Wild ramification of schemes via curves, the rank 1 case

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Wild ramification of schemes via curves, the rank 1 case

(with applications to higher class field theory)

Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften (Dr.

rer. nat.) der Fakultät f ür Mathematik der Universität Regensburg vorgelegt von

Ivan Barrientos aus Mexiko-Stadt, Mexiko 2014

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Promotionsgesuch eingereicht am: 24. M¨arz 2014

Die Arbeit wurde angeleitet von: Prof. Dr. Moritz Kerz Pr ¨ufungsausschuss:

Vorsitzender: Prof. Dr. Abels Erst-Gutachter: Prof. Dr. Kerz Zweit-Gutachter: Prof. Dr. Jannsen weiterer Pr ¨ufer: Prof. Dr. K ¨unnemann

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Chapter 1 Acknowledgements

I am deeply indebted to my advisor M. Kerz for his guidance and constant encouragement. I profited from discussions with T. Saito and L. Xiao on the state of the art of ramification theory. I also want to thank S. Kelly for beneficial comments on preliminary versions of this work. I also thank I. Zhukov for an invitation to present these results at the “Arithmetic Days 2013” conference hosted by the Steklov Institute in St. Petersburg.

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Chapter 2 Introduction

Let R be a discrete valuation ring with residue field κ, fraction field K = Frac(R), andLa finite Galois extension ofK. Ifκis perfect, there is a satisfac- tory ramification theory forL/K, see for example [Ser 68]. On the other hand, if κ is not perfect this theory is not as well-established; for a comprehensive survey on ramification see [Xia-Zhu 13].

In the 1980’s, Brylinski and Kato defined conductors when κ is not necessarily perfect and L/K is attached to a character of rank 1, see [Bry 83] and [Kat 89]. Recently, Abbes and Saito succeeded in providing a general definition of higher ramification groups in an`-adic context that agrees with the classical case of a perfect residue field; see [Abb-Sai 02] and [Abb-Sai 11]. Consequently, they also define a conductor in [Abb-Sai 11]. Moreover, their conductor agrees with that of Brylinski and Kato’s in rank 1, see [Abb-Sai 09].

In another direction, in the 1970’s Deligne had initiated a program of mea- suring ramification of sheaves along a divisor in terms of transversal curves, see [Del 76], which was further developed by Laumon in [Lau 81] and [Lau 82].

We remark that one of the principal aims of these works was to achieve new Euler-Poincar´e formulas for surfaces (over algebraically closed fields). Let us also mention work by Brylinski [Bry 83] and Zhukov [Zhu 02].

A natural question then became if one could follow Deligne’s program and express Abbes-Saito’s conductor in terms of curves. In this direction, Matsuda established results in the so-called ‘non-log’ case of Brylinski-Kato’s conductor, see [Mat 97] and [Ker-Sai 13, Coro 2.7]. There are also stronger results for the

‘non-log’ case recently obtained by T. Saito, see [Sai 13]. For a modern survey on wild ramification in the sheaf-theoretic context, see [Sai 10].

The main result of this thesis is the description of Brylinski-Kato’s ‘log’

conductor in terms of curves, see Theorem 6.1.2. In other words, we estab- lish that Abbes-Saito’s ‘log’ conductor in rank 1 is given in terms of curves.

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Our approach is a slight generalization of Deligne’s original idea of analyz- ing ramification via transversal curves and instead consideringallcurves on a scheme with special attention totangentcurves. We conjecture that our result also holds for`-adic sheaves of finite rank, see Conjecture A as well as Con- jecture B below. We hope our conjectures can be used for non-abelian higher class field theory.

As applications of our main result, we confirm an expectation of Esnault and Kerz [Esn-Ker 12, §3] in the smooth rank 1 case, see Theorem 7.2.1, and we also obtain a reformulation of the recent ‘non-log’ Existence Theorem of higher class field theory obtained by Kerz and S. Saito, [Ker-Sai 13, Coro II] in the ‘log’ case, see§8.3. In a slightly different direction, we also generalize an Isomorphism Theorem of Kerz and Schmidt [Ker-Sch 09, Thm. 8.2] by relaxing an assumption on being generically proper over a curve, see Thm.8.2.4.

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Chapter 3 Notation and definitions

3.1 Notation and definitions

Unless otherwise mentioned:

• By thedimensionof a ring or a scheme we mean the Krull dimension.

• The group ofeffective divisorson a schemeXis denoted by Div⁺(X)_.

• An effective divisor D is a simple normal crossing divisor on X if Zariski locally¹ there is a finite family of sections (fi)_i_∈_I, fi ∈ O_X such that the following two conditions hold:

(i) D=^P_i_∈_I div(fi)

(ii) for each x ∈ Supp(D), the restrictions (fi)_x that satisfy (fi)_x ∈ m_X,xform a part of a regular system of parameters inO_X,x.

• We will routinely abbreviate ‘simple normal crossing divisor’ bysncd.

• An effective divisorDis anormal crossings divisor(ncd) if (i) and (ii) hold

´etale locally.

• Avarietyis a scheme that is reduced, separated, and of finite type over a fieldk.

• Sm_k is the set of connected, smooth varieties over Spec(k).

• Acurve Cis an integral scheme of dimension 1. Acurve C ona scheme X is a closed curveC ⊂X.

• Asmooth sheaf is a locally constant constructible sheaf.

1Our definition follows [Gro-Mur 71], Def 1.8.2 and is more flexible than that of SGA 1 [XIII, 2.1] and SGA 5 [3.1.5] where thefiare required to beglobalsectionsfi∈Γ(X,O_X).

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• The set of closed points of a scheme X is denoted by |X|. The set of codimensionrpoints is denoted byX⁽^r⁾.

• Ndenotes the natural numbers; this set includes 0 (the cardinality of the empty set).

• For a noetherian local ring A, denote byA^hthe henselization ofAand ˆA its completion. We have A ⊂A^h ⊂ A.^ˆ

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Chapter 4 Background on Swan conductors

4.1 Overview

In this chapter, we begin by defining the Swan conductor at the level of finite extensions of discrete valuation fields withseparableresidue field extensions in Sec. 4.2, see Def. 4.2. In Sec 4.3 we give the definition of the conductor for a curve defined over a perfect field of finite characteristic, see Def. 4.3. We provide an example calculation of this conductor for an Artin-Schreier extension in Ex. 4.3.4. Basic lemmas for Artin-Schreier extensions are provided in Lemmas4.3.1and4.3.2.

The Brylinski-Kato filtration is introduced in Sec. 4.4from which we derive a generalized conductor for finite extensions of discrete valuation fields whose residue field extension is not necessarily separable, see Def. 4.4.4. Finally, in Sec. 4.5we define the log conductor of anF`-sheaf of rank 1 at a given generic point of a divisor on the boundary in Def. 4.5.1. As a prelude to the next chapter, where we state our two main expectations, we include in subsection 4.5.1an example that gives evidence for computing the log conductor in terms of a limit of conductors of curves tangent to a given divisor.

4.2 Conductor for discrete valuation fields with per- fect residue field

We briefly review here the classical case of Swan conductors which we later relate to the Brylinski-Kato conductor. Classically, the Swan conductor is defined for finite extensions of discrete valuation rings with separable residue field extensions, which we begin with here. Let R be a henselian discrete valuation ring of characteristic p > _{0. Let} k be the residue field of R. Put K =Frac(R). LetF be a constructibleF`- sheaf of finite rank onS=Spec(R) and ¯η = Spec(K^sep) a geometric point of S. Then the stalk M = F_η_¯ is an

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F`[Gal(K^sep/K)]-module and dim_F_`(F_η_¯) is finite. The action of Gal(K^sep/K) onF_η_¯ factors through a quotient of Gal(K^sep/K)by a closed normal subgroup Γof finite index. Fix such a quotient and write it as

G =Gal(K^sep/K)/Γ.

Let Lbe the fixed field ofΓ, i.e.

L = (K^sep)^Γ;

then G = Gal(L/K). Let R⁰ be the normalization of R in L and let k⁰ be the residue field ofR⁰.

Fori ≥0, letG_idenote thei-th (lower) ramification group ofG, i.e.

G_i =_ker[G →_Aut(R⁰/m_Lⁱ⁺¹)]_,

wherem_L is the maximal ideal of R⁰. Clearly, eachG_i is a normal subgroup of G.

We now state the main assumption of this section: assume that

the extension of residue fieldsk⁰/kofR⁰/Risseparable. (4.1) Lemma 4.2.1. Fori _0,G_iis trivial.

Proof. Letv_L denote the discrete valuation ofL. First observe that since the extension of residue fieldsk⁰/kis finite and separable, the theorem of the primi- tive element implies that there exists ¯x⁰ ∈ k^sep such thatk⁰ = k[x¯⁰]. There is a liftingx⁰ ∈ R⁰of ¯x⁰such that R⁰ = R[x⁰](see e.g. [Neu 99, II.10.4]).

We have then

Gi ={σ ∈ G : vL(σ(x⁰)−x⁰) ≥i+1}.

SinceGis a finite group, there are only finitely many valuesv_L(σ(x⁰)−x⁰) asσranges over all elements ofGand therefore

i⁰ >max

σ∈G{v_L(σ(x⁰)−x⁰)} ⇒ G_i⁰ is trivial.

SinceGacts on M, then so does eachG_i. Let M^Gⁱ denote submodule of M fixed by G_i. Let s = Spec(k) be the closed point of S = Spec(R). Then we define

Sws(M) =

∞

X

i=1

1

[G₀ : G_i]^{. dim}^F^`(M/M^Gⁱ). (4.2) Notice that this sum starts ati=1 and not ati =0 and this sum is finite by Lemma4.2.1.

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Lemma 4.2.2. The definition (4.2) does not depend on the choice ofΓ.

Proof. This result depends on a compatibility between the lower and upper higher ramification groups ofG0, which itself depends on our assumption that k⁰/kis separable; see [Ser 68, Ch IV,§3].

4.3 The case of a curve

Letkbe a perfect field andX/ka smooth curve. Fix a codimension 1 pointx ∈ X⁽¹⁾ and fix a geometric point ¯xabovex. Now consider the strict henselization ofXatx,

R =O^sh_{X, ¯}_x, and let

S=Spec(R).

Having fixed a geometric point ¯x, a choice of a separable closure ks of the residue field k(x) = O_X,x/mx is fixed. The scheme S is a strictly henselian local scheme; lets=Spec(ks)be its closed point andη =Spec(K)the generic point ofS. For a fixed separable closure K^sep ofK, let η be the corresponding geometric point ofS. LetF be a constructibleF`-sheaf of finite rank onX. The action of Gal(K^sep/K)on F_η_¯ factors through a finite quotientG =Gal(K⁰/K) of Gal(K^sep/K). Since X is of finite-type over kand x ∈ |X| (as dim(X) = 1), thenk(x)is a finite extension ofk. Thereforek(x)is perfect and soks =k(x)^sep is algebraically closed. So for R = O_{X, ¯}^sh_x and R⁰ the normalization of R inK⁰, we have that the extension of residue fields fromR⁰/Ris trivial and therefore the finite extensionR⁰/Rsatisfies (4.1). We then define

Swx(F) =Sws(F_η_¯). (4.3) We include the following Example4.3.4below, which is [Lau 81, Exemple 1.1.7], since it will fit in nicely with subsequent examples. Before doing so, we prove the following well-known results about Artin-Schreier extensions, which will be used in the sequel.

Lemma 4.3.1. Let Kbe a field of characteristic p > 0. Let a ∈ K and consider the polynomial

f(t) =t^p−t−a∈ K[t]. Then the finite extension ofK

L =K[t]/(f(t)),

is Galois with Gal(L/K)either trivial or isomorphicZ/p.

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Proof. First observe that the formal derivative of f(t) is equal to −1 which is coprime to f(t)and thereforeL/Kis separable. Next, since

t^p−_t=

p−1

Y

i=0

(_t−_i)∈ _K[_t]_,

then ifα ∈ K^sep is a root of f(t), so is α+ifor i = 0, . . .p−1. Therefore, Lis the splitting field of f(t)_{and since} L/K is separable, thenL/Kis normal. We conclude that if f(t)admits no roots inK, the extensionL/Kis non-trivial and Galois with Gal(L/K) = _Z/p.

Lemma 4.3.2. Let K be a discrete valuation field of characteristic p > 0 with uniformizer π and normalized valuation v : K^× → Z. For a fixed a ∈ K, consider the polynomial

f(t) =t^p−t−a∈ K[t].

Then ifL=K[t]/(f(t))is a non-trivial extension ofK, we have (i) L/Kis unramified if∃y ∈ Ksuch that v(a−(y^p−y)) ≥_0.

(ii) L/Kis totally ramified if∃y ∈ Kandm > 0 such thatv(a−(y^p−y)) =

−m<0, and(p,m) =1.

Proof. Let π⁰ be a uniformizer ofL. Let A, resp. B, denote the ring of integers ofK, resp. L. Recall the differentD_B/Afrom [Ser 68, Ch III]. Letα ∈ K^sepbe a root of f(t)so thatL =K[α].

(i). Suppose y ∈ _K is such that v(_a−(_y^p−_y)) ≥ 0. Then for w = _α−_y and b = a−(y^p−y) we have L = K[w] and the minimal polynomial of w is g(t) = t^p−t−b. Note that w is integral over A since v(b) ≥ 0. Now by [Ser 68, Ch. III, §6, Coro 2], D_B/A divides the principal ideal (_g⁰(_w))_{. Since} g⁰(t) = −1, then g⁰(w) = −1 and so (π⁰) does not divide D_B/_A and hence B/Ais unramified by [Ser 68, Ch. III,§5, Thm. 1].

(ii). Letwandbbe as in (i), so

w^p−w=b.

Let e = e(B/A) be the ramification index of B/A and v⁰ the normalized discrete valuation onL. We have

v⁰(b) =−m.e, and

v⁰(b) = v⁰(w^p−w) = p.v⁰(w). Sincee≤[L : K] = p, thene = pandv⁰(w) =−m.

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Remark4.3.3. In fact, in both (i) and (ii) of Lemma4.3.2we also have the con- verse statements ifKis perfect. See [Sti 09, Prop. 3.7.8]; our proof above is only a slight refinement of the proof in loc. cit.

We now turn to [Lau 81, Example 1.1.7]. For a general discussion of Artin- Schreier-Witt extensions, see Section4.4below. We point out that this example, while elementary, is slightly long. We will see a significant reduction in its required computations in Example4.4.11below.

Example 4.3.4. (cf. [Lau 81, Exemple 1.1.7]) Let k be an algebraically closed field of characteristic p > 0, and let X = _A¹_k = Spec(k[x]). Let U be the complement of the closed subvarietyV(x), that is

U =_Spec(k[x, 1/x])_.

Fix an integerm≥1 and letU⁰ →Uby the Artin-Schreier cover ofUgiven by t^p−t=x⁻^m,

i.e.

U⁰ =Spec(k[x, 1/x][t]/(t^p−t−x⁻^m)).

To calculate ramification induced byU⁰ →Ualong the boundaryV(x)_{, we} define the following (henselian) discrete valuation rings Rand R⁰ as follows.

LetRbe the henselization ofk[x]at the the prime ideal(x)and letK =Frac(R). Set

L =K[t]/(t^p−t−x⁻^m), and defineR⁰ as the integral closure ofRinL. ¹

By Lemma4.3.2(ii), the prime ideal(x)_ofRis totally ramified inR⁰/Rand by Lemma4.3.1Gal(U⁰/U) =_Z/p=_F_p. An elementσ ∈ Gal(U⁰/U)acts on tasσ(t) =t+σ. A fixed non-trivial homomorphism

ρ: Fp →_F^×_` ,

and the morphismU⁰ →Uyields a smoothF`-sheafF onU. Forj: U →Xthe open immersion ofUinto Xand 0∈ Xthe closed point ofXthat corresponds to the prime ideal(x) ≤k[x], we show that the Swan conductor

Sw0(j!F) = m⁰,

where m = m⁰.p^e with (m⁰,p) = 1 and e ≥ 0. There are two methods to see that we can reduce to the case thatU⁰ →U is given by solving the equation

T^p−T =x⁻^m⁰. (4.4)

1Recall that the integral closure of a henselian dvr in a finite extension of its fraction field is again a henselian dvr.

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The first is by the Artin-Schreier-Witt isomorphism, provided in Section4.4.1 below, which in this case reads

W₁(R)_/(F−₁)W₁(R) ∼=H¹(_Spec(R)_,_Z/p)_,

where F : R → R is the Frobenius morphisma 7→ a^pand (F−1)R ≤ W1(R) denotes the subgroup generated by{(F−₁)(b) = b^p−b; b ∈ R}. According to this isomorphism,R⁰ is given by solving

T^p−T =x⁻^m⁰^.^p^e,

wherex⁻^m⁰^.^pê is the image ofx⁻^m⁰^.pê in the quotientW1(R)/(F−1). Since (F−₁)ê(x⁻^m⁰) = x⁻^m⁰^.pê −x⁻^m⁰,

we have

x⁻^m⁰^.p^e ≡x⁻^m⁰ mod (F−1)R.

Therefore, the extension ofR⁰/R obtained from t^p−t = x⁻^m is equivalent to the extension obtained fromT^p−T =x⁻^m⁰. The second method of seeing this isomorphism is to consider the polynomial f(t) = t^p−t−x⁻^m ∈ R[t]and set

u =x⁻^m⁰^.p^e−1 +x⁻^m⁰^.p^e−2 +· · ·+x⁻^m⁰. Then

f(t+u) =t^p−t+u^p−u−x⁻^m =t^p−t−x^d, and sinceu∈ R, we have an isomorphism ofR-algebras

R[t]/(f(t)) ∼=R[t]/(f(t+u)), which gives the desired reduction.

In order to compute Sw0(j!F)by (4.2), we compute the ramification groups G_i of G = _Gal(U⁰/U). Since the order of G is prime, then either G_i = G or G_i =h0i. Therefore, we just have to compute the number of subgroupsG_ithat are non-trivial. To do so, observe that on denoting the discrete valuation ofR⁰ of by v⁰ (so v⁰(x) = p) and π⁰ a uniformizer of R⁰, we have G_i = {_σ ∈ G : v⁰(σ(π⁰)−π⁰) ≥i+1}. We show that the valuev⁰(σ(π⁰)−π⁰) is constant for σ∈ G\{0}; namely, we claim that

v⁰(σ(π⁰)−π⁰) =m⁰+1, ∀σ 6=0.

It will then follow that the number of non-trivialGi is equal tom⁰+1 and so G_j is non-trivial for m⁰ many indices j ≥ 1. Letπ⁰ = x^a.t^b where a,b ≥ 1 are such thatap−bm⁰ =1. Then,

v⁰(t) = −m⁰ <0.

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Observe that

σ(π⁰) = σ(x^a.t^b)

= x^a(σ+t)^b

= x^a(t(1+t⁻¹.σ))^b

= _π⁰(₁+t⁻¹.σ)^b

Thus,

v⁰(σ(π⁰)−π⁰) = v⁰(π⁰[(1+t⁻¹.σ)^b−1])

= v⁰(σ.π⁰[b.t⁻¹+σ.b(b−₁)

2 .t⁻²+· · ·+σ^b⁻¹.t⁻^b])

= v⁰(π⁰) +min{m⁰, 2m⁰, . . . ,bm⁰}

= v⁰(_π⁰) +m⁰

= ₁+m⁰.

Therefore, G_i = G for 0 ≤ i ≤ m⁰ and G_j = h0i for j ≥ m⁰+1 and since the moduleMisM =_F_`in this case we have

Sw0(M) =

∞

X

i=1

1

[G₀: G_i]^{. dim}^F^`(M/M^Gⁱ) =

m⁰

X

i=1

1=m⁰,

as desired.

Luckily, the computation of such a Swan conductor can be significantly simplified, as we will see in the next section. See in particular Example (4.4.11) where the above computation is essentially reduced to one line.

4.3.1 Remarks on separable residue field extensions

4.4 Conductor for general discrete valuation fields

We mostly follow the notation of [Abb-Sai 09,§9]. LetRbe a discrete valuation ring of characteristicp >0.

Let K be the fraction field of R, fix a separable closure K^sep of K. First we recall the Brylinski-Kato filtration on truncated Witt vectors over K. Let v denote the normalized discrete valuation on K. Let n ≥ 0 and denote by Wn+1(K)the ring of Witt vectors of lengthn+1 overK.

We define an increasing filtration on the additive group ofW_n+1(K)_{, which} was studied by Brylinski [Bry 83] and Kato [Kat 89].

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Definition 4.4.1. Form≥0 define the subgroup filmW_n+1(K)as filmW_n+1(K) =ⁿ(x₀, . . . ,xn) _: pⁿ⁻ⁱ.v(x_i) ≥ −m(₀≤i≤n)^o_.

We have that forx = (x₀, . . . ,xn) ∈ W_n+1(K)_, x ∈ _fil_mW_n+1(K) if and only if

m ≥sup{−pⁿ.v(x0), . . . ,−p.v(xn−1),−v(xn), 0}. (4.5) This filtration yields a conductor. Being in rank 1, we choose to define conductors at the level of characters. See Appendix9.2for the precise correspondence between characters, extensions ofK, and sheaves forZ/pⁿ-extensions. Set

H¹(K) :=lim−→

r≥1

H¹(K,Z/rZ) = H¹(K,Q/Z).

Remark4.4.2. Note that since Gal(K^sep/K)is a compact topological group (under the usual Krull topology) andQ/Zis a discrete topological group, then an element

χ∈ H¹(K) =Homcont(Gal(K^ab/K),Q/Z),

must have finite image inQ/Zand this image is therefore a finite cyclic group.

We continue to assume thatKis the fraction field of a dvrRof characteristic p > 0. The relationship between truncated Witt vectors overK and cyclicpⁿ- extensions ofKis provided by the Artin-Schreier-Witt isomorphism which we now recall. Put η = _Spec(_K) and recall that for n ≥ 1 there is a short exact sequence of sheaves overη_´et:

0−→_Z/_pⁿ −→_W_n −→^F⁻¹ _W_n −→ _0. _(4.6) Let us show thatH¹(K,Wn)vanishes.

We first briefly recall the argument establishing the additive version of Hilbert ’90 [Ser 68, Ch. X, Prop. 1], which says that H¹(G_K,K^sep) = 0. Let L/K be a Galois extension with Galois group G and write G = {σ1, . . . ,σr}. The normal basis theorem yields an elementx ∈ Lsuch that{σi(x)}₁≤i≤r is a basis forL/K. Then we have an isomorphism ofG-modules: K ∼=K⊗_Z_Z[G], implying thatKis an induced G-module (in the sense of [Ser 68, Ch. VII,§2]) and thus Hⁱ(GK,K) = 0 for each i > 0. Note that since G is finite, then in factZ[G]⊗_ZK = Hom_Z(_Z[G],K) and so K is also a co-induced G-module.

Taking colimits one immediately obtainsH¹(GK,K^sep) =0, or in our notation:

H¹(K,W1) = 0.

Next, we show that for n > 1,H¹(K,Wn) = 0. Consider the short exact sequence

0−→ Wn V

−→W_n+1 −→W_n+1/V(Wn) −→0,

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where V : Wn(K) → W_n+1(K) is the Verschiebung, i.e. it is the additive map that sends (x0, . . . ,xn−1) to (0,x0, . . . ,xn−1). This gives a filtration on

H¹(K,Wn+1)whose quotients are each trivial by the casen =1, and soH¹(K,Wn+1) = 0 as well.

Therefore, the beginning of the long exact sequence obtained from the short exact sequence (4.6) is

0−→ _Z/pⁿ −→Wn(K)−→^F⁻¹ Wn(K) −→^δ H¹(K,Z/pⁿ)→ H¹(K,Wn) =_0.

(4.7) Hence, we have a canonical isomorphism of groups

Wn(K)/(F−1) ∼=H¹(K,Z/pⁿ), (4.8) and this isomorphism classifies finite cyclic extensions ofKof orderpⁿ⁰, for n⁰ ≤n. Moreover, there is a commutative diagram

Wn(K)

V

δn //H¹(K,Z/pⁿ)

·p

Wn+1(K) ^δⁿ⁺¹ ^//H¹(K,Z/pⁿ⁺¹)

. (4.9)

Now write

filmH¹(K,Z/pⁿ) :=_δ_n(filmWn(K)). (4.10) The above filtration induces a filtration onH¹(K)via

filmH¹(K):= H¹(K)[p⁰] +filmH¹(K)[p^∞], (4.11) where

H¹(K)[p⁰] = lim−→

l≥1,(l,p)=1

H¹(K,Z/l), and

H¹(K)[p^∞] =lim−→

r≥1

H¹(K,Z/p^r).

Definition 4.4.3. Letχ∈ H¹(K). The Swan conductor ofχ, denoted by Sw(χ), is the minimal integerm≥0 for whichχ∈filmH¹(K).

Remark4.4.4. That this definition agrees with the classical definition appearing in (4.3) whenKis complete and the residue field ofKis finite²is a theorem of Brylinski [Bry 83, Coro. to Thm. 1] and the case whereKis complete and has

2such fields are traditionally calledlocal fields, [Neu 99, Ch. II,§2]

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perfect residue field is due to Kato, [Kat 89]. Note that in [Bry 83], the proof is for the Artin conductor, which differs from the Swan conductor by a potential difference of +1, cf. [Bor 02,§2.8].

We will use the following results in the sequel to justify explicit calculations of conductors using Witt vectors. Recall thatvdenotes the normalized discrete valuation onK. To begin, the Brylinski filtration inspires the following map:

Γ :Wn+1(K)→_Z,

defined by(x₀, . . . ,xn) 7→sup{−pⁿv(x₀),−pⁿ⁻¹v(x₁), . . . ,−v(xn), 0}.

Proposition 4.4.5. Let χ ∈ H¹(K,Z/pⁿ⁺¹) with Sw(χ) = N ≥ 0. Let ¯x ∈ Wn+1(K)/(F−1)correspond toχ. Supposex= (x0, . . . ,xn) ∈ Wn+1(K)satis- fiesx≡ x¯ mod (F−1) andΓ(x) = N. Furthermore, supposeN =−v(xn) >

−pⁿ⁻^m.v(xm)form = 0, . . . ,n−1. Then for eachy = (y0, . . . ,yn) ∈ Wn+1(K) we have the inequality

Γ(x+ (F−1)y) ≥_Γ(x). (4.12) We begin with a basic lemma used to prove Prop. 4.4.5.

Lemma 4.4.6. Supposexand yare elements of a discrete valuation field with uniformizerπ and normalized discrete valuationv. Then

(i) v(x) +v(y) 6=₀⇒v(x+y) = _min{v(x)_,v(y)}

(ii) x+y=0⇒v(x+y) = +_∞.

Proof. It suffices to prove (i). We may assume v(x) < v(y). Set m = v(x). There is ak∈ _Nsuch thatv(y) =m+k. We havev(x+y) = v(_π^m(1+_π^k)) = mv(π) +v(1+π^k) = m=v(x), as desired.

Next, observe that fory= (y0, . . . ,yn) ∈ W_n+1(K)we havey= (y0, . . . ,y_n−1, 0) + (0, . . . , 0,yn)and therefore

(F−₁)y= (F−₁)(y₀, . . . ,y_n−1, 0) + (F−₁)(0, . . . , 0,yn)_.

Hence, it suffices to prove Prop.4.4.5fory = (y0, . . . ,yn−1, 0)andy = (0, . . . , 0,yn). We begin with the first case.

Lemma 4.4.7. Prop.4.4.5holds fory = (y₀, . . . ,y_n−1, 0).

Proof. The proof is by induction on n ≥ 1. The base case is n = 1 which is verified as follows. Settingz0 =y0−_y₀^p_(recall_y₁=0 here) we have

~w= (x0+z0, x1+p⁻¹(x₀^p+z₀^p−(x0+z0)^p)).

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We divide the argument into two cases: case i)−p.v(z₀)≤ −v(x₁)and case ii)

−p.v(z0)>−v(x1).

For case (i),we show thatv(w1) =v(x1). Observe that v(w₁−x₁) =v





p−1

X

k=1

x₀^p⁻^kz^k₀



,

and this sum consists of p−1 monomials of degree p. Hence, the conditions

−p.v(x₀) <−v(x₁)_and−p.v(z₀)≤ −v(x₁)_{yield that}

−v(w₁−x₁)<−v(x₁), i.e.v(x1) <v(w1−x1)and sov(w1) = v(x1).

Next, for case (ii) we have

−p.v(x0) <−v(x1)<−p.v(z0),

hence v(_w₀) = _v(_z₀), which givesΓ(~_w) ≥ −_p.v(_z₀) > −_v(_x₁) and the case n=1 is complete.³

Now assume the lemma is true for Wn(K) for a fixed n > 1. Given~y ∈ Wn+1(K), letzi = yi−_y_i^p _{for 1} ≤ _i ≤ _n_(recall_y_n = 0 by hypothesis). Again, the argument is divided into two cases: Case i)−p.v(z_i−1) ≤ −v(x_i),∀i and Case ii)−p.v(zi−1) >−v(xi),∃i, where 1≤i ≤n.

For Case (i) we show that v(wn) = v(xn). To compare v(wn −_x_n) and v(xn)we make use of the following observation. Use the polynomialsS_jas in [Ser 68, II.6] to express the addition law inWn+1(K)as

~a+~_b = (S0(a0,b0),S₁(a0,a₁,b0,b₁), . . . ,Sn(~a,~_b)).

Providing the ring Z[a0, . . . ,an,b0, . . . ,bn] with the grading where ai and bi

have weight pⁱ, then S_j(a₀, . . . ,a_j,b₀, . . . ,b_j) is a homogeneous polynomial of degree p^j, forj =0, . . . ,n.

Thus, wn−xn is comprised of the following sums: for each m ∈ [0,n− 1], there are pⁿ⁻^m−1 monomials in xm,zm of polynomial degree pⁿ⁻^m. Our assumptions in this case yield that v(xn) is strictly less than each valuation v(^P_k^p₌^n−m₁ ⁻¹xm^p^n−m⁻^kz^k_m)form ∈[0,n−1], from which we conclude thatv(xn)<

v(xn−wn)_{. Hence,}v(wn) = v(xn)_.

For Case (ii), we divide the argument into two subcases. The first is if the inequality−p.v(zn−1)>−v(xn)holds. Then, we have

−p.(z_n−1) >−v(xn) >−p.v(x_n−1),

3NB: we only needed−p.v(z0)≥ −v(x1)for case (ii).

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hence,v(z_n−1) <v(x_n−1). Induction applied to(w₀, . . . ,w_n−1)givesv(w_n−1) = v(zn−1)and therefore

Γ(~w = (w0, . . . ,wn))≥ −p.v(zn−1) >−v(xn),

as desired. On the other hand, suppose−p.v(zn−1) ≤ −v(xn). Here, we need to compare the remaining z_k with xn. If for some k⁰ with 0 ≤ k⁰ ≤ n−2 the inequality−pⁿ⁻^k⁰.v(z_k⁰) >−v(xn)holds, then induction givesv(w_k⁰) = v(z_k⁰) and then

Γ(~w = (w0, . . . ,wn))≥ −pⁿ⁻^k⁰.v(z_k⁰).

Finally, if−pⁿ⁻^k⁰.v(z_k⁰) <−v(xn)for all suchk⁰, then applying a similar argument as in Case (i) above givesv(wn) = v(xn).

The following lemma completes the proof of Prop. 4.4.5. Namely, replacing xbyx+ (F−₁)(y₀, . . . ,y_n−1, 0) we show thatx+ (F−₁)(0, . . . , 0,yn) _{has the} desired property.

Lemma 4.4.8.Supposex∈W_n+1(K)satisfiesΓ(x) ≥Nand−v(xn) ≥ −pⁿ⁻^mv(xm) form =0, . . . ,n−1. Then

Γ(x+ (F−1)(0, . . . , 0,yn)) ≥ −v(xn). Proof. We have

(x₀, . . . ,xn) + (F−1)(0, . . . , 0,yn) = (x₀, . . . ,xn+y_n^p−yn).

By Lemma4.4.6,v(xn+y_n^p−yn) =min{v(xn),v(y_n^p−yn)}, ifv(xn) 6=v(y_n^p− yn). Finally, observe thatv(xn)≥min{v(xn),v(y^pn−yn)}, i.e.

−min{v(xn),v(yn^p−yn)} ≥ −v(xn), which completes the proof.

We make some additional remarks regarding ramification.

Remark4.4.9. Ifχ⁰ ∈ H¹(K)[p⁰]_{, then Sw}(_χ⁰) =0, by (4.11) above. Therefore, in the rank 1 case, our study of wild ramification is reduced to that of characters inH¹(K)[p^∞].

4.4.1 Witt vectors and extensions; computation of conductors

Lemma 4.4.10. Letχ∈ H¹(K,Z/pⁿ)and supposen⁰ ≤n. Thenχcorresponds to aZ/pⁿ⁰ extension of Kif and only if its corresponding Witt vector is in the image of the iterated Verschiebung modulo(F−1),

Vⁿ⁻ⁿ⁰ :W_n⁰(K)/(F−1) →Wn(K)/(F−1).

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Proof. This follows by the the isomorphism (4.8) and the commutative diagram (4.9).

Using the isomorphism (4.8), we can explicate p-cyclic extensions ofK as follows. IfL/Kis Galois and satisfies Gal(L/K) =_Z/pⁿ⁺¹, then there are Witt vectors

x= (x₀, . . . ,xn) ∈W_n+1(K)_, (_α₀_{, . . . ,}_α_n) ∈W_n+1(K^sep) such that

L=K[α₀, . . . ,αn] (4.13) where(_α₀, . . . ,αn) ∈W_n+1(K^sep)satisfies

(F−1)(α0, . . . ,αn) = (x0, . . . ,xn). The Swan conductor ofL/Kcan be computed as follows.

By (4.8) there is an attached character χx ∈ H¹(K). Then Sw(χx) is the minimal integerm ≥0 satisfying

x∈ (filmWn+1(K))/(F−1).

Moreover, we may also illustrate the calculation of this conductor by choos- ing a lift of the image ofxinWn+1(K)/(F−1) as follows. Choose an element y= (y₀, . . . ,yn) ∈ W_n+1(K)_{such that}

(i) y≡x mod (F−1)Wn+1(K).

(ii) There is ak ∈ [0,n]such that−pⁿ⁻^kv(y_k) = Sw(χx).

Clearly such a y exists simply because Wn+1(K) → Wn+1(K)/(F−1) is surjective.

Then with such aysatisfying (i) and (ii) above we have by (4.5) that

Sw(χx) =sup{−pⁿ.v(y0), . . . ,−p.v(yn−1),−v(yn), 0} (4.14) Note that we consider the integer 0 in the right-hand side of (4.14) since it is possible that eachy_ihas v(y_i) >0; in this case there is no wild ramification and the conductor is zero.

The isomorphism (4.8) forn+1=1 gives us a relatively explicit method of exhibiting examples of Artin-Schreier extensions of any field of characteristic p. Namely, ifKis such a field, then for a fixeda ∈K, the extension

K[T]/(T^p−T−a),

is a Z/p-extension of K if and only if a 6= (F−1)(y) = y^p−y,∀y ∈ K. We now provide examples of Artin-Schreier-Witt extensions and the computation of their conductors.

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Example 4.4.11. Let Kbe a discrete valuation field of characteristic p >0. Let m≥1 and writem= p^r.m⁰wherem⁰ ≥1 is prime topandr ≥0. Denote byπ a uniformizer ofK. Consider the polynomial

f(T) = T^p−T− ¹

π^m ∈K[T].

By Lemma4.3.1, the splitting field of this polynomial is aZ/p-extension ofK.

We now compute the Swan conductor of this extension using (4.14) inW1(K); letχ∈ _H¹(K,Z/p)be the character attached to this extension.

We have 1/π^m ≡ _1/π^m⁰ _mod (F−₁)_{. Now,} y = _1/π^m⁰ ∈ W₁(K) _{is a}

‘minimal lift’ of the image of 1/π^minW₁(K)/(F−1)in the sense of conditions (i) and (ii) above (4.14).

Therefore,

Sw(χ) =sup{−(−m⁰), 0} =m⁰.

Cf. the computations in Example4.3.4above.

Example 4.4.12. LetKbe as in Example (4.4.11) and let

¯

x = (x0,x1) = 1

π, 0

∈ W2(K). Since ¯xis not in the image of the Verschiebung

V :W1(_K)→_W₂(_K)_,

then by Lemma4.4.10x¯ defines aZ/p²-extension ofK. Letχ ∈ H¹(K,Z/p²) be the corresponding character obtained from the Artin-Schreier-Witt isomorphism (4.8) above. The conductor is computed via (4.14) and we have

Sw(_χ) = sup(−p.v(x₀),−v(x₁), 0) = sup(−p.(−1), −_{∞, 0}) = p.

We can further describe this extension as follows. Let the extension beL/K;

thenLis expressed as

L=K[_α₀_,_α₁]_, where(_α₀_,_α₁)∈ W₂(K^sep)_satisfy

(F−1)(_α₀,α1) = (x₀,x₁).

To go further, we write down the coordinates of (F−1)(α₀,α₁). In order to keep the calculations brief in this particular example, we now assume that p=2. Then the additive inverse of an element(_a₀_,_b₀) ∈_W₂(_K)is given as

−(a₀,a₁) = (−a₀,−a²₀−a₁) = (a₀,a²₀+a₁),

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where the last equality is valid since char(K) =2. We have then (F−1)(α0,α1) = (α^p₀,α₁^p)−(α0,α1)

= (α²₀,α²₁) + (α0, α²₀+α1)

= (_α²₀+_α₀, α²₀+_α²₁+_α₁+¹

2(_α⁴₀+_α²₀−(_α²₀+_α₀)²))

= (_α²₀+_α₀, α²₀+_α²₁+_α₁+_α³₀).

Therefore, the ordered pair (α0,α1) ∈ K^sep×K^sepgives a solution to the system of equations inK[_T₀_,_T₁] _:

T₀²+T0= ¹

π , T₀³+T₀²+T₁²+T1 =0.

Combining Remark 4.4.2 and Remark 4.4.9 narrows our focus to p-cyclic extensions ofK, but for further applications, such as higher class field theory, we define a conductor for any finite abelian extension ofKas follows. Given a homomorphism

χ: Gal(K^sep/K)→^M

j

Z/p^e^j,

with finite image,χthen corresponds to a finite extensionL/Kwith Gal(L/K)≤

⊕_j_Z/p^e^j. There are natural projections p_j : M

j

Z/p^e^j →_Z/p^e^j_.

Put

χ_j = p_j◦χ ∈ H¹(K,Z/p^e^j) ⊂ H¹(K,Q/Z). We have then

Definition 4.4.13. The conductor ofχ=⊕_jχ_jis defined as Sw(_χ) =_max

j {_Sw(_χ_j)}_, where the Sw(_χ_j)are defined in Definition4.4.3.

We can check that Def4.4.13is well-defined by showing that it is invariant under the automorphisms of a given Galois extension ofK. Namely, suppose thatL/Kis Galois with degree[L : K] = m>0.

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Lemma 4.4.14. Let (A,v) be a valuation ring (not necessarily discrete) with fraction fieldKand value groupΓ. Define a function

v: K^m →_Γ by

v(b1, . . . ,bm) =min

i (v(bi)). Thenvis invariant under the action of GLm(A).

Proof. Givenb = (b₁, . . . ,bm) ∈ K^m, suppose the index j₀is such thatv(b_j₀) = v(b). Then eachbj₀.biis in the ringAand by the choice ofj0, we havev(b⁻_j ¹

0 .b) = 0. Now for anyM∈ GLn(A)we have

v(Ab) = v(b⁻_j ¹

0 .Ab) +v(b_j₀) = v(Ab⁻_j ¹

0 .b) +v(b_j₀).

Therefore, it suffices to show shat ifbsatisfies v(b) = 0, thenv(Mb) = 0, for arbitrary M ∈ GLm(A). In fact, this assertion is equivalent to: ifbhas a non- zero coordinate modulom, the maximal ideal ofA, thenAbalso has a non-zero coordinate modulom. SinceM ∈ GLm(A), we have det(M) ∈ R\mand hence det(M)is non-zero modulom. The assertion is then reduced to: fork = A/m, if ¯b ∈ k^m is non-zero, then for any ¯M ∈ GLm(k), we have ¯Mb¯ 6=0∈ k^m, which is clear.

4.5 Conductor of a variety

LetX/kbe a normal variety andU ⊂Xak-smooth open subscheme such that the closed complementX\U = Eis the support of an effective Cartier divisor and letD ∈_Div⁺(X)be an effective Cartier divisor with Supp(D) ⊂E.

Denote by I the set of generic points of E; then I ⊂ X⁽¹⁾. Forλ ∈ I, let K_λ be the fraction field of the henselian discrete valuation ring O^h_X,λ. Note that the residue field ofK_λis the function field of the divisor{λ} onX.

Recall that in Def.4.4.3we defined the Swan conductor for characters over discrete valuation fields.

Letχ∈ H¹(U,Q/Z). Definition 4.5.1. We define

χ|_λ ∈ H¹(K_λ,Q/Z),

to be the pullback ofχto Spec(K_λ)induced by the canonical composition Spec(Kλ) →_Spec(O^h_X,λ)→ _X.

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The log Swan conductor ofχ|_λ, as defined by Def. 4.4.3, is denoted by Swλ(χ)_log.

Remark4.5.2. Observe that sincekis perfect andX/kis assumed to be of finite type, the residue field ofK_λhas transcendence degree equal to dim(X)−_{1 and} therefore this residue field is perfect if and only if dim(X) =1.

Definition 4.5.3. Acurve Cis a one-dimensional integral scheme and acurve C ona scheme is a closed subscheme that is a curve.

Definition 4.5.4. We letZ1(X,E)denote the set of curves ¯Con X such that ¯C is not contained inE, i.e. such that ¯C 6⊂Supp(E).

Definition 4.5.5. For ¯C ∈ Z₁(X,E)let

ΨC¯ : ¯C^N →C,^¯ denote the normalization of ¯C. We put

C¯_∞ ={z∈ |C^¯^N|: ΨC¯(z) ∈ E}. (4.15) When we want to specify a divisor D⁰ ⊂ Supp(E), we write Z1(X,D⁰), resp. ¯C∞(D⁰).

We can think of ¯C_∞ as the set of places of the global field K(C^¯) = K(C^¯^N)

“on the boundary”E. Given ¯C∈ Z1(X,E)andz∈ C^¯_∞, we writeK(C^¯)_zfor the henselization ofK(C^¯)at the place corresponding toz.

Definition 4.5.6. Given ¯C ∈ Z₁(X,E)_andz ∈ C^¯_∞, we define χ|C,z¯ ∈ H¹(K(C^¯)_z,Q/Z),

to be the restriction ofχvia

Spec(K(C^¯)_z) →C^¯ ,→X.

Since dim ¯C = 1, we do not need to emphasize a ‘log’ conductor and we un- ambiguously write

Swz(χ|C¯),

for the conductor ofχ|_C,z_¯ , which coincides with the classical Swan conductor (see Remark 4.4.4). More generally, suppose ˜C is a curve with a given finite morphism

φ: ˜C →X,

and that ˜z∈ |C^˜|is a closed point satisfyingφ(z˜) ∈ E. We will often emphasize the morphismφby writing

Swz(φ^∗χ), for the conductor ofχrestricted to ˜C.

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4.5.1 Example of computing by tangent curves

After giving examples of computing conductors by restriction to curves, we conjecture a general formula of this phenomena in5.1 and prove this conjecture in rank 1 for an sncd in Sec.6.1.

Example 4.5.7. (Artin-Schreier for a surface.) Letk=_F_pand U =Spec(k[x,y][1/y]).

The surfaceUis an open subscheme of the affine planeX =_Spec(k[x,y])_and X\U = V(y). Take D =V(y); thenD is indeed an sncd (D ∼= A¹_k) and Dhas unique generic point(y). We will consider both “fierce” and “non-fierce” wild ramification (cf. [Lau 82, §2]) to give evidence that conductors obtained from curves (locally) tangent to(y)handle both phenomena.

Let Abe the henselization of k[x,y]at(y)and K = Frac(A). We will consider Galois covers ofU with groupFp. Given a non-trivial morphism

ρ :Fp →GL₁(_F_`) =_F^×_` consider an Artin-Schreier equation overA:

t^p−t=x^a/y^b,

witha ≥0,b ≥ 1. We can further assume thatx^a/y^b is not a p^th-root inK, i.e.

thatpis prime toaorb. This equation and the characterρyield a rank 1 sheaf F _of_F_`-modules onU. Let

L =K[t]/(t^p−t−x^a/y^b),

and letB be the integral closure of Ain L; then B is also a discrete valuation ring. We will restrict F to Spec(K) via Spec(K) → X. Let k(A) = k(x) denote the residue field of Aand k(B) the residue field ofB. By definitionB/A is fierce if the extension k(B)/k(A) is inseparable, i.e. (in this case) if k(B) contains a p-th root of x. For both cases, consider curves of the form Ce ⊂ X defined by the equationy = x^e, fore > 0. EachCe is a 1-dimensional closed, irreducible, subscheme of XsinceCe ∼= A¹_k and, geometrically, as eincreases, theCe progressively become more tangent to the x-axis near the origin in the plane, i.e. to the divisor defined by D = V(y). Let j : U → X be the open immersion of U. Then j!F is a constructible F`-sheaf on X and its restric- tionj!F |_C_eis also constructible and is associated to the Artin-Schreier equation given by

t^p−t= x^a/x^be = ¹ x^be⁻^a.

Letφ: Ce ,→ Xdenote the closed immersion ofCeintoXand fix a closed point z∈ |C|such thatφ(z) ∈ D

Wild ramification of schemes via curves, the rank 1 case