FUNDAMENTAL GROUPS OF F -REGULAR SINGULARITIES VIA
F -SIGNATURE

arXiv:1606.04088v1 [math.AG] 13 Jun 2016

JAVIER CARVAJAL-ROJAS, KARL SCHWEDE, AND KEVIN TUCKER

Abstract. We prove that the local étale fundamental group of a strongly F -regular
singularity is finite (and likewise for the étale fundamental group of the complement of
a codimension ≥ 2 set), analogous to results of Xu and Greb-Kebekus-Peternell for KLT
singularities in characteristic zero. In fact our result is effective, we show that the reciprocal of the F -signature of the singularity gives a bound on the size of this fundamental
group. To prove these results and their corollaries, we develop new transformation rules
for the F -signature under finite étale-in-codimension-one extensions. As another consequence of these transformation rules, we also obtain purity of the branch locus over
rings with mild singularities (particularly if the F -signature is > 1/2). Finally, we generalize our F -signature transformation rules to the context of pairs and not-necessarily
étale-in-codimension-one extensions, obtaining an analog of another result of Xu.

1. Introduction
In [Kol11, Question 26] J. Kollár asked whether if (0 ∈ X) is the germ of a KLT
singularity, then π1 (X \ {0}) is finite. In [Xu14] C. Xu showed that this holds for the
étale local fundamental group, in other words, for the profinite completion of π1 (X \ {0}).
Building on this result, [GKP13] proved the finiteness of the étale fundamental groups of
the regular locus of KLT singularities (see also [TX15]). Over the past few decades, we
have learned that KLT singularities are closely related to strongly F -regular singularities
in characteristic p > 0, see [HW02, Har98]. Hence it is natural to ask whether their local
étale fundamental groups are also finite. We show that this is indeed the case. In fact,
we find an upper bound for the size of the fundamental group in terms of a well studied
invariant for measuring singularities in characteristic p > 0, the F -signature s(R).
Theorem A (Theorem 5.1). Let (R, m, k) be a normal F -finite and strongly F -regular
strictly Henselian1 local domain of prime characteristic p > 0, with dimension d ≥ 2.
Then the étale fundamental group of the punctured spectrum of R, i.e. π1 := π1ét Spec◦ (R) ,
is finite. Furthermore, the order
 of π1 is at most 1/s(R) and is prime to p. The same
ét
also holds for π1 Spec(R) \ Z where Z ⊆ Spec R has codimension ≥ 2.
Observe, that unlike the characteristic zero situation, our characteristic p > 0 result is
effective, we give an explicit bound on the size of the π1 . It is also worth noting that we are
working with the étale fundamental group, not the tame fundamental group. Indeed, for
R strongly F -regular, any finite étale in codimension 1 local extension (R, m, k) ⊆ (S, n, ℓ)

The first named author was supported in part by the NSF FRG Grant DMS #1265261/1501115.
The second named author was supported in part by the NSF FRG Grant DMS #1265261/1501115
and NSF CAREER Grant DMS #1252860/1501102.
The third named author was supported in party by NSF Grant DMS #1419448.
1This just means it is Henselian with separably closed residue field.
1

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JAVIER CARVAJAL-ROJAS, KARL SCHWEDE, AND KEVIN TUCKER

must be tame everywhere. This was already implicitly observed in [ST14] but we make
it precise here. Indeed, we note that p cannot divide [K(S) : K(R)] if the residue fields
are equal Corollary 2.11.
The technical tool where F -regularity is used in our proof is a transformation rule for F signature under finite étale-in-codimension-1-morphisms. The F -signature was introduced
implicitly in [SVdB97] and explicitly in [HL02]. Roughly speaking, it measures how many
different ways R ֒→ F∗e R splits as e goes to infinity. Explicitly, if R has perfect residue
field and F∗e R = R⊕ae ⊕ M as an R-module, where M has no free R-summands, then
ae
s(R) = lim e dim R . Here are three quick facts:
e→∞ p
◦ The limit s(R) exists [Tuc12].
◦ s(R) > 0 if and only if R is strongly F -regular [AL03].
◦ s(R) ≤ 1.
Note that there have been a number of transformation rules for F -signature under finite
maps in the past. However, they were generally only an inequality (that went the wrong
way for our purposes), or assumed that S is flat over R (or made other assumptions about
R and S). See for instance [HL02, Yao06, HY09, Tuc12].
Theorem B (Theorem 3.1). Let (R, m, k) ⊆ (S, n, ℓ) be a module-finite local extension of
F -finite d-dimensional normal local domains in characteristic p > 0, with corresponding
extension of fraction fields K ⊆ L. Suppose R ⊆ S is étale in codimension 1, and that R
is strongly F -regular. Then if one writes S = R⊕f ⊕ M as a decomposition of R-modules
so that M has no nonzero free direct summands, then f = [ℓ : k] ≥ 1 and the following
equality holds:
[L : K]
s(S) =
· s(R).
[ℓ : k]
Below, before Theorem D, we discuss how to still get precise transformation rules of
F -signature even when R ⊆ S is not necessarily étale in codimension 1.
By applying Theorem B in the case k = ℓ, we see that s(S) = [L : K] · s(R). Since
s(S) ≤ 1, we immediately see that [L : K] ≤ 1/s(R). In other words, the reciprocal of
the F -signature s(R) gives an upper bound on the generic rank of a finite local étale in
codimension 1 extension with the same residue field. Theorem A then follows. We also
obtain characteristic p > 0 corollaries similar to some of those in [GKP13].
Because our bound on the size of the étalefundamental group is effective, we immediately obtain a new result on purity of the branch locus.
Theorem C (Corollary 3.3). Suppose Y −
→ X is a finite dominant map of F -finite
normal integral schemes. If s(OX,P ) > 1/2 for all P ∈ X then the branch locus of
Y −
→ X has no irreducible components of codimension ≥ 2, in other words it is a divisor.
In [BST12], the notion of F -signature of pairs was introduced. In Theorem 4.4, we
obtain an analogous result to Theorem B in the context of pairs. Indeed, if (R, ∆) is
a strongly F -regular pair, then this can be interpreted as follows. The reciprocal of
s(R, ∆) gives an upper bound on the generic rank of a finite local extension (R, m) ⊆
(S, n) such that π ∗ ∆ − Ram ≥ 0 (here Ram is the ramification divisor on Spec S and
π : Spec S −
→ Spec R is the induced map). By taking cones, this immediately yields
the following characteristic p > 0 analog of the second main result of [Xu14]. Here note

FUNDAMENTAL GROUPS OF F -REGULAR SINGULARITIES VIA F -SIGNATURE

3

that globally F -regular varieties are an analog of log-Fano varieties in characteristic zero
[SS10].
Theorem D (Corollary 4.8). Suppose that (X, ∆) is a globally F -regular projective pair
over an algebraically closed field of characteristic p > 0. There is a number n such
that every finite separable cover π : Y −
→ X with π ∗ ∆ − Ram ≥ 0 has generic rank
[K(Y ) : K(X)] ≤ n.
Acknowledgements: The authors would like to thank János Kollár, Christian Liedtke,
Linquan Ma, Lance Miller, Mircea Mustaţă, Stefan Patrikis and David Speyer for valuable
and inspiring conversations. We would like to thank Stefan Kebekus Chenyang Xu and
Lance Miller for valuable comments on a previous draft. We would especially like to thank
David Speyer for sharing an early preprint of [Spe15] with us and allowing us to include
Lemma 2.14 which is a special case of his result.
2. Preliminaries
Convention 2.1. Throughout this paper, all rings will be assumed to be Noetherian. They
will all be characteristic p > 0 unless otherwise stated and they will all be F -finite. All
schemes will be assumed to be Noetherian and separated. If R is an integral domain, then
K(R) will denote the fraction field of R (likewise with K(X) if X is an integral scheme).
Given a finite separable map of normal integral schemes f : Y −
→ X, we use Ram to
denote the ramification divisor on Y .
2.1. Maps and divisors. First we fix some notation. Given a Weil divisor D on X =
Spec R, we use R(D) to denote Γ X, OX (D) .

Definition-Proposition 2.2 (Maps and Divisors). Suppose that R ⊆ S is a finite inclusion of normal domains and A, B are divisors on Spec R and Spec S respectively with π :
Spec S −
→ Spec R the canonical map. Then any nonzero element ϕ ∈ HomR S(B), R(A)
yields an effective divisor Dϕ ∼ (KS − B) − π ∗ (KR − A). If ϕ, ϕ′ ∈ HomR S(B), R(A)
are such that Dϕ = Dϕ′ then ϕ and ϕ′ are S-unit multiples of each other.
Proof. We first notice that



HomR S(B), R(A) ∼
= HomR S B + π ∗ (KR − A) , R(KR )



∼
= HomS S B + π ∗ (KR − A) , S(KS ) ∼
= S KS − B − π ∗ (KR − A) .

Since ϕ is a section of that reflexive sheaf, it yields a divisor Dϕ of zeroes linearly equivalent
to KS − B − π ∗ (KR − A) as claimed. Two sections yield the same divisor if and only if
they are S-unit multiples of each other.


Lemma 2.3. Suppose we have a finite inclusion of normal domains R ⊆ S ⊆ T and we
have divisors A, B, C on Spec R, Spec S, Spec T respectively and maps
β : T (C) −
→ S(B), γ : S(B) −
→ R(A).
Then Dγ◦β = Dβ + π ∗ Dγ where π : Spec T −
→ Spec R is the induced map.
Proof. We work locally and assume that R is a DVR and that S and T are semi-local
Dedekind domains and hence are PIDs. Since R, S, T are all PIDs, T (C) ∼
=
= T, S(B) ∼
R.
By
applying
these
isomorphisms
uniformly,
we
may
assume
that
C
=
B, R(A) ∼
=

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JAVIER CARVAJAL-ROJAS, KARL SCHWEDE, AND KEVIN TUCKER

0, B = 0, A = 0. Let Φ ∈ HomR (S, R) be an S-module generator, Ψ ∈ HomS (T, S) be a
T -module generator, and observe that Φ ◦ Ψ ∈ HomR (T, R) is a T -module generator by,
for instance [Sch09, Lemma 3.9]. Write γ( ) = Φ(s · ), β( ) = Ψ(t · ). This implies
that Dγ = divS (s) and that Dβ = divT (t). We observe that γ ◦ β( ) = Φ ◦ Ψ(st · ) and
so
Dγ◦β = divT (st) = divT (t) + divT (s) = Dβ + π ∗ Dγ
as desired.

2.2. F -signature.
Definition 2.4 (F -signature, [HL02] [BST12]). Suppose that (R, m, k) is a d-dimensinal
F -finite local ring with α = α(R) such that pα = [k : k p ]. If we write F∗e R = R⊕ae ⊕ M
where M has no free R-summands, in other words ae is the maximal rank of a free
R-summand of F∗e R, then the F -signature of R, is equal to
ae
s(R) := lim e(d+α) .
e→∞ p
More generally, if ∆ ≥ 0 is a Q-divisor on Spec R and if we let a∆
e denote the maxe
imal rank of a free R-summand
of F∗ R whose corresponding projection maps lie in

HomR F∗e R(⌈(pe − 1)∆⌉), R ⊆ HomR (F∗e R, R), then the F -signature of the pair (R, ∆)
is equal to
a∆
e
s(R, ∆) := lim e(d+α)
.
e→∞ p

The elements of HomR F∗e R(⌈(pe − 1)∆⌉), R ⊆ HomR (F∗e R, R) form what is called a
Cartier algebra since they are closed under composition, [Sch11, Bli13]. Indeed in [BST12],
F -signature with respect to general Cartier algebras is defined and studied. Below, we
discuss F -signature with respect to an object that is not quite a Cartier algebra (but
which will otherwise be convenient for us).
Consider (R, m, k) an F -finite normal local domain with full Cartier algebra C = CR .
If
a Q-divisor, we can form the Cartier
subalgebra C ∆ ⊆ C , with C ∆ =

L∆ ≥ ∆0 is L
e
⌈(pe − 1)∆⌉ , R . However, it is frequently natural to
e≥0 Ce =
e≥0 HomR F∗ R L

L
•
instead consider G = G ⌊p ∆⌋ = e≥0 Ge = e≥0 HomR F∗e R(⌊pe ∆⌋), R . This object is
not generally a Cartier algebra since given two maps


ϕ ∈ HomR F∗e R(⌊pe ∆⌋), R = Ge , and ψ ∈ HomR F∗f R(⌊pf ∆⌋), R = Gf
we can compose and obtain


ϕ ◦ F∗e ψ : F e+f R ⌊pf ∆⌋ + pf ⌊pe ∆⌋ −
→ F∗e R(⌊pe ∆⌋) −
→ R.

However, ⌊pf ∆⌋ + pf ⌊pe ∆⌋ is not always ≥ ⌊pe+f ∆⌋ and so G is not closed under composition.
Observe that if ∆ ≥ 0 satisfies ⌊∆⌋ = 0, then we have
(2.4.1)

Ge = HomR

⌊pe ∆⌋ ≤ ⌈(pe − 1)∆⌉ so that

F∗e R(⌊pe ∆⌋), R ⊇ HomR F∗e R(⌈(pe − 1)∆⌉), R = Ce∆ .

furthermore, we have equality if (pe − 1)∆ is an integral Weil divisor.

Setting 2.5. Suppose (R, m) is an F -finite normal local domain of dimension d and ∆ ≥ 0
is a Q-divisor on Spec R such that ⌊∆⌋ = 0. We define Ge as above.

FUNDAMENTAL GROUPS OF F -REGULAR SINGULARITIES VIA F -SIGNATURE

5

Definition 2.6. With notation as in Setting 2.5, we set
Gens = {ϕ ∈ Ge | Image(ϕ) ⊆ m}
IeG = {r ∈ R | ϕ(F∗e r) ∈ m, for all ϕ ∈ Ge }.
We also define aGe to be the maximal number of free summands of F∗e R whose associated
projection homomorphisms belong to Ge .
We observe that



aGe = λR Ge /Gens = peα λR R/IeG

by [BST12, Lemma 3.6], where α = α(R) = logp [k 1/p : k] .

(2.6.1)

Lemma 2.7. [cf. [BST12, Lemma 4.17]] With notation as in Setting 2.5
lim

e→∞

aGe
pe(d+α)

= s(R, ∆).

Moreover, the same result holds even if we replace Ge by the asymptotically small perturbation of it:
 
Ge′ := HomR F∗e R ⌊pe ∆⌋ + D , R
for any other effective Weil divisor D.
Proof. Choose 0 6= c ∈ R such that
(F∗e c) · Ce∆ ⊆ (F∗e c) · Ge ⊆ Ce∆ ⊆ Ge .
Indeed, any c with div(c) ≥ ⌈∆⌉ will work. Obviously we have that

IeG :R c = {r ∈ R | ϕ(F∗e r) ∈ m, for all ϕ ∈ (F∗e c) · Ge }.

Thus, the above inclusions (F∗e c) · Ge ⊆ Ce∆ ⊆ Ge induce inclusions of ideals:

IeG ⊆ Ie∆ ⊆ IeG :R c .

Now simply apply Lemma 2.8 below.
For the last assertion in the statement of the lemma, we just choose a 0 6= c ∈ R such
that div(c) ≥ ∆ + D, then we are going to have (F∗e c) · Ge ⊆ Ge′ ⊆ Ge , thus we argue as
above, including running Lemma 2.8.

Lemma 2.8. Let (R, m, k) be an F -finite normal local domain of dimension d. Suppose we
e
have a pair of sequences of finite colength ideals {Ie }e and {Je }e such that m[p ] ⊆ Ie ⊆ Je
holds for all e. Additionally assume that there is a 0 6= c ∈ R such that Ie ⊆ Je ⊆ (Ie : c)
for all e. Then, it follows that:
1
λR (Je /Ie ) = 0.
e→∞ pde
lim

In particular,
1
1
λR (R/Je ) = lim de λR (R/Ie )
de
e→∞ p
e→∞ p
provided that any (then both) of the limits exists.
lim

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JAVIER CARVAJAL-ROJAS, KARL SCHWEDE, AND KEVIN TUCKER

Proof. Consider the four term exact sequence
·c

0−
→ (Ie : c)/Ie −
→ R/Ie −→ R/Ie −
→ R/(Ie + cR) −
→ 0.
From this we conclude cf. [Vra08, Page 8],



λR (Je /Ie ) ≤ λR (Ie : c)/Ie = λR R/(Ie + cR)

e
Since m[p ] ⊆ Ie there is a constant C such that λR (R/(Ie + cR) ≤ Cpe(d−1) . The result
then follows.


We conclude this subsection with a brief recollection of strongly F -regular singularities
and globally F -regular projective varieties.
Definition 2.9. Suppose that (R, m) is a normal local ring and ∆ ≥ 0 is a Q-divisor. We
say that (R, ∆) is strongly F -regular if s(R, ∆) > 0. This is equivalent by [AL03, BST12]
to the assertion that for every 0 6= c ∈ R, there exists some e > 0 and ϕ ∈ Ce∆ with
ϕ(F∗e cR) = R.
Suppose X is a normal projective variety over an algebraically closed field and ∆ ≥ 0
is
on X. For any ample line bundle L on X, form the section ring R =
La Q-divisor
0
i
H
(X,
L
) and let ∆R be the corresponding Q-divisor. We say that (X, ∆) is
i≥0
globally F -regular if (R, ∆R ) is strongly F -regular, this is independent of the choice of L
[SS10].
2.3. Notes on trace. We believe the following easy lemma is well known to experts but
we do not know an easy reference
Lemma 2.10. Suppose that (R, m) ⊆ (S, n) is a finite extension of normal local rings.
Then Tr(n) ⊆ m.
Proof. Choose A a divisorial discrete valuation ring with uniformizer a ∈ A of K(R)
centered over V (m) ⊆ Spec R. Let B be the normalization of A inside K(S). Note
that B is a 1-dimensional semi-local ring with maximal ideals b1 , . . . , bl . Note that each
bi ∩ S = n. Let KB/A be the relative canonical divisor/ramification divisor of A ⊆ B and
observe that2 Tr B(KB/A ) ⊆ A. Now, notice that

Tr a · B(KB/A ) ⊆ hai
T
by linearity of trace. We claim that i bi ⊆ a · B(KB/A ). We can check this locally on
B. Indeed, localize at a bi to obtain a DVR B ′ = Bbi with uniformizer b. Then write
a = ubn where u is a unit of B ′ . We also know that KB′ /A = m div(b) where n − 1 ≤ m
(with equality in the case of tame ramification). B ′ (KB′ /A ) = b1m B ′ and so
a
a · B ′ (KB′ /A ) = m B ′ = bn−m B ′ ⊇ bB ′
b
where the last containment holds since n − m ≤ 1. In conclusion
\ 

Tr
bi ⊆ Tr a · B(KB/A ) ⊆ hai.
i

Next choose x ∈ n, then x ∈
2This

T

i

bi and so Tr(x) ∈ aA ∩ R = m as claimed.



follows from the assertion that the section Tr ∈ HomA (B, A) corresponds to the ramification divisor,
see [ST14].

FUNDAMENTAL GROUPS OF F -REGULAR SINGULARITIES VIA F -SIGNATURE

7

The following corollary can also be viewed as a result on purity of the branch locus.
We also believe it is well known to experts but we do not know a reference.
Corollary 2.11. Suppose that (R, m) is a splinter (for instance, if it is strongly F regular). Then there is no finite local extension (R, m) ⊆ (S, n) of normal domains with
corresponding fraction fields K ⊆ L such that
(a) the residue fields R/m = S/n = k are equal,
(b) p [L : K], and
(c) R ⊆ S is étale in codimension 1.
Proof. Suppose that there is such an extension. By (c), we know that Tr : S −
→ R is
surjective, see [ST14, Proposition 7.4]. By Lemma 2.10 Tr(n) ⊆ m and so we have an
induced surjective map
Tr : S/n −
→ R/m.
But this is a k-linear map between 1-dimensional k-vector spaces by (a), and hence an
isomorphism. Consider 1 ∈ R ⊆ S. We know Tr(1) = [L : K] · 1 = 0 by (b). Hence
Tr(1) = 0 which is a contradiction.

Note in the above proof, if the condition that R is a splinter is removed, then condition
(c) can be replaced by the condition that Tr : S −
→ R is surjective, and then the result is
well known to experts (with the same proof).
Definition 2.12. Let R ⊆ S be a finite separable extension of normal domains, we say
that a direct R-summand M of S is a Tr-summand if M ∼
= R and the associated projection
∼
=
linear map ρ : S −
→ M −→ R is such that Dρ ≥ Ram, in other words, ρ = Tr(s · ) for
some s ∈ S, equivalently ρ ∈ Tr ·S inside the S-module HomS (S, R). This is independent
of the choice of the isomorphism M ∼
= R.
We will use the next lemma frequently.
Lemma 2.13. Suppose (R, m, k) ⊆ (S, n, ℓ) is a separable finite extension of normal local
domains. If Tr : S −
→ R is surjective, then [ℓ : k] is equal to the number of simultaneous
free R-summands of S whose projection maps are multiples of Tr, in other words the
number of Tr-summands. In particular if K = K(R), L = K(S) and [ℓ : k] = [L : K],
then S is a free R-module.
Proof. Consider D = Tr ·S ⊆ HomR (S, R). Notice that hϕ ∈ D | ϕ(S) ⊆ mi = Tr ·S by
Lemma 2.10. By [BST12, Lemma 3.6], the number of free summands whose projection
maps are multiples of Tr is equal to


Tr ·S
lengthR
= lengthR (ℓ) = [ℓ : k]
Tr ·n
which proves the first statement. For the second, if we have equality, then we have a
surjective map S −
→ R⊕[ℓ:k]. But S is torsion free of generic R-rank [L : K] = [ℓ : k]. 
Let us now briefly discuss tame ramification. In the case that K(R) ⊆ K(S) is Galois,
the surjectivity of the trace map Tr : S −
→ R has been coined cohomologically tamely
ramified by M. Kerz and A. Schmidt; see for example [KS10, Claim 1, Theorem 6.2].
They actually proved in that article that this is in fact the strongest among all the notion
of tameness, including the one in [GM71].

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JAVIER CARVAJAL-ROJAS, KARL SCHWEDE, AND KEVIN TUCKER

What we want to observe next is that in the cases we consider in this paper, if Tr :
S−
→ R is surjective, then the extension of residue fields k ⊆ ℓ is a separable extension.
This is implicit in the proof of [KS10, Claim 1, Theorem 6.2] but we give a careful proof
in our setting. First we recall a very special case of a result of David Speyer. We thank
David Speyer for sharing a preliminary draft of a paper with us.
Lemma 2.14 ([Spe15]). Suppose that (R, m) ⊆ (S, n) is a finite local extension of F finite normal domains with Tr : S −
→ R surjective. Further suppose that ϕR : F∗e R −
→R
is an R-linear map that extends to ϕS : F∗e S −
→ S. If m is ϕR -compatible, then n is
ϕS -compatible.
Proof. We give a short proof here for the convenience of the reader. We have the commutative diagram [ST14, Proposition 4.1]
F∗e S

ϕS

//

S

F∗e Tr

Tr



F∗e R



//

ϕR

R.


We know that ϕR F∗e Tr(n) ⊆ ϕR (F∗e m) ⊆ m. Hence, we also have that

Tr ϕS (F∗e n) ⊆ m.

Thus ϕS (F∗e n) ( S, since Tr is onto, and hence ϕS (F∗e n) ⊆ n as desired.



Lemma 2.15. Suppose (R, m, k) ⊆ (S, n, ℓ) is a finite local extension of F -finite normal
domains that is étale in codimension 1 and such that R is strongly F -regular. Let K =
K(R) and L = K(S). In this case k ⊆ ℓ is separable and [ℓ : k] [L : K].
Proof. We begin with a claim.
Claim 2.16. There exists a surjective map ϕR : F∗e R −
→ R such that ϕR (F∗e m) ⊆ m.
Proof of claim. Choose a surjective ψ : F∗e R −
→ R (which exists since R is F -split). Then
there exists a smallest j > 0 such that ψ(F∗e mj ) ⊆ m. If j = 1, we may take ϕ = ψ,
so assume j > 1. By hypothesis ϕ(F∗e mj−1 ) = R. Choose z ∈ mj−1 with ϕ(F∗e z) = 1.
Form the map ϕR (F∗e ) = ψ F∗e (z · ) , it has the desired properties. This proves the
claim.

Since R ⊆ S is étale in codimension 1, ϕR extends to a map ϕS by [ST14, Theorem
5.7]. Therefore we have the commutative diagram where the vertical maps are inclusions.
F∗e R


ϕR

//

_





F∗e S

R _

ϕS

//

S.

FUNDAMENTAL GROUPS OF F -REGULAR SINGULARITIES VIA F -SIGNATURE

9

We notice that n is ϕS -compatible by Lemma 2.14. Thus by modding out the bottom row
by the compatible ideal n and the top row by m = n ∩ R we obtain the diagram
F∗e k

ϕk

// k
_

_



F∗e ℓ



ϕℓ

//

ℓ.

It follows from [ST14, Proposition 5.2] that k ⊆ ℓ is separable since ϕk is surjective and
hence not the zero map. This proves the first result.
After completion, the second part follows from the standard claim below.
Claim 2.17. Suppose (R, m, k) ⊆ (S, n, ℓ) is a finite local inclusion of complete F -finite
normal local rings with k ⊆ ℓ separable. Let K = K(R) and L = K(S). Then
[ℓ : k] [L : K].
Proof of claim. Since the extension of coefficient fields is separable, we can choose k ⊆ ℓ
a containment of coefficient fields as well. Now consider R ⊗k ℓ. Obviously this maps to
S. Hence we have a factorization
R−
→ R ⊗k ℓ −
→ S.
Now R ⊗k ℓ is a finite extension of R with unique maximal ideal m ⊗k ℓ. Also, since ℓ/k is
geometrically normal, R ⊗k ℓ is normal by [Sta16, Tag 06DF]. It follows that R ⊗k ℓ is a
normal local domain and a finite extension of R with a map to S. The map R ⊗k ℓ −
→ S is
also finite and must be injective because if it had a kernel, S would be finite over a lower
dimensional ring. Hence R ⊗k ℓ ⊆ S. Let T = K(R ⊗k ℓ). Then we have K ⊆ T ⊆ L
and [T : K] = [ℓ : k] (since R ⊗k ℓ is a free R-module of rank [ℓ : k]). This finishes the
claim.

It also finishes the proof of Lemma 2.15.



Remark 2.18. It is not difficult to generalize Lemma 2.15 to the context of pairs if R ⊆ S
is not necessarily étale in codimension 1. The only places where étale in codimension 1 is
used is when we extend ϕR to ϕS . Indeed, if one assumes that (R, ∆) is strongly F -regular
and that π ∗ ∆ − Ram is effective, the proof works without change. We will not need this
generalization in what follows, however.
2.4. Local fundamental groups of singularities. The study of local fundamental
groups of (normal) singularities has a long history, having early origins in the study of
resolution of singularities in positive characteristic based on the work of S. Abhyankar
and others (this mostly related to fundamental groups of a curve singularities). It also
goes back for example to the work of D. Mumford [Mum61] in which it is proven that for
the analytic germ of a normal complex surface, regularity or smoothness is equivalent to
the triviality of the local fundamental group. The same principle was generalized to the
algebraic setting by H. Flenner in [Fle75], see also [CS93, Corollary 5]. This however is
known to be false in positive characteristic by examples given by M. Artin in [Art77], see
also [CS93]. Our results are focused on higher dimensions however.
By local fundamental group of a singularity here we understand and mean the étale
fundamental group of the punctured spectrum of a strictly local3 normal domain (R, m, k),
3Also

called strictly Henselian, which means Henselian with separably closed residue field.

10

JAVIER CARVAJAL-ROJAS, KARL SCHWEDE, AND KEVIN TUCKER


i.e. π1ét Spec◦ (R) , as defined in [Gro63, Exposé V] or well in [Mur67]. Notice that since
R is a domain, its punctured spectrum is connected, thereby it makes sense to talk about
its étale fundamental group.
As customary for normal schemes, we choose the generic point as our base point x̄, i.e.
our base point is going to be the field extension x̄ : K ֒→ K sep , the or some separable
closure of K the fraction field of R.
We will need to compare étale coverings of a normal connected scheme X and the
étale coverings of an open connected subscheme U of it, say U = X \ Z with Z ⊆
X of codimension ≥ 2. Then we need to observe the following two Galois categories
are equivalent cf. [Art77] for the local case we care about: the category G of normal
schemes finite over X which are étale except possibly above Z; and the category FEt/U
of étale coverings of U. We clearly have a functor G −
→ FEt/U given by restriction to
−1
U, i.e. (f : Y −
→ X) 7−→ (f : f (U) 7−→ U), or as more classically described in this
context Y /X 7−→ (Y ×X U)/U. In fact, this functor gives rise to an equivalence of Galois
categories. This is nothing but a consequence of Zariski’s Main Theorem, see for instance
[Mil80, Theorem 1.8, Chapter 1]. For if we have an étale covering V −
→ U, then the
composition or extension V −
→ U ⊆ X is a quasi-finite morphism, hence there exists a
scheme Y together with a finite morphism Y −
→ X such that V can be realized as an open
subscheme of Y and the quasi-finite morphism above V −
→ X factors as V ⊆ Y −
→ X.
In other words, any étale covering V −
→ U can be realized as the restriction of a finite
map Y −
→ X which is étale everywhere except possibly above the complement of U, that
is étale on V . Let us use the terminology, any étale cover of U extends to a cover of X.
The above observation will be of great help for us inasmuch as by definition π1 (U)
is the fundamental group classifying or pro-representing the Galois category FEt/U, but
we use the equivalence above to classify instead the category G . In case X = Spec(R)
with R a singularity as above, this category is the same as the category of module-finite
(semi-local) inclusions of normal rings R ⊆ S (where the morphism are the morphism of
R-algebras) which are étale except possibly at the prime ideals lying over the closed Z
(or m when U is the punctured spectrum of R). Hence, we can work in a ring-theoretic
setting.
Another advantage is obtained in the understanding of the Galois coverings. For remember that in order to compute π1 one takes the directed projective system of connected/minimal and Galois elements R ⊆ S in G , and then considers its associated
projective limit
π1 = lim AutG S.
←−
However, since the base normal domain R is Henselian, the connected/minimal elements of
G are precisely the local domains in G , this is precisely the equivalence between (a) and (b)
in [Mil80, Theorem 4.2, Chapter 1]. The Galois extensions here are those R ⊆ S with finite
Galois extensions of fraction fields, moreover in that case AutG S = HomR (S, K sep ) =
Gal(L/K) where L is the fraction field of S.
We finish this series of remarks by pointing out that for general normal schemes there
is a particularly nice way to describe the computation of the respective étale fundamental
group. Say S is a normal and connected scheme, consequently integral, so if we choose
the base point to be a fixed separable closure of the fraction field K, we can take Xi to
be the normalization of X in Ki where the Ki are the finite Galois extension of K inside

FUNDAMENTAL GROUPS OF F -REGULAR SINGULARITIES VIA F -SIGNATURE

11

of our fixed separable closure such that Xi is unramified over X, then
π1 = lim Gal(Ki /K).
←−
3. F -Signature goes up under the presence of ramification
Suppose (R, m, k) ⊆ (S, n, ℓ) is a module-finite local extension of F -finite characteristic
p > 0 d-dimensional local domains with corresponding extension of fraction fields K ⊆ L.
According to [Tuc12, Corollary 4.13] if one writes S = R⊕f ⊕ M as a decomposition of
R-modules so that M has no nonzero free direct summands, then the following inequality
relating the F -signatures holds:
f · s(S) ≤ [L : K] · s(R).
In the theorem below, we show that if the extension above is étale in codimension 1 and
if R is strongly F -regular, then equality holds. We will also observe that if ℓ = k, then
f = 1. So that if there is any ramification (i.e. the extension is not étale everywhere)
then the F -signature of S would be at least twice the F -signature of R.
Theorem 3.1. Let (R, m, k) ⊆ (S, n, ℓ) be a module-finite local extension of F -finite ddimensional normal local domains in characteristic p > 0, with corresponding extension
of fraction fields K ⊆ L. Suppose R ⊆ S is étale in codimension 1, and that R is strongly
F -regular. Then if one writes S = R⊕f ⊕ M as a decomposition of R-modules so that
M has no nonzero free direct summands, then f = [ℓ : k] ≥ 1 and the following equality
holds:
[L : K]
s(S) =
· s(R).
[ℓ : k]
Proof. We notice that if R is strongly F -regular, so is S by [Wat91, Theorem 2.7], so both
R and S have positive F -signature by [AL03]. Also see [ST14, Lemma 3.5, Lemma 3.6].
By [ST14, Proposition 4.8], it follows that the trace map Tr : S −
→ R generates the
S-module HomR (S, R). Moreover, by [ST14, Corollary 7.7], the trace map is surjective
(this following from R being strongly F -regular). By Lemma 2.13 f = [ℓ : k].
Having the above in mind, we follow the proof of [Tuc12, Corollary 4.13], trying to
improve the estimates there by using these stronger conditions. We notice that


α(R) := logp [k 1/p : k] = logp [ℓ1/p : ℓ] =: α(S),
and denote by be the maximal rank of a free R-module appearing in a direct sum decome
e
position of S 1/p . If one writes a decomposition S 1/p = S ⊕ae (S) ⊕ Ne as S-modules where
Ne does not admit a free direct summand as S-module, then one also gets a decomposition
e
of S 1/p as an R-module
ae (S)
e
S 1/p = R⊕f ⊕ M
⊕ Ne = Rf ·ae (S) ⊕ M ⊕ae (S) ⊕ Ne .

From this one concludes that be ≥ f · ae (S). However, equality might not hold because
of the possibility of free R-summands coming from Ne . We will show that this cannot
happen under our stronger hypotheses. In fact, if N is any S-module with no nonzero
free direct S-summands, we will show it has no free direct R-summands as well. Indeed,
by [Sch09, Lemma 3.9] one has that any R-linear map N −
→ R is going to admit a
Tr
factorization through the trace map, i.e. N −
→ S −→ R (for this we are making use of the
condition HomR (S, R) = Tr ·S). Then, by virtue of the inclusion Tr(n) ⊆ m Lemma 2.10,

12

JAVIER CARVAJAL-ROJAS, KARL SCHWEDE, AND KEVIN TUCKER

if N −
→ R is surjective so has to be the factor N −
→ S, so that any free R-summand of
N would give rise to a free S-summand.
In conclusion, we have that be = f ·ae (S). By dividing through by pe(d+α(R)) = pe(d+α(S))
and letting e −
→ ∞ one obtains the desired equality (this by making use of [Tuc12,
Theorem 4.11]).

The following corollary will be one of the key ingredients to show the finiteness of
the étale fundamental group of a strongly F -regular singularity. It reflects how the F signature imposes strong conditions for the existence of non trivial coverings of a punctured spectrum.
Corollary 3.2. With the same setup as in Theorem 3.1, if the inclusion is not étale
everywhere, then s(S) ≥ 2s(R).
Proof. Since in our case, [ℓ : k] | [L : K] by Lemma 2.15, it is sufficient to show that
[L : K] > [ℓ : k]. But if we have equality, then S is a free R-module by Lemma 2.13 and
hence R ⊆ S is étale everywhere by [AK70, Chapter VI, Theorem 6.8].

Now we state and prove our promised purity of the branch locus result. Compare with
[Zar58, Nag58, Kun69, Gro05, Cut95].
Corollary 3.3 (Purity of the branch locus for rings with mild singularities). Suppose
Y −
→ X is a finite dominant map of F -finite normal integral schemes. If s(OX,P ) >
1/2 for all P ∈ X then the branch locus of Y −
→ X has no irreducible components of
codimension ≥ 2, in other words it is a divisor.
Proof. We work locally with X = Spec R and Y = Spec S. Let P ⊆ R be a minimal
prime of the locus where R ⊆ S is not étale, i.e. a branch point. Suppose however
 Qthe
c
height of P is at least 2. Localizing at P and completing, we may form RP , m, k ⊆ Si
where the Si are complete normal local domains. Since the original RP ⊆ SP is not étale,
cP ⊆ Si is not étale. Switching notation, let R = R
cP
at least one of the finite inclusions R
and S = Si so that we have a finite inclusion (R, m, k) ⊆ (S, n, ℓ) with s(R) > 1/2. We
notice that this extension is étale in codimension 1 but not étale. But this contradicts
Corollary 3.2 and the fact that s(S) ≤ 1.

Remark 3.4. Kunz began using Frobenius to study singularities because he observed that
if F∗e R is a free R-module, then purity of the branch locus holds for R. On the other hand,
s(R) > 1/2 can be interpreted as saying that F∗e R is more than half-free as an R-module
(at least for e ≫ 0). In other words, the above says that if F∗e R is more than half-free as
an R-module for e ≫ 0, then purity of the branch locus holds.
We rephrase this in one more way, and point out that we can do slightly better in
characteristic 2.
Corollary 3.5. Suppose (R, m, k) is a local F -finite strongly F -regular domain with fraction field K = K(R). Then ⌊1/s(R)⌋ is an upper bound on the size of [L : K]/[ℓ : k],
where L = K(S) and (R, m, k) ⊆ (S, n, ℓ) is an extension of normal local domains that
is étale in codimension 1. In the case that k is separably closed, ⌊1/s(R)⌋ is an upper
bound on the maximum size of [L : K] over any such étale-in-codimension-one extension.
Returning to the case of a general choice of k, if
(i) s(R) > 1/2 or,

FUNDAMENTAL GROUPS OF F -REGULAR SINGULARITIES VIA F -SIGNATURE

13

(ii) s(R) > 1/3, p = 2 and k is separably closed,
then there is no proper finite local extension of normal domains (R, m) ⊆ (S, n) that is
étale in codimension 1 but not étale (N.B. this implies triviality of the respective local
fundamental group).
Proof. Suppose that (R, m, k) ⊆ (S, n, ℓ) is a finite local extension that is étale in codimension 1 and with g := [L : K]/[ℓ : k] > ⌊1/s(R)⌋. Hence by Theorem 3.1, we obtain
s(S) = g · s(R) > 1 a contradiction. This proves the first statement. Next if k is separably closed, then k = ℓ by Lemma 2.15 which proves the second. Part (i) is just a
local restatement of Corollary 3.3. For (ii), Corollary 2.11 implies that g > 2. The result
follows.

Example 3.6. It is known that the F -signature of Fp Jx0 , ..., x3 K/(x20 + · · ·+ x23 ) is 3/4, see
[WY05, Theorem 3.1], [HL02, Proof of Theorem 11], [Tuc12, Proof of Proposition 4.22].
Of course, it is well known that this complete intersection satisfies purity of the branch
locus [Gro05].
Corollary 3.7. Suppose that L is an ample line bundle onL
a globally F -regular projective
variety X over an algebraically closed field k. Write R = i≥0 H 0 (X, L i ). If L = A m
for another line bundle A , with p 6 | m, then m ≤ 1/s(R).
L
Proof. We have an inclusion R ⊆ S = i≥0 H 0 (X, A i ) which, while not graded, simply
multiplies degrees by m. We notice that this inclusion is étale in codimension 1, see for
instance [SS10, Lemma 5.7]. Furthermore, it is easy to see that it has generic rank m.
The result follows immediately by completion (or Henselization) and Theorem 3.1.

4. Behavior of F -signature of pairs under finite morphisms
In this section we generalize the formula of Theorem 3.1 to maps which are not necessarily étale in codimension 1. We include it in a separate section because of its much
more technical proof. We will see that in compensation, the formalism of divisors provides a way to include the presence of codimension-1 ramification into the transformation
formula, so that we can recover the formula without pairs in the case of extensions that
are étale in codimension 1.
As in the proof of Theorem 3.1, one of the main ingredients will be the capability
of factoring through the trace maps. The following two lemmas are formulated here as
generalization of the sort of factorization in [Sch09, Lemma 3.9], we do it in two steps;
we first add pure ramification with Lemma 4.1 and secondly the presence of pairs with
Lemma 4.2.
Lemma 4.1. Suppose that R ⊆ S is a finite separable extension of normal domains with
ramification divisor Ram on Spec S. Then the image of the map induced by composition


HomS F∗e S(Ram), S(Ram) × HomR S(Ram), R −
→ HomR (F∗e S, R)

contains HomR F∗e S(Ram), R ⊆ HomR (F∗e S, R). In particular, any map ϑ : F∗e S −
→R
which factors through a map F∗e S(Ram) −
→ R also factors through Tr : S(Ram) −
→ R.
Proof. We know
HomR F∗e S(Ram), R



∼
= HomR



F∗e S(Ram)





⊗S S, R ∼
= HomS F∗e S(Ram), HomR (S, R)

∼
= HomS F∗e S(Ram), S(Ram) .



14

JAVIER CARVAJAL-ROJAS, KARL SCHWEDE, AND KEVIN TUCKER


Given an element α ∈ HomS F∗e S(Ram), HomR (S, R) it is identified with a map β ∈
HomR F∗e S(Ram), R through Hom −⊗ adjointness by composing with the evaluationat-1 map HomR (S, R) −
→ R. In our case,
ϑ ∈ HomR F∗e S(Ram), R)

yields a map ψ ∈ HomS F∗e S(Ram), S(Ram) such that, since Tr : S(Ram) −
→ R corresponds to the evaluation-at-1 map HomR (S, R) −
→ R, we know that ϑ = Tr ◦ψ.

Lemma 4.2. Suppose that R ⊆ S is a finite separable extension of normal domains with
π : Y := Spec S −
→ X := Spec R the corresponding morphism of schemes, and that ∆ ≥ 0
is a Q-divisor on X such that π ∗ ∆ − Ram is effective. Then any map

ϑ ∈ HomR F∗e S(⌊pe π ∗ ∆⌋ + Ram), R ⊆ HomR (F∗e S, R)

when restricted to ϑ : F∗e S −
→ R factors through Tr : S −
→ R.


Proof. Obviously HomR F∗e S(⌊pe π ∗ ∆⌋+Ram), R ⊆ HomR F∗e S(Ram), R and so we can
obtain a map

ψ ∈ HomS F∗e S(Ram), S(Ram)
with ϑ = Tr ◦ψ by Lemma 4.1.
Claim 4.3. ψ is contained in



HomS F∗e S(⌊pe π ∗ ∆⌋ + Ram), S(Ram) ⊆ HomS F∗e S(Ram), S(Ram) .

Proof of claim. Viewing ϑ as a section of HomR F∗e S(Ram), R , we obtain a divisor Dϑ
with



F∗e S(Dϑ ) ∼
= HomR F∗e S(Ram), R ∼
= HomS F∗e S(Ram), S(Ram) ∼
=F∗e S (1 − pe )(KS − Ram)

=F∗e S (1 − pe )π ∗ KR .
and hence Dϑ ∼ (1 − pe )π ∗ KR . Furthermore, by construction Dϑ ≥ ⌊pe π ∗ ∆⌋. We have
the composition
F∗e S(Ram)

ψ

//

S(Ram)

Tr

//
99

R.

ϑ

Since in this diagram, Tr : S(Ram) −
→ R corresponds to the zero divisor, we see that ψ cor
responds to the same divisor as ϑ by Lemma 2.3. In other words ψ ∈ HomS F∗e S(Ram), S(Ram)
yields the divisor Dψ ≥ ⌊pe π ∗ ∆⌋. This just means that

ψ ∈ HomS F∗e S(⌊pe π ∗ ∆⌋ + Ram), S(Ram) .
This proves the claim.



By twisting and using the projection formula, we view ψ as an element contained in
HomS F∗e S(⌊pe π ∗ ∆⌋ + (1 − pe )Ram), S and so by restriction, we obtain

ψ : F∗e S ⌊π ∗ ∆ + (pe − 1)(π ∗ ∆ − Ram)⌋ −
→ S.

Further restricting to F∗e S (since π ∗ ∆ − Ram is effective) proves the lemma.



FUNDAMENTAL GROUPS OF F -REGULAR SINGULARITIES VIA F -SIGNATURE

15

Theorem 4.4. Suppose that (R, m, k) ⊆ (S, n, ℓ) is a finite separable extension of normal
local domains with ramification / relative canonical divisor Ram. Let π : Y = Spec S −
→
X = Spec R be the corresponding morphism of schemes. Suppose that ∆X ≥ 0 is a Qdivisor on X such that ∆Y = π ∗ ∆X − Ram ≥ 0. Let f be the maximal number of direct
Tr-summands of S, then
f · s(S, ∆Y ) = [L : K] · s(R, ∆X ).
Proof. First note that if Tr(S) ( R, then f = 0. Furthermore, in this case, by [ST14,
Corollary 6.26] τ (R, ∆X ) = Tr(τ (S, ∆Y )) ⊆ Tr(S) ( R and so (R, ∆X ) is not strongly
F -regular and s(R, ∆X ) = 0. Thus the equality holds. We may henceforth assume that
Tr(S) = R so that (S, ∆Y ) is strongly F -regular if and only if s(R, ∆X ) is by [ST14,
Corollary 6.31]. Hence one side is zero if and only if the other is, and so we may assume
that s(R, ∆X ), s(S, ∆Y ) > 0. Finally, again note that since Tr(S) = R, f = [ℓ : k] by
Lemma 2.13. We additionally remark that we can also assume without lost of generality
that ⌊∆X ⌋ = 0, otherwise the equality we plan to show would become trivially 0 = 0.
Recall that by [BST12, Lemma 3.6] one has that


[ℓ : k] = f = λR S/hs ∈ S | ρ(s) ∈ m, for all ρ ∈ Tr ·Si = λR Tr ·S/hρ ∈ Tr ·S | ρ(S) ⊆ mi .
As in Section 2.2, write


Ge∆X = HomR F∗e R(⌊pe ∆X ⌋), R ⊆ HomR (F∗e R, R) = CeR

and set IeG = {r ∈ R | ϕ(F∗e r) ⊆ m, for all ϕ ∈ Ge∆X }. This set Ge∆X corresponds to the
maps ϕ ∈ CeR which factors through a map F∗e R(⌊pe ∆X ⌋) −
→ R via the natural inclusion
R ⊆ R(⌊pe ∆X ⌋). One defines and interprets Ge∆Y in the same fashion.
In order to recover an appropriate analog of [Tuc12, Theorem 4.11] we introduce
 
De := HomR F∗e S ⌊pe π ∗ ∆X ⌋ + Ram , R ⊆ HomR (F∗e S, R).
Let JeD be the R-submodule of S

JeD := s ∈ S | ϑ(F∗e s) ∈ m, for all ϑ ∈ De

and write




be := λR F∗e S/JeD = peα(R) λR S/JeD .

The result now follows immediately from the two lemmas, Lemma 4.5 and Lemma 4.6
below.

Lemma 4.5. With notation as in the proof of Theorem 4.4,
be
lim e(d+α(R)) = [L : K] · s(R, ∆X ).
e→∞ p
Proof. The proof of this is, mutatis mutandis, the same as the proof of [Tuc12, Theorem
4.11]. Indeed, one has the following equalities,

1
[L : K] · s(R, ∆X ) = rankR (S) · s(R, ∆X ) = rankR (S) · lim de eHK IeG ; R
e→∞ p

1
= lim de eHK IeG ; S
e→∞ p

1
= lim de λR S/IeG S .
e→∞ p

16

JAVIER CARVAJAL-ROJAS, KARL SCHWEDE, AND KEVIN TUCKER

The second equality holds in virtue of [Tuc12, Corollary 3.7] cf. [BST12, Corollary 3.17]
and Lemma 2.7. The next equality follows for example from [Mon83, Lemma 1.3]. The
last equality still follows from [Tuc12, Corollary 3.7].
It only remains to verify that


1
1
(4.5.1)
lim de λR S/IeG S = lim de λR S/JeD .
e→∞ p
e→∞ p
For this purpose we utilize Lemma 2.8. First observe that



R ⌊pe ∆X ⌋ ⊆ S π ∗ ⌊pe ∆X ⌋ ⊆ S ⌊pe π ∗ ∆X + Ram⌋ .
∈
For any rs ∈ I G S and any ϑ ∈ De , write ϑ′ (F e ) = ϑ(F e s ). Then ϕ′ := ϑ′ |
e

Ge∆X

so that

∗

ϑ(F∗e rs)

=ϕ

′

(F∗e r)

∗

R ⌊pe ∆X ⌋

∈ m. Hence

IeG S ⊆ JeD .

(4.5.2)

Now let 0 6= b ∈ R be such that divS (b) ≥ Ram. Then observe that for any Q-divisor
D on Spec R, divS (b) ≥ Ram ≥ ⌊π ∗ D⌋ − π ∗ ⌊D⌋.4 It follows that


(4.5.3)
b2 S ⌊pe π ∗ ∆X ⌋ + Ram ⊆ S π ∗ ⌊pe ∆X ⌋ .

Next choose a free module G = R⊕[L:K] ⊆ S and a 0 6= c ∈ R such that cS ⊆ G. It follows
that
(4.5.4)

∗∗
∗∗
⊕[L:K]
cS ⌊pe π ∗ ∆X ⌋ = cS ⊗R R(⌊pe ∆X ⌋) ⊆ G ⊗R R(⌊pe ∆X ⌋) = R(⌊pe ∆X ⌋)

where ∗∗ denotes reflexification as an R-module (or equivalently, since it can be viewed
as S2-ification, as an S-module where appropriate). Note the equalities and containments
can be checked in codimension 1 where they are obvious. Putting (4.5.3) and (4.5.4)
together yields:

⊕[L:K]
cb2 S ⌊pe π ∗ ∆X ⌋ + Ram ⊆ R(⌊pe ∆X ⌋)
.

Now choose y ∈ JeD and


⊕[L:K] 
ϕ ∈ HomR F∗e R ⌊pe ∆X ⌋
, R ⊆ HomR (F∗e G, R).

Then, viewing things at the level of the field of fractions, ϕ(F∗e cb2 ·
ϕ(F∗e cb2 y) ∈ m. It follows that

IeG S ⊆ JeD ⊆ IeG S : cb2

) ∈ De and so

where the first containment was shown in (4.5.2). Now (4.5.1) follows from Lemma 2.8.


Lemma 4.6. With notation as in the proof of Theorem 4.4,
be
lim e(d+α(R)) = f · s(S, ∆Y ).
e→∞ p
4To

see this work locally with a separable extension of DVRs R ⊆ S with uniformizers r and s such that
r = usn . Observe if D = λ divR (r), then
⌊π ∗ D⌋ ≤ π ∗ D = π ∗ {λ} divR (r) + π ∗ ⌊D⌋ = n{λ} divS (s) + π ∗ ⌊D⌋
which implies that ⌊π ∗ D⌋ ≤ ⌊n{λ} divS (s) + π ∗ ⌊D⌋ and so and ⌊n{λ} divS (s)⌋ ≤ Ram.

FUNDAMENTAL GROUPS OF F -REGULAR SINGULARITIES VIA F -SIGNATURE

17

Proof. By applying the second part of Lemma 2.7, we can compute s(S, ∆Y ) using

∆
Ge′ = G ′ e Y = HomR F∗e S(⌊pe ∆Y ⌋ + Ram), S ,
′

with the associated splitting numbers a′e = aGe .
We are going to prove that the following equality holds: be = f · a′e . Taking limits will
then prove the lemma. We prove first that be ≥ f · a′e . This can be verified by observing
that the composition of an element ψ ∈ Ge′ with a map ρ ∈ Tr ·S gives an element ϑ := ρ◦ψ
in De . Indeed, Dψ ≥ ⌊pe ∆Y ⌋ + Ram and Dρ ≥ Ram and so by Lemma 2.3, we see that



Dϑ ≥ ⌊pe ∆Y ⌋+Ram +pe Ram = ⌊pe π ∗ ∆X −pe Ram⌋+pe Ram +Ram = ⌊pe π ∗ ∆X ⌋+Ram.
It follows that ϑ ∈ De as claimed. We can certainly construct f · a′e distinct R-summands
by such compositions and so be ≥ f · a′e .
Now we prove that be ≤ f · a′e . Indeed, given any ϑ ∈ De , Lemma 4.2 provides
 a
factorization ϑ = Tr ◦ψ where ψ belongs to HomS F∗e S(⌊pe π ∗ ∆X ⌋ + Ram), S(Ram) by
Claim 4.3. However, as observed above ⌊pe ∆Y ⌋+(pe +1)Ram = ⌊pe π ∗ ∆X ⌋+Ram, whence
ψ restricts by twisting by S(Ram) to a map ψ : F∗e S(⌊pe ∆Y ⌋ + Ram) −
→ S, which is by
′
definition an element of Ge . Next note that Lemma 2.10 ensures that ψ is surjective
whenever ϑ is. But now, if N ⊆ F∗e S is an S-module summand of F∗e S with no free
S-summands whose projection maps are in Ge′ , then N has no free R-summands with
corresponding projections in De . The result follows.


Corollary 4.7. Suppose (R, m) is a strictly Henselian F -finite normal local domain. If
∆ ≥ 0 is a Q-divisor on Spec R such that (R, ∆) is strongly F -regular, then ⌊1/s(R, ∆)⌋ is
an upper bound on the maximal generic rank of a finite separable local extension (R, m) ⊆
(S, n) so that π ∗ ∆ − Ram is effective where π : Spec S −
→ Spec R is the induced map.
Proof. Suppose that (R, m) ⊆ (S, n) is a finite separable local extension such that ∆S =
π ∗ ∆ − Ram of generic rank g > ⌊1/s(R)⌋. Hence g > 1/s(R, ∆). But s(S, ∆S ) =
gs(R, ∆) > 1 and the result follows.

The following should be viewed as a characteristic p > 0 analog of [Xu14, Proposition
1]. First however, recall that if X is projective variety over k = k, ∆ L
is a Q-divisor and
L is an ample line bundle, then we can form the the section ring R = n≥0 H 0 (X, L n ).
We have a canonical k ∗ -bundle map Spec R \ V (R>0 ) −
→ X. We can then pull back ∆
from X and obtain a unique corresponding divisor ∆R on Spec R. See [SS10, Section 5].
Corollary 4.8. Suppose that (X, ∆X ) is a globally F -regular projective
variety over an
L
algebraically closed field k. Suppose L is ample on X and that R = n≥0 H 0 (X, L n ) with
∆R the corresponding divisor on R. Then 1/s(R, ∆R ) is an upper bound on the generic
rank of a finite separable cover π : Y −
→ X with Y normal such that ∆Y = π ∗ ∆X −RamY /X
is effective. In particular, if ∆ = 0, then 1/s(R) is an upper bound on a finite separable
étale-in-codimension-1 cover of any open set U ⊆ X whose complement has codimension
≥ 2 in X.

L
L
Proof. Write S = n≥0 H 0 (Y, π ∗ L n ) = n≥0 H 0 X, (π∗ OY ) ⊗ L n . We have a graded
finite inclusion R ⊆ S with associated µ : Spec S −
→ Spec R. Since (X, ∆X ) is globally
F -regular, (R, ∆R ) is strongly F -regular by [SS10, Proposition 5.3]. Next observe that
the generic rank of π∗ OY over OX is the same as the generic rank of S over R since
⊕m
if OX
⊆ π∗ OY has torsion cokernel, so does the corresponding inclusion R⊕m ⊆ S.

18

JAVIER CARVAJAL-ROJAS, KARL SCHWEDE, AND KEVIN TUCKER

Finally, observe that RamS/R simply corresponds to RamY /X and so ∆S = µ∗ ∆R −RamS/R
corresponds to ∆Y and ∆S is effective if and only if ∆Y is. The corollary now follows
immediately by completion (or Henselization) and Corollary 4.7.


5. Finiteness of the fundamental group of a strongly F -regular
singularity
In this section we prove our main result, namely; finiteness of the étale fundamental
group of a strongly F -regular singularity. Let (R, m, k) be a normal F -finite and strongly
F -regular strictly local domain of prime characteristic p > 0. We will demonstrate finiteness of the étale fundamental group of U ′ ⊆ U := Spec◦ (R) where U ′ is the complement
of a closed subset Z of X := Spec(R) through m of codimension at least 2. In particular,
since strongly F -regular rings are normal, the singular locus has codimension at least 2,
and we can take U ′ = Ureg as the regular locus.
Theorem 5.1. Let (R, m, k) be a normal F -finite and strongly F -regular strictly local
domain of prime characteristic p > 0, with dimension d ≥ 2. Let Z ∋ m be a closed
subscheme of X := Spec(R) of codimension at least 2 with complement U.Then the étale
fundamental group of U, i.e. π1 := π1ét U, x̄ , is finite. Furthermore, the order of π1 is at
most 1/s(R) and is prime to p. For example, Z = {m} and U = Spec◦ (R).
Proof. First note that since R is a normal domain, U = X \ Z is always connected. Next,
notice that since the trace is surjective, all the coverings we consider are (cohomologically)
tamely ramified, in particular; given we have taken the hypothesis k = k sep , all the residue
field extensions in this category are trivial, see Lemma 2.15.
In this way, to show that π1 is finite one just has to show that a sequence of module-finite
local inclusions
(R, m, k) ⊆ (S1 , n1 , k) ⊆ (S2 , n2 , k) ⊆ (S3 , n3 , k) ⊆ · · ·
in which the consecutive inclusions (Si , ni , k) ⊆ (Si+1 , ni+1 , k) are étale in codimension 1,
and the extensions (R, m, k) ⊆ (Si , ni , k) are all Galois, stabilizes; by this we mean that
Si = Si+1 for all i ≫ 1. In fact, what we have is that all but finitely many consecutive
inclusions are étale everywhere, this is a direct consequence of the Corollary 3.2: because if
this is not the case, then the F -signature of the local rings Si will eventually be arbitrarily
large as i grows (since we started with s(R) > 0), but the F -signature of any ring is wellknown to be at most 1. Hence, eventually one has equalities since in this setting étale-ness
guarantees equality (the extension will be free because it is étale, but at most 1 = [ℓ : k]
free summand is allowed or possible because the extensions of residue fields are all trivial).
For the upper bound on the order of π1 , just noticed that in this case there will be a
Galois extension (R, m) ⊆ (S ⋆ , n⋆ ) representing G . This is going to dominate any other
such extension. To get such (S ⋆ , n⋆ ), let (S, n) be a maximal element in a chain as above
(this corresponds to a maximal element a chain of étale extensions over U). If it does not
dominate some other extension (R, m) ⊆ (T, o) ⊆ K sep which is also étale over U, then
both (S, m) and (T, o) can be dominated by a larger extension also étale over U. Thus
we may take S ⋆ = S.

FUNDAMENTAL GROUPS OF F -REGULAR SINGULARITIES VIA F -SIGNATURE

19

We now have π1 = AutG S ⋆ = Gal(L⋆ /K), the second equality holds because of the
Galois condition, thereby
1
,
#π1 = # AutG S ⋆ = [L⋆ : K] ≤
s(R)
the inequality at the end is just given by Corollary 3.5. By Corollary 2.11, it also follows
the order of the group is prime to p.

Remark 5.2. We remark we need to assume our local ring expressing the singularity is
strictly Henselian and not just Henselian since otherwise the associated fundamental group
would contain Gal(k sep /k), which might easily be infinite, for instance; for perfect fields
the separable closure coincides with the algebraic closure. For example, this is infinite
for k = Fp . Nonetheless, under the hypothesis #Gal(k sep /k) < ∞ the same result would
follow.
Remark 5.3. As we have observed several times before, the surjectivity of the trace imposes
strong tameness on the ramification, namely cohomological tameness. So that the étale
fundamental group we dealt with in Theorem 5.1 is actually the same as the/any tame
fundamental group. In fact, as we noticed, the order of the group is prime to p.
From the proof of Theorem 5.1 we get the following statement, which is a local and
positive characteristic analog for [GKP13, Theorem 1.1].
Scholium 5.4. Let (R, m, k) be a strongly F -regular local domain. In any chain
(R, m) ⊆ (S1 , n1 ) ⊆ (S2 , n2 ) ⊆ (S3 , n3 ) ⊆ · · ·
of module-finite local étale in codimension 1 inclusions of normal local domains all but
finitely many of the extensions are étale everywhere.
The following corollary is the local positive characteristic analog of [GKP13, Theorem
1.5]. However, in our local case it follows from a general result for singularities with finite
fundamental group. So that it is not that interesting as in the global case.
Corollary 5.5. Suppose that (R, m) is a strictly local F -finite strongly F -regular domain.
Then inside any fixed separable closure of its fraction field K, there exists a unique largest
finite local étale in codimension 1 extension (R, m) ⊆ (S, n) of normal domains so that
(S, n) has no non-trivial finite local étale in codimension 1 extension (S, n) ( (T, o), i.e.
with trivial fundamental group.
Compare the following with [GKP13, Theorem 1.10].
Corollary 5.6. Suppose (R, m) ⊆ (S, n) as in Corollary 5.5 with π : Spec S −
→ Spec R
the induced morphism. If D is a Q-Cartier Weil divisor on Spec R such that n, the index
of D, is not divisible by p, then n ≤ 1/s(R) and n | [K(S) : K(R)]. Furthermore, π ∗ D is
Cartier on Spec S.
Proof. If D is Q-Cartier with index n, not divisible by p, then the cyclic cover R ⊆
R ⊕ R(−D) ⊕ · · · ⊕ R(−(n − 1)D) = T is étale in codimension 1 by [Wat91, Example
2.8]. Now, T is a domain by [TW92, Corollary 1.9] hence local since R is Henselian. The
extension R ⊆ T has generic rank n and the first statements follows. Note that π ∗ D is
still Q-Cartier but it must be Cartier since otherwise we could take a cyclic cover on S
which has no non-trivial étale in codimension 1 covers.


20

JAVIER CARVAJAL-ROJAS, KARL SCHWEDE, AND KEVIN TUCKER

By taking cones we also have the following.
Corollary 5.7. Suppose that X is a projective globally F -regular variety over an algebraically closed field and that Z ⊆ X is a closed subset of codimension ≥ 2. Let U = X \Z.
Then π1 := π1ét U) is finite of order prime to p.
Proof. We can bound the degree of any finite étale cover of U as before in Corollary 4.8
and prove that its order is relatively prime to p. The result then follows exactly as in
Theorem 5.1.

Remark 5.8. In the case that dim X ≤ 3 and p ≥ 11, if X is globally F -regular it is
also rationally chain connected by [GLP+ 15]. Hence by [CL04], if it is also smooth, the
étale fundamental group is finite of order prime to p (see also [Kol95, Theorem 4.13]
and [GKP13, Section 11.2]). On the other hand, Corollary 5.7 can also be thought of as
evidence that (smooth) globally F -regular varieties are rationally chain connected in all
dimensions.
It is natural to ask whether the characteristic p > 0 results of this paper imply the
characteristic zero results of [Xu14] and some of the results of [GKP13]. Unfortunately,
we do not know how to reduce local algebraic fundamental groups to characteristic p ≫ 0
by spreading out. However, if we had a positive answer to the following question, a
number of characteristic zero results would immediately follow.
Question 5.9. Suppose that (R, m) is the local ring of a singularity in characteristic zero.
Consider a family of characteristic p > 0 reductions (Rp , mp ). Is5
lim sup s(Rp ) > 0?
p→∞

Examples seem to suggest that this is the case. Of course, this is only a special case of
the following question which a number of people have already considered.
Question 5.10. Does limp→∞ s(Rp ) have a geometric interpretation, or at least some geometric lower bounds?
Indeed, answers to this question would yield effective versions of many of the results of
[Xu14] and [GKP13].
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Department of Mathematics, University of Utah, Salt Lake City, UT 84112
E-mail address: carvajal@math.utah.edu
Department of Mathematics, University of Utah, Salt Lake City, UT 84112
E-mail address: schwede@math.utah.edu
Department of Mathematics, University of Illinois at Chicago, Chicago, IL 60607
E-mail address: kftucker@uic.edu

