΦeR

:

F∗e ωR

−→ ωR denotes the trace of the e-iterated Frobenius, see Section 2.3.

Proof. By the usual theory of test elements (cf. [Sch11, proof of Theorem 3.18]), we have
for some power cn of c that
XX
τ (ωX ; Γ) =
φ(F∗e cn ωR )
e≥0 φ

F -SINGULARITIES VIA ALTERATIONS

21

where the inner sum ranges over all φ ∈ HomR (F∗e ωR (⌈(pe − 1)Γ⌉), ωR ). Furthermore, note
it is harmless to replace n by any larger integer n + k. The reason the statement does not
follow immediately is that, as div(g) may not be reduced, we may have ⌈t(pe − 1)⌉ div(g) ≥
e
⌈t(pe − 1) div(g)⌉. Choose k such that div(ck ) + (pe − 1)Γ ≥ div(g⌈t(p −1)⌉ ) for all e ≥ 0,
and set N = n + k.
e
Now, the map ψ( ) = ΦeR (g⌈t(p −1)⌉ · ) ∈ HomR (F∗e ωR (⌈(pe − 1)Γ⌉), ωR ) appears
in the sum above. This implies the containment ⊇ for our desired equality. Furthermore, for any φ ∈ HomR (F∗e ωR (⌈(pe − 1)Γ⌉), ωR ), it is clear then that φ(F∗e cN ωR ) ⊆
e

F∗e Φe (F∗e cn g⌈(p −1)⌉ ωR ). Thus, we have the containment ⊆ as well.
We now describe the transformation rule for the parameter test module under finite maps.
Proposition 4.4 (cf. [ST10]). Given a finite map f : Y −→ X of normal F -finite integral
schemes and a Q-Cartier Q-divisor Γ on X, we have

Trf f∗ τ (ωY ; f ∗ Γ) = τ (ωX ; Γ) .

Proof. Without loss of generality, we may assume that X = Spec R and Y = Spec S are
affine and that Γ = t div(g) for some g ∈ R.
→ ωS be the corresponding traces of Frobenius
→ ωR and ΦeS : F∗e ωS −
Let ΦeR : F∗e ωR −
and set JS = Image(Trf : f∗ ωS −
→ ωR ) ⊆ ωR . Now, Lemma 4.3 above implies that there
exists an element c ∈ R such that both
X
e
τ (ωR ; Γ) =
ΦeR (F∗e cg⌈t(p −1)⌉ JS )
e≥0

τ (ωS ; f ∗ Γ) =

X

ΦeS (F∗e cg⌈t(p

e −1)⌉

ωS ) .

e≥0

The idea for the remainder of the proof is to apply Trf ( ) to τ (ωS , f ∗ Γ), noting that
Trf (f∗ ΦeS ( )) = ΦeR (F∗e Trf ( )) since trace is compatible with composition (shown precisely in (2.23)). Therefore,
 

X
e
Trf (f∗ τ (ωS ; f ∗ Γ)) = Trf f∗ 
ΦeS (F∗e cg⌈t(p −1)⌉ ωS )
=

X
e≥0

=

X
e≥0

=

X



e≥0

Trf f∗ ΦeS (F∗e cg⌈t(p

e −1)⌉


f∗ ωS )



e
ΦeR F∗e Trf (cg⌈t(p −1)⌉ f∗ ωS )
ΦeR (F∗e cg⌈t(p

e −1)⌉

JS )

e≥0

= τ (ωR ; Γ)
which completes the proof.



To reduce the main theorem of this paper to Theorem 3.2, we need a variant of the cyclic
covering construction, cf. [TW92] or [KM98, Section 2.4].
Lemma 4.5. Suppose that X is a normal integral scheme and Γ is a Q-Cartier Q-divisor
on X. Then there exists a finite separable map g : W −→ X from a normal integral scheme
W such that g ∗ Γ is a Cartier divisor.

F -SINGULARITIES VIA ALTERATIONS

22

Proof. We may assume that X = Spec R is affine and that nΓ = divX (f ) for some non-zero
non-unit f ∈ R. We view f ∈ K = K(R) and suppose that α is a root of the polynomial
xn + f x + f in some separable finite field extension L of K. Let S be the normalization of R
inside L so that we have a module-finite inclusion R ⊆ S. Set π : Y = Spec S −
→ Spec R =
n = −f (α + 1) we
X. Further observe
that
S
contains
α
since
α
is
integral
over
R.
Since
α
p
have α, α + 1 ∈ hα + 1i, so that α + 1 is a unit. Thus, n divY (α) = divY (f ) = π ∗ nΓ, and
so π ∗ Γ = divY (α) is Cartier as desired.

Theorem 4.6. Given a normal integral F -finite scheme X with a Q-divisor ∆ such that
KX + ∆ is Q-Cartier, there exists a finite separable map f : Y −→ X from a normal F -finite
integral scheme Y such that


Trf
∗
(4.7)
τ (X; ∆) = Image f∗ ωY (−⌊f (KX + ∆)⌋) −−→ K(X) .
Alternatively, if X is of finite type over a F -finite (respectively perfect) field, one may take
f : Y −→ X to be a regular (respectively separable) alteration.
Before proving this theorem, we state several corollaries.
Corollary 4.8. Assume that X is a normal variety over an F -finite field k. If (X, ∆) is
a log-Q-Gorenstein pair, then using the images of the trace map as in Proposition 2.18, we
have
!
\
Tr
f
(4.9) τ (X; ∆) =
Image hdim Y −dim X Rf∗ ωY (−⌊f ∗ (KX + ∆)⌋) −−→ K(X)
f :Y −
→X
where f : Y −→ X ranges over all maps from a normal variety Y that are either:
(a) finite dominant maps
(b) finite separable dominant maps
(c) alterations (i.e. generically finite proper dominant maps)
(d) regular alterations
(e) proper dominant maps
(f) proper dominant maps from regular schemes
or, if additionally k is perfect,
(g) regular separable alterations.
Furthermore, in all cases the intersection stabilizes (i.e. is equal to one of its members).
Corollary 4.10. Assume X is a normal integral F -finite scheme with a Q-divisor ∆ such
that KX + ∆ is Q-Cartier. Then using the images of the trace map as in Proposition 2.18,
we have
!
\
q
Tr
f
q
(4.11)
τ (X; ∆) =
Image h− dim X Rf∗ ωY (−⌊f ∗ (KX + ∆)⌋) −−−→ K(X)
f :Y −
→X
where the intersection runs over all proper dominant maps f : Y −→ X from a normal integral scheme Y such that f ∗ (KX + ∆) is Cartier. Furthermore, once again, the intersection
stabilizes.
Proof of Theorem 4.6. We make use the identification of τ (X; ∆) = τ (ωX ; KX + ∆) from
Lemma 2.39 and will prove the statement for the latter. For (4.7), first take a separable finite
map f ′ : X ′ −
→ X such that Γ := f ′∗ (KX +∆) is Cartier by Lemma 4.5. By Proposition 4.4,

Trf ′ f∗′ τ (ωX ′ ; Γ) = τ (ωX ; KX + ∆) = τ (X; ∆).

F -SINGULARITIES VIA ALTERATIONS

23

Now, using Theorem 3.2 we may fix a separable finite map h : Y −
→ X ′ such that we
→ ωX ′ ). The projection formula then gives the equality
have τ (ωX ′ ) = Image(Trh : h∗ ωY −
→ ωX ′ ), and (4.7) now follows after applying Trf ′ .
τ (ωX ′ ; Γ) = Image(Trh (h∗ ωY (−h∗ Γ) −
For the remaining statement when X, if we are given a composition of alterations f ◦ g :
g

f

Z−
→Y −
→ X, it follows that
Trf ◦g

Image(f∗ g∗ ωZ (−⌊g∗ f ∗ (KX + ∆)⌋) −−−−→ K(X))
Trf

⊆ Image(f∗ ωY (−⌊f ∗ (KX + ∆)⌋) −−−→ K(X)) .
Thus, the second statement immediately follows from the first by taking a further regular
(separable) alteration using [dJ96, Theorem 4.1].

Proof of Corollary 4.8. For equation (4.9), we have the containment τ (X; ∆) ⊆ from Proposition 4.1
or Proposition 4.2. The result then follows from equation (4.7).

Proof of Corollary 4.10. Finally, for equation (4.11) we still have the containment τ (X; ∆) ⊆
from Proposition 4.2. On the other hand, if Y has the same dimension as X, then it is readily seen from the spectral sequence argument used in the proof of Proposition 2.13 that




q
Trf q h− dim X Rf∗ ωY (−⌊f ∗ (KX + ∆)⌋) = Trf hdim Y −dim X Rf∗ ωY (−⌊f ∗ (KX + ∆)⌋) .
so that equality holds in (4.11) for f : Y −
→ X as in Theorem 4.6.



Remark 4.12. If R is an F -finite Q-Gorenstein splinter ring (i.e. any module-finite extension
R ⊆ S splits as a map of R-modules), then Corollary 4.8 above gives that τ (X) = OX ,
implying that R is strongly F -regular (since by τ (X) we always mean the big test ideal).
This recovers the main result of [Sin99].
5. Nadel-type vanishing up to finite maps
Among the most sorely missed tools in positive characteristic birational algebraic geometry (in comparison to characteristic zero) are powerful cohomology vanishing theorems.
Strong additional assumptions (e.g. lifting to the second Witt vectors) are required to
recover the most basic version of Kodaira vanishing, and even under similar assumptions
the most powerful variants (e.g. Kawamata-Viehweg or Nadel-type vanishing) cannot be
proven. By applying the results and ideas of B. Bhatt’s dissertation (see also [Bha12]), we
derive here variants of Nadel-type vanishing theorems. These are strictly weaker than what
one would hope for as we only obtain the desired vanishing after a finite covering. Notably
however, we need not require a W2 lifting hypothesis.
Before continuing, we recall the following well known Lemma.
Lemma 5.1 (cf. [Bha10]). Suppose that X is a Noetherian scheme and that we have a
q
q
q
≥0
map of objects f : A −→ B within Dcoh
(X). Then f factors through h0 B if and only if
τ >0 (f ) is the zero map.
Proof. Certainly if f factors through h0 B , then τ >0 (f ) = 0 since τ >0 (h0 B ) = 0. For
the converse direction, suppose τ >0 (f ) = 0. Consider the diagram
q

h0 A

q

/ Aq

q

q

q

/ τ >0 A q

+1

/

0



h0 B

/B



/ τ >0 B q

+1

/

F -SINGULARITIES VIA ALTERATIONS

Since A −
→ B
≥0
Dcoh
(X)
q

q

−
→ τ >0 B

24

is zero, we also have the following diagram of objects in

q

A

id

q



h0 B

q

+1

/ Aq

/0


/Bq


/ τ >0 B q

.
/

+1

/


We begin with a lemma which can be viewed as a kind of “Grauert-Riemenschneider
vanishing [GR70] up to finite maps.” Using the notation from below, note that in the
special case where W is smooth (or a tame quotient singularity) it has been shown only
recently by A. Chatzistamatiou and K. Rülling [CR11] that hi Rπ∗ ωX is zero for i > 0.
Lemma 5.2. Suppose that π : X −→ W is an alteration between integral schemes of characteristic p > 0. Then there exists a finite map g : U −→ X such that the trace map
(5.3)

τ >− dim X R(π ◦ g)∗ ωU −→ τ >− dim X Rπ∗ ωW
q

q

is zero and furthermore that the trace map
(5.4)

τ >0 R(π ◦ g)∗ ωU −→ τ >0 Rπ∗ ωW

is zero. As a consequence hi R(π ◦ g)∗ ωU −→ hi Rπ∗ ωX is zero for all i > 0.
Proof. It is harmless to assume that π is birational (simply take the normalization of W
inside the fraction field of X) and also that W is affine.
q
q
→ ωX factors through
First, choose a finite cover a : X ′ −
→ X such that a : ωX ′ −
ωX [dim X] by [Bha10, Proposition 5.4.2]. Set X ′ −
→ W′ −
→ W to be the Stein factorization
′
of π ◦ a (thus ℓ : W −
→ W is finite). By [Bha10, Theorem 5.0.1], there exists a finite cover
→ R(π ◦ a ◦ b)∗ OX factors through
→ X ′ from a normal X such that R(π ◦ a)∗ OX ′ −
b:X −
→ W to be the Stein-factorization of π ◦ a ◦ b (thus m : W −
→ W′
(π ◦ a ◦ b)∗ OX . Set X −
f−
is finite and (π ◦ a ◦ b)∗ OX = (l ◦ m)∗ OW ). Choose a further cover n : W
→ W such that
q
q
f
→ τ >− dim X (ωW ) is zero by [Bha10, Proposition 5.4.2]. By making W
τ >− dim X (n∗ ω f ) −
W
e−
larger if necessary, we additionally assume that c : X
→ X, the normalization of X in the
f
fraction field of W , satisfies the condition that R(π ◦ a ◦ b)∗ OX −
→ R(π ◦ a ◦ b ◦ c)∗ OXe factors
again
using
[Bha10,
Theorem 5.0.1].
through (π ◦ a ◦ b ◦ c)∗ OXe ∼
= (l ◦ m ◦ n)∗ OW
f
e
X

c

b

/ X′

a

/X
π



f
W

/X


n

/W



m

/ W′



ℓ

/W

Putting this all together, we have the following factorization of Rπ∗ OX −
→ Rπ∗ OXe :
→ (π ◦ a ◦ b)∗ OX
Rπ∗ OX −
→ R(π ◦ a)∗ OX ′ −
−
→ R(π ◦ a ◦ b)∗ OX −
→ R(π ◦ a ◦ b ◦ c)∗ OXe .
→ (π ◦ a ◦ b ◦ c)∗ OXe −
We now note that the term R(π ◦ a ◦ b)∗ OX in the factorization can be removed yielding:
→ (π ◦ a ◦ b)∗ OX −
Rπ∗ OX −
→ R(π ◦ a)∗ OX ′ −
→ R(π ◦ a ◦ b ◦ c)∗ OXe .
→ (π ◦ a ◦ b ◦ c)∗ OXe −
which by factoring along the lower part of the above diagram yields
→ (ℓ ◦ m ◦ n)∗ OW
→ R(π ◦ a ◦ b ◦ c)∗ OXe .
→ (ℓ ◦ m)∗ OW −
Rπ∗ OX −
→ R(π ◦ a)∗ OX ′ −
f −

F -SINGULARITIES VIA ALTERATIONS

25

Now we apply the functor RH omX ( , ωW ) to this factorization and obtain:
q

α

− (ℓ ◦ m)∗ ωW ←−− (ℓ ◦ m ◦ n)∗ ωW
Rπ∗ ωX ←
− R(π ◦ a)∗ ωX ′ ←
− R(π ◦ a ◦ b ◦ c)∗ ωXe
f ←
q

q

q

q

q

e the first statement
Note we don’t need R on ℓ, m and n since they are finite. Setting U = X,
of the theorem, (5.3) now follows since τ >− dim W (α) = 0 based on our choice of n. However,
Rπ∗ ωX ←
− R(π ◦ a)∗ ωX ′
q

q

factors through Rπ∗ ωX [dim X]. Precomposing with the natural map R(π ◦ a ◦ b ◦ c)∗ ω e ←
X

R(π ◦ a ◦ b ◦ c)∗ ωXe [dim X] and taking cohomology yields (5.4).
q

Theorem 5.5. Suppose that π : X −→ S is a proper morphism of F -finite integral schemes
of characteristic p > 0 with X normal. Further suppose that L is a Cartier divisor on
X and that ∆ is a Q-divisor on X such that L − (KX + ∆) is a π-big and π-semi-ample
Q-Cartier Q-divisor on X. Then there exists a finite surjective map f : Y −→ X from a
normal integral F -finite scheme Y such that:
(a) The natural trace map
f∗ OY (⌈KY + f ∗ (L − (KX + ∆))⌉) −→ OX (⌈KX + L − (KX + ∆)⌉)
has image τ (X; ∆) ⊗ OX (L).
(b) The induced map on cohomology
hi R(π ◦ f )∗ OY (⌈KY + f ∗ (L − (KX + ∆))⌉) −→ hi Rπ∗ τ (X; ∆) ⊗ OX (L)
is zero for all i > 0.
Proof (cf. [Bha10]). Certainly by Theorem 4.6 we can assume that (a) holds for some surjective finite map f ′ : Y ′ −
→ X (and every further finite map). On Y ′ , we may also as′∗
sume that f ∆ is integral and f ′∗ (KX + ∆) is Cartier, and we may further assume that
→W
OY ′ (f ′∗ (L − (KX + ∆))) is the pull-back of a line bundle L via some map π : Y ′ −
over S such that L is ample over S. Note, f ′∗ (L − (KX + ∆)) is still big so we may assume
that W has the same dimension as Y ′ (and thus also the same dimension as X).
Y′

f′

/X

π



W

ρ


/S

Since we only need now to prove (b), replacing X by Y ′ we may assume that KX + ∆ is
Cartier and that OX (L − (KX + ∆)) is the pull-back of some relatively ample line bundle
L via an alteration g : X −
→ W over S with structural map ρ : W −
→ S.
Y

h

/ X := Y ′
❍❍
❍❍ π
❍❍
g
❍❍
❍❍

$/

W

ρ

S

By Lemma 5.2, there exists a finite cover h : Y −
→ X such that R(g ◦ h)∗ ωY −
→ Rg∗ ωX
factors through g∗ ωX . Choose n0 > 0 such that hi Rρ∗ (g∗ ωX ⊗ L n ) = 0 for all i > 0 and
all n > n0 (by Serre vanishing). Also choose an integer e > 0 such that pe > n0 . Consider

F -SINGULARITIES VIA ALTERATIONS

now Y
ρ

h

/X

Fe

g

/W

Fe
h

/W

ρ

26

g
h
/ S which can also be expressed as Y −−
→ X −−→

g

Fe

ρ

W −−→ S −−−→ S and Y −−→ X −−→ W −−−→ W −−→ S. Finally consider the factorization:

e
hi R(ρ ◦ F e ◦ g ◦ h)∗ (ωY ⊗ h∗ g∗ (F e )∗ L ) = hi R(ρ ◦ F e )∗ L p ⊗ R(g ◦ h)∗ ωY

e
−
→ F∗e hi R(ρ)∗ (L p ⊗ g∗ ωX )


e
e
−
→ F∗e hi R(ρ)∗ (L p ⊗ Rg∗ ωX ) = hi (Rρ∗ )F∗e (L p ⊗ Rg∗ ωX )
−
→ hi ((Rρ∗ )(L ⊗ Rg∗ F∗e ωX ))

−
→ hi ((Rρ∗ )(L ⊗ Rg∗ ωX )) = hi Rπ∗ (g∗ L ⊗ ωX ) .
This map is zero since the second line is zero by construction.



Corollary 5.6. Let X be a projective variety over an F -finite field k. Suppose that L is
a Cartier divisor on X, and that ∆ is a Q-divisor such that L − (KX + ∆) is a big and
semi-ample Q-Cartier Q-divisor. Then there exists a finite surjective map f : Y −→ X from
a normal variety Y such that:
(a) The natural trace map
f∗ OY (⌈KY + f ∗ (L − (KX + ∆)⌉) −→ OX (⌈KX + L − (KX + ∆)⌉)
has image τ (X; ∆) ⊗ OX (L).
(b) The induced map on cohomology



H i Y, OY (⌈KY + f ∗ (L − (KX + ∆))⌉) −→ H i X, τ (X; ∆) ⊗ OX (L)

is zero for all i > 0.

6. Transformation rules for test ideals under alterations
The fact that the test ideal can be computed via alterations suggests a transformation
rule for test ideals under alterations. We derive this transformation rule in this section. We
first state a definition.
Definition 6.1. Suppose that X is a normal variety over a field k and Γ is a Q-Cartier
Q-divisor. For example, one might take Γ = L − (KX + ∆) where L is a Cartier divisor and
KX + ∆ is Q-Cartier. We define

\
 Trf

T 0 (X, Γ) :=
Image H 0 (Y, OY ⌈KY + f ∗ Γ⌉) −−−→ H 0 X, OX (⌈KX + Γ⌉)
f: Y−
→X
where f runs over all finite dominant maps f : Y −
→ X such that Y is normal and equidimensional.
Alternately, if X is any (not necessarily normal or even reduced) F -finite d-dimensional
equidimensional scheme of finite type over a field k and L is any line bundle on X, then
we define

\
 Trf

T 0 (X, L ) :=
Image H 0 Y, ωY ⊗ f ∗ L −−−→ H 0 X, ωX ⊗ L
f: Y−
→X

where f runs over all finite dominant maps f : Y −
→ X such that Y is normal and equidimensional of dimension d. In both cases the maps are induced by the trace map as described
in Proposition 2.18.

Remark 6.2. We expect that the reader has noticed the upper-script 0 in the definition of
T 0 (X, Γ). We include this for two reasons:

F -SINGULARITIES VIA ALTERATIONS

27

(i) It serves to remind the reader that T 0 (X, Γ) is a submodule of H 0 (X, OX (⌈KX + Γ⌉)).
(ii) In the future, it might be reasonable to extend this definition to higher cohomology
groups. For example, Theorem 5.5 is a vanishing theorem for appropriately defined
higher T i (X, Γ).
First we make a simple observation:
Lemma 6.3. For any finite dominant map f : Y −→ X between proper normal varieties
over a field k with Γ a Q-Cartier Q-divisor on X, then T 0 (Y, f ∗ Γ) is sent onto T 0 (X, Γ)
via the trace map


β : H 0 Y, OY (⌈KY + f ∗ Γ⌉) −→ H 0 X, OX (⌈KX + Γ)⌉) .

Proof. If Y is proper over a field k, then H 0 Y, OY (⌈KY + f ∗ Γ⌉) is a finite dimensional
vector space, and so T 0 (Y, f ∗ Γ) is the image of a single map


→ H 0 Y, OY (⌈KY + f ∗ Γ⌉) ,
H 0 Y ′ , OY ′ (⌈KY ′ + g∗ f ∗ Γ⌉) −

0 X, O (⌈K + Γ⌉) yields
for some finite cover g : Y ′ −
→ Y . Composing
with
the
map
to
H
X
X

the inclusion T 0 (X, Γ) ⊆ β T 0 (Y, f ∗ Γ) .
For the other inclusion, we simply notice that given any finite dominant map h : W −
→ X,
we can find a finite dominant map U −
→ X which factors through both h and f .

Lemma 6.4. Suppose that X is as in Definition 6.1. Observe that H 0 (X, ωX ⊗ L ) ∼
=
q
H0 (X, ωX [− dim X] ⊗ L ). Furthermore, T 0 (X, ωX ⊗ L ) is:


\
Trf q
q
q
0
∗
0
Image H (Y, ωY [− dim X] ⊗ f L ) −−−−→ H (X, ωX [− dim X] ⊗ L ) .
f :Y −
→X
where the intersection runs over all finite dominant maps f : Y −→ X such that Y is normal
and equidimensional.
Proof. First note that the isomorphism H 0 (X, ωX ⊗ L ) ∼
= H0 (X, ωX [− dim X] ⊗ L ) follows
q
from analyzing the spectral sequence computing H0 (X, ωX [− dim X] ⊗ L ). The second
statement then follows immediately.

q

Mimicking Lemma 6.3, we also obtain:
Lemma 6.5. For any finite dominant map f : Y −→ X between proper equidimensional
schemes of finite type over a field k with L a line bundle on X, then T 0 (Y, ωY ⊗ f ∗ L ) is
sent onto T 0 (X, ωX ) via the trace map


q
q
H0 Y, ωY [− dim X] ⊗ f ∗ L −→ H0 X, ωX [− dim X] ⊗ L .
Proof. The proof is identical to the proof of Lemma 6.3.



T 0 (X, L

By Theorem 4.6, if X = Spec R is affine and L = 0, then
− (KX + ∆)) is just
the global sections of τ (X; ∆). Inspired by this, we demonstrate that we may also use
alterations in order to compute T 0 (X, L − (KX + ∆)).
First, suppose that f : Y −
→ X is an alteration and recall from Proposition 2.18 that we
have a map
Trf : f∗ ωY (−⌊f ∗ (KX + ∆)⌋) −
→ ωX (−⌊KX + ∆⌋) .
Twisting by a Cartier divisor L and taking cohomology leads us to a map
(6.6)

Ψ : H 0 (Y, ωY (⌈f ∗ (L − KX − ∆)⌉)) = H 0 (X, f∗ ωY (⌈f ∗ (L − KX − ∆)⌉))
−
→ H 0 (X, ωX (⌈L − KX − ∆⌉)) .

F -SINGULARITIES VIA ALTERATIONS

28

Theorem 6.7. Suppose that X is a F -finite normal variety over k and that ∆ is a Q-divisor
such that KX + ∆ is Q-Cartier. Finally set L to be any Cartier divisor. Then
T 0 (X, L − KX − ∆)


\
Trf
=
Image H 0 (Y, ωY (⌈f ∗ (L − KX − ∆)⌉)) −−−→ H 0 (X, ωX (⌈L − KX − ∆⌉))
f :Y −
→X
where f runs over all alterations f : Y −→ X and the maps in the intersection are as in (6.6).
T
Proof. Certainly we have the containment T 0 (X, L − (KX + ∆)) ⊇ f : Y −
→X (. . .) since
this latter intersection intersects more modules than the one defining T 0 (X, L − (KX + ∆)).
We need to prove the reverse containment.
β

α

Fix an alteration f : Y −
→ X. Set Y −−→ W = Spec(f∗ OY ) −−→ X to be the Stein
factorization of f . Certainly
τ (ωW , β ∗ (L − KX − ∆)) ⊆ α∗ OY (⌈KY + f ∗ (L − KX − ∆)⌉).
Choose a finite cover h : U −
→ W such that Trh : OU (⌈KU − h∗ β ∗ (L − KX − ∆)⌉) −
→
OW (⌈KW − β ∗ (L − KX − ∆)⌉) has image τ (ωW , β ∗ (L − KX − ∆)).
Y ❇
❇

❇❇ f
❇❇
❇❇

/X
/W
α

U

h

β

We now have the following factorization:
H 0 (U, OU (⌈KU + h∗ β ∗ (L − KX − ∆)⌉)) −
→ H 0 (X, β∗ τ (ωW , β ∗ (L − KX − ∆)))
−
→ H 0 (X, β∗ α∗ OY (⌈KY + f ∗ (L − KX − ∆)⌉))
= H 0 (X, f∗ OY (⌈KY + f ∗ (L − KX − ∆)⌉))
= H 0 (Y, ωY (⌈f ∗ (L − KX − ∆))⌉)
−
→ H 0 (X, ωX (⌈L − KX − ∆⌉))
And so we obtain the desired inclusion.



Theorem 6.8. Suppose that f : Z −→ X is an alteration between normal F -finite varieties
over a field k, ∆ is a Q-divisor on X such that KX + ∆ is Q-Cartier and L is any Cartier
divisor on X. Additionally suppose that either X is affine or proper over k.
Then



0
∗
Ψ T Z, f (L − (KX + ∆)) = T 0 X, L − (KX + ∆)

where Ψ is the map described in (6.6).
In particular, if f is proper and birational and X is affine, then this yields a transformation rule for the test ideal under proper birational morphisms.

Proof. Certainly T 0 (X, L − (KX + ∆)) ⊇ Ψ T 0 (Z, f ∗ (L − (KX + ∆))) in either case by
Theorem 6.7.
If X is proper over k, then so is Z and so H 0 (Z, OZ (⌈KZ + f ∗ (L − (KX + ∆))⌉)) is a
finite dimensional k-vector space. It follows that T 0 (Z, f ∗ (L − (KX + ∆))) is the image of
a single map
 Trg
H 0 Z ′ , OZ ′ (⌈KZ ′ + g∗ f ∗ (L − KX − ∆)⌉) −−−→ H 0 (Z, OZ (⌈KZ + f ∗ (L − KX − ∆)⌉))

F -SINGULARITIES VIA ALTERATIONS

29

for some finite cover g : Z ′ −
→ Z. However, f ◦ g : Z ′ −
→ X is also an alteration, and it
0
follows that T (X, L − (KX + ∆)) is contained in the image of
 Trf ◦g
H 0 Z ′ , OZ ′ (⌈KZ ′ + g ∗ f ∗ (L − KX − ∆)⌉) −−−−→ H 0 (X, OX (⌈KX + L − KX − ∆⌉))

whose image is clearly Trf T 0 (Z, f ∗ (L − (KX + ∆))) .
On the other hand, suppose X = Spec R is affine and observe that, without loss of
generality, we may assume that KX + ∆ is effective. Set S = H 0 (Y, OY ) and define
α

β

Y −−→ X ′ = Spec S −−→ X to be the Stein factorization of f . Now, the global sections H 0 (Z, OZ (⌈KZ + f ∗ (L − (KX + ∆))⌉)) can be identified with elements of ωS (L)
because we assumed that KX + ∆ ≥ 0. In particular, we have τ (ωS ,β ∗ (L − KX − ∆)) ⊆
H 0 (Z, OZ (⌈KZ −f ∗ (L−(KX +∆))⌉)). But Trβ τ (ωS , β ∗ (L−KX −∆)) = τ (R, L−KX −∆)
and the other containment follows.

Remark 6.9. In order to generalize the above result to arbitrary schemes, it would be helpful
to know that the intersection defining T 0 (X, Γ) stabilized in general.

7. Surjectivities on cohomology
In this section we show how the vanishing statements obtained in [Bha10] and in Section 5,
combined with the ideas of Section 6, can be used to construct global sections of adjoint line
bundles. We are treating this current section as a proof-of-concept. In particular, many of
the statements can be easily generalized. We leave the statements simple however in order
to demonstrate the main ideas. Consider the following prototypical application of Kodaira
vanishing.
Example 7.1. Suppose that X is a smooth projective variety in characteristic zero and that
D is an effective Cartier divisor on X. Set L to be an ample line bundle on X. We have
the following short exact sequence
0−
→ ωX ⊗ L −
→ ωX (D) ⊗ L −
→ ωD ⊗ L |D −
→0.
Taking cohomology gives us
0−
→ H 0 (X, ωX ⊗ L ) −
→ H 0 (X, ωX (D) ⊗ L ) −
→ H 0 (D, ωD ⊗ L |D ) −
→ H 1 (X, ωX ⊗ L ).
Kodaira vanishing implies that H 1 (X, ωX ⊗ L ) is zero and so
H 0 (X, ωX (D) ⊗ L ) −
→ H 0 (D, ωD ⊗ L |D )
is surjective.
Consider now the same example (in characteristic zero) but do not assume that X is
smooth.
Example 7.2. Suppose that X is a normal projective variety in characteristic zero and that
D is a reduced Cohen-Macaulay Cartier divisor on X. These conditions are enough to imply
that the natural map ωX (D) −
→ ωD is surjective.
e −
Set L to be an ample (or even big and nef) line bundle on X. Choose π : X
→ X to
e
e
be a log resolution of (X, D) and set D to be the strict transform of D on X. We have the

F -SINGULARITIES VIA ALTERATIONS
b (X):
following diagram of exact triangles in Dcoh


/ π∗ ω e (D)
e ⊗ π∗L
/ π∗ ω e ⊗ π ∗ L
0
X
X

Rπ∗ ωXe ⊗ π ∗ L

0


/ ωX ⊗ L



/ Rπ∗ ω e (D)
e ⊗ π∗L
X

/ ωX (D) ⊗ L



30

/ π∗ ω e ⊗ π ∗ L |D
D



/ Rπ∗ ω e ⊗ π ∗ L |D
D

/ ωD ⊗ L |D



/0

+1

/

/0

The vertical equalities and the top right surjection are due Grauert-Riemenschneider vanishing [GR70]. Because X is not smooth, H 1 (X, ωX (L)) is not necessarily zero, see for example
e ω e ⊗ π ∗ L ) is zero by Kawamata-Viehweg
[AJ89]. However, H 1 (X, (π∗ ωXe ) ⊗ L ) = H 1 (X,
X
∗
vanishing since π L is big and nef, [Kaw82, Vie82]. Thus we have the surjection:
e ⊗L) −
→ H 0 (X, (π∗ ωDe ) ⊗ L |D ) ,
H 0 (X, (π∗ ωXe (D))

between submodules of H 0 (X, ωX (D) ⊗ L ) and H 0 (D, ωD ⊗ L |D ).
Interestingly, π∗ ωDe is independent of the choice of embedding of D into X since π∗ ωDe is
the multiplier module for D by definition. Even more, H 0 (D, π∗ ωDe ⊗ L |D ) only depends
on the pair (D, L |D ).
Furthermore, using the method of proof of Theorem 8.3 below, it is easy to see that
\

Image H 0 (E, ωE ⊗ f ∗ L |D ) −
→ H 0 (D, ωD ⊗ L |D )
H 0 (D, π∗ ωDe ⊗ L |D ) =
f : E−
→D

where the intersection runs over all regular alterations f : E −
→ D. In light of Theorem 6.7
0
the subspace H (D, π∗ ωDe ⊗ L |D ) may be viewed as an analog of T 0 (D, ωDe ⊗ L |D ). This
inspires the remainder of the section.
In characteristic p > 0, Kodaira vanishing does not hold even on smooth varieties [Ray78].
However, we have the following corollary of [Bha10], cf. Corollary 5.6.
Theorem 7.3. Suppose that D is a Cartier divisor on a normal proper d-dimensional
variety X and L is a big and semi-ample line bundle on X. Using the natural map
γ : H 0 (X, ωX (D) ⊗ L ) −→ H 0 (D, ωD ⊗ L |D )
one has an inclusion
T 0 (D, ωD ⊗ L |D ) ⊆ γ(T 0 (X, ωX ⊗ L (D))).
In particular, if T 0 (D, ωD ⊗ L |D ) 6= 0, then H 0 (X, ωX (D) ⊗ L ) 6= 0. And if T 0 (D, ωD ⊗
L |D ) = H 0 (D, ωD ⊗ L |D ) then γ is surjective.
Proof. Using Lemma 6.4, set f : Y −
→ X to be a finite cover of X such that T 0 (X, ωX ⊗
L (D)) is equal to


Trf q
q
q
∗
0
0
Image H (Y, ωY [−d] ⊗ f L (D)) −−−−→ H (X, ωX [−d] ⊗ L (D))

q
noting that H0 (X, ωX [−d] ⊗ L (D)) ∼
= H 0 (X, ωX ⊗ L (D)). By [Bha10, Proposition 5.5.3],
there exists a finite cover g : Z −
→ Y such that H d−1 (Y, f ∗ L −1 ) −
→ H d−1 (Z, g∗ f ∗ L −1 ) is
q
q
1−d
∗
∗
the zero map. Therefore the dual map H (Z, g f L ⊗ ωY ) −
→ H1−d (Y, f ∗ L ⊗ ωX ) is
zero.

F -SINGULARITIES VIA ALTERATIONS

31

Set DY = f ∗ D and DZ = g∗ f ∗ D. Note that DY or DZ may not be normal or even
reduced, even if D is. They are however equidimensional. Then there is a map between
long exact sequences:
0

/

q

H−d (Z, g∗ f ∗ L ⊗ ωZ )

/

/

q

H−d (Z, g∗ f ∗ L ⊗ ωZ (DZ ))

H1−d (DZ , g∗ f ∗ L |D ⊗ ωD )

/

q

Z

H1−d (Z, g∗ f ∗ L ⊗ ωZ )

q

β


0

/


q

H−d (Y, f ∗ L ⊗ ωY )

/

/

q

H−d (Y, f ∗ L ⊗ ωY (DY ))

H1−d (DY , f ∗ L |D ⊗ ωD )

q

Y

ν

0

0

/

q

/

H−d (X, L ⊗ ωX )

H 0 (X, L ⊗ ωX )

/


/

H1−d (Y, f ∗ L ⊗ ωY )

q


/

q

/

δ

α




0



H−d (X, L ⊗ ωX (D))

γ

H 0 (X, L ⊗ ωX (D))

γ



H1−d (D, L |D ⊗ ωD )

q

/

/

H1−d (X, L ⊗ ωX )

q

H 0 (D, L |D ⊗ ωD )

where the vertical equalities are obtained from the spectral sequences computing the middle
lines. Note we identify f with its restriction f |DY and g with g|DZ
Choose the h : E −
→ DZ to be normalization of the (DZ )red and notice we have a map
H 0 (E, ωE ⊗ h∗ g∗ f ∗ L |D ) = H1−d (E, ωE ⊗ h∗ g∗ f ∗ L |D )
q

−
→ H1−d (DZ , ωDZ ⊗ g ∗ f ∗ L |D )
q

α◦β

−−−→ H1−d (D, ωD ⊗ L |D ) = H 0 (D, ωD ⊗ L |D ).
q

The image of this map contains T 0 (D, ωD ⊗ L |D ), and thus the image of α ◦ β also contains
T 0 (D, ωD ⊗L |D ). Therefore, if we view d ∈ T 0 (D, ωD ⊗M ) as an element of H1−d (D, L |D ⊗
q
q
ωD ), it must have some pre-image z ∈ H1−d (DZ , f ∗ L |D ⊗ ωDZ ). The diagram implies that
q
δ(β(z)) = 0. Thus β(z) is an image of some element in y ∈ H 0 (X, f ∗ L ⊗ ωY (DY )). It
follows that ν(y) ∈ T 0 (X, ωX ⊗ L (D)) and so d = γ(ν(y)) which completes the proof. 
Remark 7.4. If we knew that the intersection defining T 0 (X, L − KX − ∆) stabilized, then
the previous result could be generalized to arbitrary equidimensional schemes (not just
those which are of finite type over a field). Even without this hypothesis, the argument
above still implies that T 0 (D, ωD ⊗ L |D ) ⊆ γ(H 0 (X, ωX (D) ⊗ L )). The same statement
holds for Theorem 7.6.
Remark 7.5. We expect that one can obtain more precise surjectivities involving characteristic p > 0 analogs of adjoint ideals. In particular, T 0 (X, ωX ⊗ L ) is not the right analog
e appearing in Example 7.2 above.
of the term H 0 (X, π∗ ωXe (D))
We also show that this method can be generalized with Q-divisors.

Theorem 7.6. Suppose X is a normal F -finite variety which is proper over a field k and
that D is a Cartier divisor on X. Additionally, suppose that ∆ is a Q-divisor on X with
no common components with D and such that KX + ∆ is Q-Cartier. Finally suppose that
L is a Cartier divisor on X such that L − (KX + D + ∆) is big and semi-ample. Then the
natural map
H 0 (X, OX (⌈KX + D + L − (KX + D + ∆)⌉) = H 0 (X, OX (⌈L − ∆⌉)
γ

−−→ H 0 (D, OD (⌈KD + L|D − (KD + ∆|D )⌉)) = H 0 (D, OD (L|D − ⌊∆⌋|D ).
yields an inclusion


T 0 (D, L|D − (KD + ∆|D )) ⊆ γ T 0 (X, D + L − (KX + D + ∆))

F -SINGULARITIES VIA ALTERATIONS

32

noting that T 0 (D, L|D − (KD + ∆|D )) ⊆ H 0 (D, OD (⌈L|D − ⌊∆⌋|D ⌉)).
Proof. Let us first point out that γ is induced from the restriction map
OX (⌈KX + D + L − (KX + D + ∆)⌉)
−
→ OD (⌈KX + D + L − (KX + D + ∆)⌉|D ) = OD (KD + L|D − (KD + ⌊∆⌋|D )).
Now choose a finite cover h : W −
→ X, with W normal, such that h∗ (KX + ∆) is an integral
∗
Cartier divisor and set DW = h D. Note DW is not necessarily normal or even reduced (it
is however unmixed and thus equidimensional). We have the diagram


/ h∗ ωD h∗ (L − (KX + D + ∆))
h∗ ωW h∗ (D + L − (KX + D + ∆))
W
ξ



ωX (⌈D + L − (KX + D + ∆)⌉)


/ ωD (⌈L|D − (KD + ∆)⌉|D )

of which we take global sections
and then apply the method of Lemmas
6.3 and 6.5 to


0
∗
conclude that the image ξ T DW , ωDW ⊗ OW (h (L − (KX + D + ∆))) is equal to


T 0 D, L|D − (KD + ∆|D ) ⊆ H 0 D, OD (KD + L|D − (KD + ∆|D ))
(7.7)

(7.8)
⊆ H 0 D, ωD (⌈L|D − (KD + ∆)⌉|D ) .

Likewise T 0 (X, D + L − (KX + D + ∆)) is the image of T 0 (W, h∗ (D + L − (KX + D + ∆))).
Thus it is sufficient to show that via the map


H 0 W, ωW (h∗ (D + L − (KX + D + ∆))) −
→ H 0 DW , ωDW (h∗ (L − (KX + D + ∆))) ,

each element of T 0 DW , ωDW ⊗ OW (h∗ (L − (KX+ D + ∆))) is the image of an element
of T 0 W, ωW ⊗ OW (h∗ (D + L − (KX + D + ∆))) . We have just reduced to the setting of
Theorem 7.3 and the result follows.

8. Transformation rules for multiplier ideals
It still remains to be proven that the multiplier ideal in characteristic zero can be characterized as in our Main Theorem, for which we need to explore the behavior of multiplier
ideals under proper dominant maps. We further analyze the behavior of multiplier ideals
under alterations in arbitrary characteristic, which leads to an understanding of when (and
why) the classical characteristic zero transformation rule (2.41) for the multiplier ideal under
finite maps may fail in positive characteristic.
Suppose that A is a normal ring in arbitrary characteristic and that (X = Spec A, ∆)
is an affine pair such that KX + ∆ is Q-Cartier. As discussed in Section 2.4, J (X; ∆) is
only known to be quasi-coherent assuming a theory of resolution of singularities is at hand.
Nonetheless, it is always a sheaf of (fractional) ideals, and in this section we will use the
notation J (A; ∆) := H 0 (X, J (X; ∆)) to denote the corresponding ideal of global sections.
An important and useful perspective, largely in the spirit of [Lip94], is to view Definition 2.28
as a collection of valuative conditions for membership in the multiplier ideal J (A; ∆).
Specifically, suppose E is a prime divisor on a normal proper birational modification
θ: Z −
→ X. After identification of the function fields K = Frac(A) = K(X) = K(Z),
E gives rise to a valuation ordE centered on X. The valuation ring of ordE is simply the
local ring OZ,E . Thus, we have that J (A; ∆) can be described as the fractional ideal inside
of K(X) given by

F -SINGULARITIES VIA ALTERATIONS

J (A; ∆) =

\

OZ,E (⌈KZ − θ ∗ (KX + ∆)⌉)

θ : Z−
→X
Prime E on Z

=

(

33

f ∈K

ordE (f ) ≥ ordE (⌊θ ∗ (KX + ∆) − KZ ⌋)
for all θ : Z −
→ X and all prime E on Z

)

We now show how to generalize the characteristic zero transformation rule for multiplier
ideals under finite maps (2.41) so as to incorporate the trace map as in (2.42). Furthermore,
by working in arbitrary characteristic, we also recover both of these transformation rules in
positive characteristic for separable finite maps of degree prime to the characteristic.
Theorem 8.1. Suppose that π : Spec B = Y −→ X = Spec A is a finite dominant map of
normal integral schemes of any characteristic and that (X, ∆X ) is a pair such that KX +∆X
is Q-Cartier. Define ∆Y = ∆X − Ramπ . Then
Tr(J (B; ∆Y )) ⊆ J (A; ∆X ) .
Furthermore, if the field trace map Tr : Frac(B) −→ Frac(A) satisfies Tr(B) = A (e.g. the
degree of π is prime to the characteristic), then
Tr(J (B; ∆Y )) = J (B; ∆Y ) ∩ K(A) = J (A; ∆X ).
Proof. Suppose f ∈ J (B; ∆Y ). Fix a prime divisor E on a normal proper birational modification θ : Z −
→ X. Consider the discrete valuation ring R = OZ,E , viewed as a subring
of K(A), and let r ∈ K(A) be a uniformizer for R. Denote by S the integral closure of R
inside of K(B). Then S can also be realized in the following manner. Let W be the normal
scheme fitting into a commutative diagram
W

η

/Y

ρ

π





Z

θ

/X

where ρ is finite and η is birational (that is, take W to be the normalization of the relevant
irreducible component of Y ×X Z). Let E1 , . . . , Ek be the prime divisors on W mapping
T
onto E. Then we have S = ki=1 OW,Ei , where again we have considered each OW,Ei as a
subring of K(B). In particular, for g ∈ K(B), we have g ∈ S if and only if ordEi (g) ≥ 0 for
all i = 1, . . . , k.
Let Φ be a generator for the rank one free S-module HomR (S, R). If we write Tr : S −
→R
as Tr( ) = Φ(c · ), we know from [ST10, Proposition 4.8] that divW (c) = Ramρ so that
ordEi (c) = ordEi (KW − ρ∗ KZ ) .
Let λE = ordE (⌈KZ − θ ∗ (KX + ∆X )⌉) and consider g = cr λE f . Since f ∈ J (B; ∆Y ), it
follows that ordEi (f ) + ⌈KW − η ∗ (KY + ∆Y )⌉ ≥ 0 and hence
ordEi (g) ≥ ordEi (KW − ρ∗ KZ + ρ∗ ⌈KZ − θ ∗ (KX + ∆X )⌉ − ⌈KW − η ∗ (KY + ∆Y )⌉)
= ordEi (ρ∗ ⌈−θ ∗ (KX + ∆X )⌉ − ⌈−η ∗ (KY + ∆Y )⌉)
= ordEi (ρ∗ ⌈−θ ∗ (KX + ∆X )⌉ − ⌈−η ∗ (KY + π ∗ ∆X − (KY − π ∗ KX ))⌉)
= ordEi (ρ∗ ⌈−θ ∗ (KX + ∆X )⌉ − ⌈ρ∗ (−θ ∗ (KX + ∆X ))⌉)
≥ 0.

F -SINGULARITIES VIA ALTERATIONS

34

It now follows that g ∈ S, and thus Φ(g) = r λE Tr(f ) ∈ R. In other words, we have
ordE (Φ(g)) = ordE (Tr(f )) + ordE (r λE )
= ordE (Tr(f )) + ordE (⌈KZ − θ ∗ (KX + ∆X ⌉) ≥ 0
and we conclude that Tr(f ) ∈ J (A; ∆X ) and thus Tr(J (B; ∆Y )) ⊆ J (B; ∆X ) as desired.
Note that every divisorial valuation ν : K(B) \ {0} −
→ Z centered on Y can be realized as
ν = ordEi as in the setup above. Indeed, the restriction ν to K(A) gives rise to a discrete
valuation ring whose residue field has transcendence degree (dim(Y ) − 1) = (dim(X) − 1)
over Λ; see [Bou98, Chapter VI, Section 8]. By Proposition 2.45 in [KM98], this valuation
ring can be realized as OZ,E for some prime divisor E on θ : Z −
→ X as above, so that
ν = ordEi for some i.
Let us now argue that J (A; ∆X ) · π∗ OY ⊆ π∗ J (B; ∆Y ). Suppose h ∈ J (A; ∆X ). We
have by assumption
0 ≤ ordE (h) + ordE (⌈KZ − θ ∗ (KX + ∆X )⌉)
whence it follows from [Har77, Chapter IV, Proposition 2.2], that
0 ≤ ordEi (h) + ordEi (ρ∗ ⌈KZ − θ ∗ (KX + ∆X )⌉)
≤ ordEi (h) + ordEi (Ramρ ) + ordEi (⌈ρ∗ (KZ − θ ∗ (KX + ∆X ))⌉)
= ordEi (h) + ordEi (⌈KW − η ∗ (KY + ∆Y )⌉) .
Thus, we conclude h ∈ J (B; ∆Y ) and hence J (A; ∆X ) · π∗ OY ⊆ π∗ J (B; ∆Y ).
Now assume in addition that the trace map is surjective, i.e. Tr(B) = A. We have, using
the surjectivity of trace at the third inequality, that
J (A; ∆X ) ⊆ (J (A; ∆X )·π∗ OY )∩OX ⊆ π∗ J (B; ∆Y )∩OX ⊆ Tr (π∗ J (B; ∆Y )) ⊆ J (A; ∆X ).
This necessitates equality throughout, which completes the proof.



We now complete the proof of our main theorem.
Corollary 8.2. Suppose that (X, ∆) is a pair in characteristic zero. Then


\
Tr
J (X; ∆) =
Image π∗ OY (⌈KY − π ∗ (KX + ∆)⌉) −−−→ K(X)
π:Y −
→X
where π ranges over all alterations with Y normal, Tr : K(Y ) −→ K(X) is the field trace,
and we have that KY = π ∗ KX + Ramπ wherever π is finite.
Proof. Because of the existence of resolution of singularities in characteristic zero, we may
assume that each Y is smooth and that KY − π ∗ (KX + ∆) is supported on a simple normal
crossings divisor on Y . First consider a finite map f : W −
→ X with W normal (note that
we are in characteristic zero, so the map is separable). If we pick a representative KX such
that OX (KX ) = ωX and also pick KW = f ∗ KX + Ramf , then we recall from Example 2.2
that the field trace Tr : K(W ) −
→ K(X) induces a map Tr : OW (KW ) −
→ OX (KX ) which is
identified with the Grothendieck trace Trf : f∗ ωW −
→ ωX .
Fix any proper dominant map π : Y −
→ X. Factor π through a birational map ρ : Y −
→W
and a finite map f : W −
→ X with W normal via Stein factorization. Thus π∗ ωY −
→ ωX
factors through Tr : f∗ ωW −
→ ωX . It follows that
π∗ OY (KY − π ∗ (KX + ∆)) −
→ K(X)
factors through Tr : K(W ) −
→ K(X). Furthermore, ρ∗ OY (KY −π ∗ (KX +∆)) is by definition
the multiplier ideal J (W ; ∆W ) where ∆W = f ∗ ∆ − Ramf , since Y is smooth. Thus,

F -SINGULARITIES VIA ALTERATIONS

35



Image π∗ OY (KY − π ∗ (KX + ∆)) −
→ K(X) is simply Tr(ρ∗ J (W ; ∆W )). But that is equal
to J (X; ∆) by Theorem 8.1.

Finally, we turn our attention to the behavior of the multiplier ideal under proper dominant maps in characteristic zero.
Theorem 8.3. If (X, ∆) is a log-Q-Gorenstein pair in characteristic zero, then


\
Tr
J (X; ∆) =
Image hdim Y −dim X Rπ∗ OY (⌈KY − π ∗ (KX + ∆)⌉) −−−π→ K(X)
π:Y −
→X
where the intersection runs over all proper dominant maps from normal varieties Y . Note
the map to K(X) is induced from the trace map as in Proposition 2.18.
Proof. We may restrict our maps π : Y −
→ X to those maps where Y is regular and which
factor through a fixed regular alteration f : Z −
→ X such that f ∗ (KX + ∆) is an integral
Cartier divisor. Thus by Corollary 8.2, it is sufficient to show that
hdim Y −dim X R(f ◦ ρ)∗ OY (KY − π ∗ (KX + ∆)) −
→ h0 Rf∗ ωZ (−f ∗ (KX + ∆))
is surjective on global sections. However, by the statement of [Kov00, Theorem 2], first
correctly proved in full generality in [Bha10, Theorem 4.1.3], the natural map OZ −
→ Rρ∗ OY
has a left inverse in the derived category, and thus so does
Rf∗ Rρ∗ (ωY [dim Y ] ⊗ OY (π ∗ (KX + ∆))) −
→ Rf∗ (ωZ [dim Z] ⊗ OZ (f ∗ (KX + ∆))) .
In particular, taking −dth cohomology yields the desired surjection on global sections.



9. Further questions
This theory suggests a large number of potential directions for further inquiry. We
highlight a few of the more obvious ones below.
Question 9.1 (Mixed characteristic). What can be said in mixed characteristic? In particular, does the intersection from our Main Theorem stabilize for schemes in mixed characteristic?
We have learned that M. Hochster and W. Zhang have made progress on this question
in low dimensions for isolated singularities.
Question 9.2 (Adjoint ideals). Can one develop a characteristic theory analogous to the
theory of adjoint ideals, cf. [Laz04, 9.3.E] or [Tak08], described via alterations or finite
covers?
The characterization of test ideals, as well as F -regular and F -rational singularities suggests the following:
Question 9.3 (F -pure singularities). Can F -pure (or F -injective) singularities likewise be
described by alterations?
Finally, we consider the following question:
Question 9.4 (Effectivity of covers and alterations). Given a pair (X, ∆), how can we identify
finite covers (or alterations) π : Y −
→ X such that

Tr
τ (X; ∆) = Image ⌈π∗ OY (KY − π ∗ (KX + ∆)⌉) −−−π→ K(X) ?
In other words, can we determine when the intersection from our Main Theorem stabilizes?

F -SINGULARITIES VIA ALTERATIONS

36

We do not have a good answer to this question. The key point in our construction is
repeated use of the Equational Lemma [HH92, HL07, SS12]. The procedure in that Lemma
is constructive. However, this is not very satisfying. It would be very satisfying and likely
useful if one had a different geometric or homological criterion for identifying π : Y −
→X
as in the question above.
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Institut für Mathematik, Johannes Gutenberg-Universität Mainz, 55099 Mainz, Germany
E-mail address: blicklem@uni-mainz.de
Department of Mathematics, The Pennsylvania State University, University Park, PA,
16802, USA
E-mail address: schwede@math.psu.edu
Department of Mathematics, University of Utah, Salt Lake City, UT, 84112, USA
E-mail address: kevtuck@math.utah.edu

