p ᡶ Teichm¨uller ྸᛯƷኰʼ §1

V. p

^ᡶ

Teichm¨uller

^{ྸᛯƷኰʼ}

§1. Witt

^࿢Ʊ

Frobenius

k

^{Ǜ ೅ૠ}

p

^{Ʒ˳ ƱƢǔᲢ}

p

^{ƸእૠᲣŵƢǔƱŴ}

Φ

_k

: k x → x

^p

∈ k

ƸŴੑƚምƩƚưƳƘŴឱƠም ƱNjɲᇌƢ ǔŵƭLJǓŴ˳Ʒ แӷ׹ Ǜܭ፯ƠƯƍǔŵ ƜƷݧƷƜƱǛ

Frobenius

^{ݧ ƱԠƿŵ}

Φ

_k ^Ƹ

࣏ƣ ҥݧ ƴƳǔƕŴμݧ ƴƳǔƱƖŴ

k

^Ǜ ܦμ˳ ƱԠƿŵ

ƞƯ

k

ƕ ܦμ˳ ưƋǔƱˎܭƠǑƏŵ᩼᝟

ૢૠ

N

^{Ǜ ชƑ܌ ƱƠŴ}

k

^{Ǜ ঺Ў ƱƢǔș} ǯȈȫᲷž

Witt

^șǯȈȫſ

(λ

0

, λ

1

, λ

2

, . . . , λ

_n

, . . . )

ᲢƨƩƠ

λ

_n

∈ k

^{Უμ˳ƔǒƳǔᨼӳ}

W (k)

ƴƋǔ ᐯ໱Ƴ࿢ನᡯ ǛλǕǔƜƱƕӧᏡư Ƌǔŵ࿢ನᡯƷଢᅆႎƳཎܭƸᩊƠƍƨNJŴ ƜƜưƸႾဦƢǔŵƜƷǑƏƴƠƯࢽǒǕ ƨ࿢ƷƜƱǛ

Witt

^{࿢ ƱԠƿŵ}

2

Witt

^࿢

W (k)

^Ʒನ঺Ƹ

k

^{ƴ᧙ƠƯ ᧙৖ႎ} ưƋǔŵƭLJǓŴܦμ˳Ʒแӷ׹

k

₁

→ k

₂ ƴݣƠƯݣࣖƢǔ

Witt

^࿢Ʒแӷ׹

W (k

₁

) → W (k

₂

)

ƕࡽƖឪƜƞǕǔŵཎƴŴ

Φ

_k

: k →

^∼

k

^ǑǓ

Witt

^{࿢Ʒᐯࠁӷ׹}

Φ

_W(k)

: W (k) → W (k)

ƕᛔݰƞǕǔŵ

̊Ჴ

Witt

^{࿢ƷஇNjؕஜႎƳ̊Ƹ}

p

^ᡶૢૠ࿢

Z

_p ^def

= lim ←−

_n

Z/p

ⁿ

Z

ưƋǔŵƭLJǓŴ

Z

_p ^Ƹ

W (F

_p

)

^ƴᐯ໱ƴӷ

׹ưƋǔŵɥƷᡞಊᨂƴႇئƢǔ

Z/p

ⁿ

Z

^Ʒ ӲŷƷᢿЎՠƨƪ

p

^j

Z/p

^j+1

Z

^ƸŴž

p

^{j Ǜٳ} ƢſƜƱƴǑƬƯ

F

_p ^{ƱӷɟᙻưƖǔŵžƜ} ƷǑƏƳ

F

_p ^{ƨƪſƸദƴ}

Witt

^șǯȈȫƴ ЈƯƘǔ঺ЎƨƪƴݣࣖƠƯƍǔǘƚƩƕŴ

3

Z/p

ⁿ

Z

Ʒžฆ೅ૠႎƳನᡯſƔǒƲƷǑƏ ƴƠƯਁЈƞǕŴž

Witt

^{șǯȈȫƷ̾ŷƷ}

঺ЎſƱƍƏ ЎᘷƞǕƨᘙᅆ ƴᣐፗƞǕǔ ƔŴƱƍƏƜƱƴƸ ขƍᜋᲢNjƠƘƸȭȞ ȳᲹᲣƕƋǔŵ

Z

_p Ʒ̊ƔǒNjਖ਼ยƞǕǔǑƏƴŴ

k

^ƕദ೅

ૠƷ˳Ტ࿢ᲣưƋǔƴNj᧙ǘǒƣŴ

W (k )

^Ƹ

ૢ؏ ưƦƷՠ˳Ƹ ೅ૠ

0

^{Ʒ˳ᲢᲷ̊ƑƹŴ}

Z

_p ^ƷئӳŴ

p

^ᡶૠ˳

Q

_p^{ᲣƴƳǔŵƭLJǓŴ}

Witt

^࿢

W (k)

^{Ƹദ೅ૠƷ˳}

k

^ƴݣƢǔŴ ž೅ૠ

0

ǁƷ೅แႎƳਤƪɥƛſƱᙸǔƜƱ

ƕưƖǔŵ

ž೅แႎƳਤƪɥƛſƸŴ̾ŷƷΨƴݣƠƯ Nj܍נƢǔŵ

λ ∈ k

^ƴݣƠƯŴ

X

^p

= Φ

_W(k)

(X )

Ǜ฼ƨƠŴƔƭ

0

^ഏ঺Ўƕ

λ

^ƴƳǔǑƏƳ Ψ

∈ W (k)

ƸŴܱƸ ՗ɟƭ܍נ ƢǔŵƜƷ Ψ

[λ] ∈ W (k)

^Ƹ

λ

^Ʒ

Teichm¨uller

^ˊᘙΨ ƱԠƹǕǔŵ

4

ƜƷ

Teich

ˊᘙΨƴƸNjƏɟƭƷᙸ૾ƕƋ

ǔŵ

0

^ഏ঺Ўƕ

λ

^{ƴƳǔ ˓ॖƷ}

Λ ∈ W (k)

ƴݣƠƯ

Ψ : Λ → Φ

⁻¹_W(k)

(Λ

^p

)

ƱƍƏદ˺Ǜ଀ƢƱŴӷơƘ

0

^ഏ঺Ўƕ

λ

ƴƳǔǑƏƳ

W (k)

^{ƷΨƕưƖǔŵ࿢}

W (k)

ƴƸ

p

ᡶˮႻ ƕλƬƯƍǔƕŴƜƷદ˺Ǜ ӒࣄƠƯࢽǒǕǔΨƨƪƷಊᨂ

n→∞

lim Ψ

ⁿ

(Λ)

ƸŴ

Teich

^ˊᘙΨ

[λ]

^{ƴƳǔŵᲢᚰଢᲴųų}

Λ = [λ] + p

ⁿ

· W (k ) = ⇒

ųųųųųų

Λ

^p

= [λ

^p

] + p

ⁿ⁺¹

· W (k)

^ŵᲣ

׆Ljƴ

Teich

^ˊᘙΨƷž

Teichm¨uller

^ſƱŴ ȪȸȞȳ᩿Ʒ

Teich

^ྸᛯƷž

Teichm¨uller

^ſ ƸŴᲢᐻԛขƍƜƱƴᲛᲣӷɟʴཋ ưƋǔŵ

5

§2.

^{ยעዴƷ࠹˴Ʊ೅แႎ}

Frob

^ਤƪɥƛ ȪȸȞȳ᩿Ʒᛅƴ৏ǖƏŵɥҞ࠯᩿Ʒ࠹˴

ǛˊᘙƢǔྵᝋƱƠƯž

P SL

₂

(R)

^Ʒɟࢲૠ ᢿЎ፭ư්ƢſƱƍƏNjƷƕƋǔŵƜƷǑ Əƴž්ƠƨſƱƖƷ៽ᢊƸŴžยעዴſƴ Ƴǔŵ

ɟ૾ŴȪȸȞȳ᩿ƷദЩನᡯNjžɟࢲૠଈ Ʒɶư්ƢſƱƍƏྵᝋǛᙸƨŵƜǕƸȪȸ Ȟȳ᩿ƷžȢǸȥȩǤᆰ᧓ſᲷž

Teich

^ᆰ᧓ſ ϋƷžȕȭȸſᲷžยעዴƷ࠹˴ ƱᙸǔƜ ƱƕưƖǔŵ

6

ƜƷǑƏƳžยעዴƷ࠹˴Ʒ

p

^{ᡶ༿ſƸŴദ} ƴ

§1

^Ʒ

Ψ

^Ŵұƪž

p

ʈϙ΂ƷǑƏƳϙ΂ſư ƋǔŵƜƷǑƏƳϙ΂ƷƜƱǛ

Frobenius

^ਤƪɥƛ

ƱԠƿŵɟᑍƷദ೅ૠٶಮ˳ƩƱ

Witt

^࿢Ʒ ǑƏƳ೅แႎƳਤƪɥƛNj ƳƚǕƹŴ

Ψ

^Ʒ ǑƏƳ೅แႎ

Frobenius

^{ਤƪɥƛNj Ƴƍŵ} ɟ૾Ŵӑ୺ႎȪȸȞȳ᩿ƸŴ̊Ƒƹ ݧࢨᆰ᧓ Ʒɶƴ؈NJᡂlj ƜƱƴǑƬƯˊૠႎƳ૾ᆉ

ࡸᲷ ٶ᪮ࡸưܭ፯ƞǕǔ ƜƱƕЎƔǔŵƜ ƷǑƏƴӑ୺ႎȪȸȞȳ᩿ƴݣࣖƢǔǑƏ Ƴˊૠٶಮ˳ᲢᲷٶ᪮ࡸưܭ፯ƞǕƨ࠹˴

ႎݣᝋᲣǛ

ӑ୺ႎˊૠ୺ዴ

ƱԠƿŵӑ୺ႎˊૠ୺ዴƸŴᙐእૠ˳ƷǑ ƏƳ೅ૠ

0

Ʒ˳ƩƚưƳƘŴദ೅ૠƷ˳ Ǎ

Witt

࿢ƷǑƏƳฆ೅ૠƷ࿢ ƷɥưNjᎋݑƢ ǔƜƱƸӧᏡưƋǔŵ

7

LJƨ ӑ୺ႎˊૠ୺ዴƷȢǸȥȩǤ NjᲢ᭗ഏ ΨᲣƷˊૠٶಮ˳ᲢƴƪǐƬƱƠƨ੆Ǔƕ˄

ƍƯƍǔNjƷᲣǛܭ፯ƠƯƍǔŵഏƷኽௐƸ

p

^ᡶ

Teichm¨uller

ྸᛯ ƷؕஜܭྸưƋǔŵ ܭྸᲴ

Z

_p ɥƷӑ୺ႎˊૠ୺ዴƷȢǸȥȩǤ Ʒ ^∃žᘮᙴſǍŴƦƷɥƷ୍ᢄႎƳӑ୺ႎˊ ૠ୺ዴƷƋǔ ^∃žᘮᙴſƷɥƴ

೅แႎƳ

Frobenius

^ਤƪɥƛ ƕ܍נƢǔŵ

8

VI. p

ᡶᢒǢȸșȫ࠹˴Ʒኰʼ

§1.

ˊૠ୺ዴƷૠᛯႎؕஜ፭

ȪȸȞȳ᩿Ʒ᧓Ʒޅ৑ႎƳஊᨂϙ΂ƴƭƍƯ ᎋƑǑƏŵƜƷǑƏƳϙ΂Ƹ ޅ৑ႎƳദЩ ᧙ૠ Ǎ șǭኢૠ ƕƲƷǑƏƴϙƞǕǔƔ ǛᙸǔƜƱƴǑƬƯਵƑǔƜƱƕưƖǔŵ

C[[z

]] → C[[z ]]

ཎƴŴžޅ৑ႎƴƸӷ׹ſưƋǔƱƍƏࣱឋ ƸŴ

z

→

f (z) = c

₁

· z + c

₂

· z

²

+ . . . + c

_n

· z

ⁿ

+ . . .

c

₁

= 0

ƔƲƏƔƱƍƏவˑƱ ӷ͌ ưƋǔŵ

9

ƜƷǑƏƴžޅ৑ႎӷ׹ſƱƍƏಒࣞǛș ǭኢૠǛNjƬƯܭࡸ҄ƢǔƱŴ˓ॖƷ˳

k

ƷɥƷșǭኢૠ࿢Ʒ᧓Ʒแӷ׹

k[[t

]] → k[[t]]

ƴਘࢌƢǔƜƱƕưƖǔŵƜƷǑƏƴਘࢌƞ Ǖƨžޅ৑ႎӷ׹ſƷಒࣞǛ

´etale

^{ᲢǨǿȸȫᲣ} ƱԠƿŵ

k

^{Ǜ˓ॖƷ˳ƱƠ}

X

^Ǜ

k

^{ɥƷ ӑ୺ႎˊૠ୺ዴ} ƱƠǑƏŵ

X

ɥƷ ஊྸ᧙ૠ ǛᎋƑǔƜƱƴ ǑƬƯ

X

^{Ʒ ᧙ૠ˳}

K

_X

ƱƍƏ˳ƕࢽǒǕǔŵƜƷ˳Ʒˊૠ᧍Ѽ

K

_X ǛᢠƿƱŴ

K

_X ^{Ʒ ዌݣǬȭǢ፭}

G

_KX ^def

= Gal(K

_X

/K

_X

)

ƱƍƏ иஊᨂ፭ ƕܭLJǔŵ

K

_X ^ƷɶƴŴ

k

Ʒˊૠ᧍Ѽ

k ⊆ K

_X ^{ƕ࣏ƣԃLJǕǔŵ}

10

ࢼƬƯ

k

^{Ʒ ዌݣǬȭǢ፭}

G

_KX

G

_k ^def

= Gal(k/k)

ƱƍƏ

G

_KX ^{Ʒՠ፭NjܭLJǔŵ}

X

^{ƷᲢ᧍Უໜ}

x

^ƴݣƠƯ

K

_X ^Ǜ

x

^ƴƓƍƯ žܦͳ҄ſƢǔƜƱƴǑƬƯ

k

₁

[[t]]

^ᲢƨƩƠ

[k

₁

: k ] < ∞

Უƴӷ׹Ƴșǭኢૠ࿢ƕܭLJ ǔŵӷಮƳƜƱƸᲢ

K

_X ^ϋƷᲣ

K

_X ^Ʒ˓ॖ

Ʒ ஊᨂഏਘٻ

K

⊆ K

_X ^{ƴNjᚕƑǔŵࢼƬ} ƯƜƷǑƏƴƠƯࢽǒǕǔಮŷƳ࿢แӷ׹

k

₁

[[t]] → k

₁

[[t]]

ƕμƯžǨǿȸȫſưƋǔ ƱƍƏவˑǛᛢƢƜƱƴǑƬƯ

G

_KX

Π

_X

( G

_k

)

ƱƍƏᲢ

G

_k ƴμݧƢǔᲣՠ፭ƕܭLJǔŵƜ Ʒиஊᨂ፭

Π

_X ^Ǜ

X

^{Ʒžૠᛯႎؕஜ፭ſƱ} Ԡƿŵ

G

_k ǁƷμݧƷఋǛƱǔƜƱƴǑƬƯ ᐯ໱ƳܦμኒЗ

1 → Δ

_X

→ Π

_X

→ G

_k

→ 1

ƕࢽǒǕǔŵ

11

§2. Grothendieck

^{ʖे׹Ʒܭྸ}

έᆉƷܦμኒЗưƸŴ

Δ

_X ^Ƹ

X

ƷȢǸȥȩǤƴǑǒƳƍ

ƱƍƏ᣻ᙲƳࣱឋǛ฼ƨƠƯƍǔŵLJƨ

k = k

^ƷƱƖ

Δ

_X

= Π

_X ^{ƱƳǔŵ̊ƑƹŴ}

k = C

ƷƱƖ

X

^{Ƹ ӑ୺ႎȪȸȞȳ᩿}

X

^ƴݣࣖƠ ƯƍƯ

Π

_X ^Ƹ

X

Ʒ ɦᢿˮႻ୺᩿ Ʒž୍ᡫ Ʒؕஜ፭ſƷ иஊᨂܦͳ҄ ƴᐯ໱ƴ ӷ׹

ƴƳǔŵ

ȪȸȞȳ᩿Ʒᛅƴ৏ǖƏŵžׄᚌƷחſ

Squr(X )

ƴႇئƢǔׄᚌƨƪƸŴž

X

^{Ʒɶƴ؈NJᡂLJ} Ǖƨ ؕᄽ˳

C

ƷݱƞƳdzȔȸ ᲷˊᘙᎍſƱ ᙸǔƜƱƕưƖǔŵ

12

ഏƴ

k

^ƕ

p

^{ᡶ˳ Ჷ}

p

^ᡶૠ˳

Q

_p ^{Ʒஊᨂഏਘ} ٻưƋǔƱˎܭƠǑƏŵƜƷǑƏƳ˳

k

^Ʒ ዌݣǬȭǢ፭

G

_k ^ƷǢȸșȫ҄

G

^ab_k ^ƷɶƴŴ Ტޅ৑᫏˳ᛯ ƴǑǓᲣ

k

^{× Ʒ ݱƞƳdzȔȸ}

ƕλƬƯƍǔŵƠƔNj

X

^ƷᲢ

k

^ஊྸᲣໜ

x

^Ǜ ᢠƿƱμݧ

Π

_X

G

_k ^ƷǻǯǷȧȳ

G

_k

→

Π

_X ^{ƕܭLJǓŴƦƷ΂}

G

_k

⊆ Π

_X ^Ǜ

δ ∈ Δ

_X ƴ˺ဇƞƤǔƱŴž

G

_k

· δ ⊆ Π

_X^ſƱƍƏ žؕᄽ˳ƷݱƞƳdzȔȸſƕ

Π

_X ^ƷɶƴưƖ ǔŵ

Δ

_X ƕȪȸȞȳ᩿ƷɦᢿˮႻ୺᩿Ʒž

p

ᡶ༿ſƴ࢘ƨǔƱᎋƑǔƱŴƜǕƸέᆉƷ

Squr(X )

^ƷᛅƷ

p

ᡶ༿ ƱᙸǔƜƱƕưƖǔŵ

13

ഏƷኽௐƸ

p

ᡶᢒǢȸșȫ࠹˴ ƷؕஜႎƳ ܭྸưƋǔƕŴ

Squr(X )

^{ƴ᧙ƢǔžദЩನᡯ}

ࣄΨƷܭྸſƷ

p

ᡶ༿ƱᙸǔƜƱƕưƖǔŵ ܭྸᲴ

X , Y

^Ƹ

p

^ᡶ˳

k

^{ɥƷ ӑ୺ႎˊૠ୺ዴ} ƱƢǔŵƢǔƱŴ

G

_k ɥƷиஊᨂ፭Ʒٳᢿӷ׹

Π

_X

→

^∼

Π

_Y

Ʊ

k

^{ɥƷˊૠ୺ዴƷӷ׹}

X →

^∼

Y

ᲢƨƩƠžٳᢿſƱƸžϋᢿᐯࠁӷ׹ƱƷӳ঺

ǛᨊƍƯſƷॖᲣƸ

1

^ݣ

1

^{ƴݣࣖƠƯƍǔŵ}