The Lefschetz Fixed Point Theorem
Frederik Vercauteren
[email protected]
University of Bristol
1
Overview
• Homology and cohomology
• Intersection of cycles
• The Lefschetz Fixed Point Theorem
• A good p-adic cohomology for the affine line
• Monsky-Washnitzer cohomology
2
Homology
• Chain complex K is a sequence {Cn , ∂n }n∈Z of Abelian groups
∂n−1
∂n+1
∂
∂n+2
n
· · · ←− Cn−1 ←−
Cn ←− Cn+1 ←− · · ·
and boundary maps (homomorphisms) such that ∂n ∂n+1 = 0.
• Since ∂n ∂n+1 = 0 one has Im ∂n+1 ⊂ Ker ∂n and
Hn (K) := Ker ∂n /Im ∂n+1
is the n-th homology group of K.
• Example: singular homology.
3
Singular Homology
• n-simplex: convex hull of n + 1 points x0 , . . . , xn not in
n − 1-dimensional subspace.
• Standard n-simplex σn : x0 = (1, 0, . . . , 0), . . . , xn = (0, 0, . . . , 1).
• A singular n-simplex of a topological space X is continuous
function φ : σn → X.
• For each 0 ≤ i ≤ n we obtain a singular n − 1-simplex
(∂ (i) φ)(t0 , . . . , tn−1 ) = φ(t0 , . . . , ti−1 , 0, ti , . . . , tn−1 )
• Boundary operator ∂ is given by
∂n = ∂ (0) − ∂ (1) + · · · + (−1)n ∂ (n)
4
Singular Homology
• Let Sn (X) be free abelian group with basis singular n-simplices
X
Sn (X) = {
nφ · φ | nφ 6= 0 finitely many }
φ
• By linearity ∂n : Sn (X) ← Sn−1 (X) and ∂n ◦ ∂n+1 = 0.
• Element c ∈ Sn (X) is n-cycle if ∂n (c) = 0.
• Element d ∈ Sn (X) is n-boundary if d = ∂(e) for e ∈ Sn+1 (X).
• n-th singular homology group
Hn (K) := Ker ∂n /Im ∂n+1
5
Singular Homology
6
Cohomology
• Cochain complex is a sequence {C n , dn }n∈Z of Abelian groups
dn−2
dn−1
d
dn+1
n
· · · −→ Cn−1 −→ Cn −→
Cn+1 −→ · · ·
and coboundary maps or differentials such that dn dn−1 = 0.
• Since dn dn−1 = 0 one has Im dn−1 ⊂ Ker dn and
H n (K) := Ker dn /Im dn−1
is the n-th cohomology group of K.
• Example: algebraic de Rham cohomology.
7
Algebraic de Rham Cohomology
• X smooth, affine variety over K of char 0 with coordinate ring
A := K[x1 , . . . , xn ]/(f1 , . . . , fm )
• Module of Kähler differentials Ω1A/K generated by dg with g ∈ A
Ω1A/K
•
m
X
∂fi
∂fi
= (A dx1 + · · · + A dxn )/(
A(
dx1 + · · · +
dxn )) .
∂x
∂x
1
n
i=1
ΩiA/K
=
Vi
Ω1A/K and di : ΩiA/K → Ωi+1
A/K exterior diff.
• Since di+1 ◦ di = 0 we get the de Rham complex ΩA/K
d
d
d
1
2
0
Ω2A/K −→
Ω3A/K · · ·
0 −→ A −→
Ω1A/K −→
8
• i-th de Rham cohomology group of is defined as
i
HDR
(A/K) := Ker di /Im di−1
9
Intersection of Cycles
10
Intersection of Cycles
11
Intersection of Cycles
• Let A and B two cycles that intersect tranversely at point p.
• The intersection number of A and B is
X
#(A · B) =
ıp (A · B)
p∈A∩B
• Intersection index ıp (A · B) ∈ {−1, +1} depends on orientation.
• #(A · B) only depends on homology classes of A and B!
• General: intersection number defines pairing
Hk (M, Z) × Hn−k (M, Z) → Z
• Poincaré: for any k-cycle A on M there is closed (n − k)-form ϕA
Z
#(A · B) =
ϕA
B
12
The Lefschetz Fixed Point Theorem
• Let M be compact oriented manifold of dimension n and
f : M → M an endomorphism.
• The Lefschetz number of f is defined as
L(f ) =
n
X
i
(−1)i Trace(f∗ |HDR
(M )) .
i=0
• A point p ∈ M is called a fixed point of f is
f (p) = p
• Question: what is #{p ∈ M | f (p) = p}?
13
The Lefschetz Fixed Point Theorem
• Diagonal ∆ ⊂ M × M and graph Γf = {(p, f (p))|p ∈ M } of f .
fixed point = intersection of ∆ and Γf
14
The Lefschetz Fixed Point Theorem
• If f has only nondegenerate fixed points then
X
#(∆ · Γf )M ×M =
ıf (p)
f (p)=p
• The Lefschetz Fixed Point Formula
P
P
i
i
f (p)=p ıf (p) = L(f ) =
i (−1) Trace(f∗ |HDR (M ))
• Proof:
Z
#(∆ · Γf )M ×M =
ϕ∆
Γf
• ϕ∆ Poincaré dual of homology class of diagonal.
15
The Lefschetz Fixed Point Theorem
• Corollary 1: #{p ∈ M : f (p) = p} ≥ |L(f )|.
• Corollary 2: If L(f ) 6= 0, then f has a fixed point.
• Theorem: for analytic cycles V and W of compact complex
manifold meeting transversally ıp (V · W ) = +1.
• Lefschetz Fixed Point Theorem: Let M be a compact complex
analytic manifold and f : M → M an analytic map. Assume that
f only has isolated nondegenerate fixed points then
#{p ∈ M | f (p) = p} = L(f ) =
16
P
i
i
(−1)
Trace(f
|H
∗
DR (M ))
i
A p-adic Cohomology of the Affine Line
• Frobenius F : Fp → Fp : x 7→ xp then x ∈ Fp iff F (x) = x.
• Consider C : xy − 1 = 0 with coordinate ring A = Fp [x, 1/x], then
r
Nr = #C(F ) = # fixed points of F = pr − 1
pr
• Construct de Rham cohomology in characteristic p?
– Only possible to compute Nr (mod p).
– Ω1 (A) := A dx/(d A) is infinite dimensional.
– xk dx with k ≡ −1 (mod p) cannot be integrated.
17
p-adic numbers
• p-adic norm | · |p of r 6= 0 ∈ Q is
|r|p = p−ρ ,
r = pρ u/v,
ρ, u, v ∈ Z,
p 6 | u, p 6 | v.
• Field of p-adic numbers Qp is completion of Q w.r.t. | · |p ,
∞
X
ai pi ,
ai ∈ {0, 1, . . . , p − 1},
m ∈ Z.
m
• p-adic integers Zp is the ring with | · |p ≤ 1 or m ≥ 0.
• Unique maximal ideal M = {x ∈ Qp | |x|p < 1} = pZp and
Zp /M ∼
= Fp .
18
A p-adic Cohomology of the Affine Line
First attempt: lift situation to Zp and try again?
• Consider two lifts to Zp
A1 = Zp [x, 1/x]
and
A2 = Zp [x, 1/(x(1 + px))]
• A1 and A2 are not isomorphic; both x and 1 + px invertible in A2 .
dx
dx
1
1
(A1 /Qp ) = h dx
(A
/Q
)
=
h
• HDR
i
and
H
,
2
p
DR
x
x 1+px i.
• Frobenius does not always lift:
– Example: A = F3 [x]/(x2 − 2) and A = Z3 [x]/(x2 − 2)
19
A p-adic Cohomology of the Affine Line
Second attempt: use p-adic completion.
X
∞ ∼ ∞ ∼
A1 = A2 = {
αi xi ∈ Zp [[x, 1/x]] |
i∈Z
lim αi = 0}
|i|→+∞
1
(A∞ /Qp ) is again infinite dimensional!
• However: HDR
•
P
i pi−1
ip x
is in A
∞
but integral
P
i
p
x
is not.
i
• Convergence property lost in integration.
20
A p-adic Cohomology of the Affine Line
Third attempt: consider the dagger ring or weak completion
X
†
A ={
αi xi ∈ Zp [[x, 1/x]] | ∃² ∈ R>0 , δ ∈ R : vp (αi ) ≥ ²|i| + δ}
i∈Z
• Note: A†1 is isomorphic to A†2 , since 1 + px invertible in A†1 .
∞
X
1
=
(−1)i pi xi
1 + px
i=0
21
A p-adic Cohomology of the Affine Line
• Monsky-Washnitzer := de Rham cohomology of A† ⊗ Qp
• H 1 (A/Qp ) = (A† ⊗ Qp )dx/(d(A† ⊗ Qp )) and clearly for k 6= −1
xk+1
x dx = d(
)
k+1
k
• Conclusion: H 1 (A/Qp ) has basis
dx
x
• Lifting Frobenius F to A† : infinitely many possibilities
F (x) ∈ xp + pA†
• Examples: F1 (x) = xp or F2 (x) = xp + p
22
A p-adic Cohomology of the Affine Line
• Action of F1 on basis dx
x is given by
µ ¶
d(xp )
d(F1 (x))
dx
dx
=
F1 ∗
=
=
p
x
F1 (x)
xp
x
• Action of F2 on basis dx
x is given by
µ ¶
dx
d(F2 (x))
d(xp + p)
pxp−1
p
dx
F2 ∗
=
=
= p
dx =
x
F2 (x)
xp + p
x +p
1 + px−p x
−p −1
P∞
• Power series expansion: (1 + px ) = i=0 (−1)i pi x−ip ∈ A†
!
̰
µ ¶
i+1
i−1
X (−1) p
dx
dx
x−ip
F2 ∗
=p
+d
x
x
i
i=1
23
A p-adic Cohomology of the Affine Line
• Action of F1 and F2 are equal on H 1 (A/Qp )!
dx
dx
F∗ ( ) = p
⇒ F∗−1
x
x
µ
dx
x
¶
1 dx
=
p x
• Lefschetz Trace formula applied to C gives
#C(F ) = Trace
pr
¡
(pF∗−1 )r |H 0 (C/Qp )
¢
− Trace
• Conclusion:
#C(Fpr ) = pr − 1
24
¡
(pF∗−1 )r |H 1 (C/Qp )
¢
Monsky-Washnitzer cohomology
• X smooth affine variety over Fq with coordinate ring A.
• Exists A := Zq [x1 , . . . , xn ]/(f1 , . . . , fm ) with A ⊗Zq Fq ∼
=A
• Dagger ring or weak completion A† is defined
A† := Zq hx1 , . . . , xn i† /(f1 , . . . , fm )
with Zq hx1 , . . . , xn i† overconvergent power series
(
X
I
vp (αI )
aI x ∈ Zq [[x1 , . . . , xn ]] | lim inf
>0
|I|
|I|→∞
)
I
• M-W cohomology is the de Rham cohomology of A† ⊗ Qq .
25
Monsky-Washnitzer cohomology
• Definition only depends on A and not on choices made!
• Every morphism G : A → B lifts to G : A† → B † .
• Induced map on H i (A/Qq ) → H i (B/Qq ) only depends on G.
• Cohomology groups H i (A/Qq ) are finite dimensional.
• Lefschetz trace formula: for X of dimension d
Nr =
d
X
i
¡
(−1) Tr (q
d
F∗−1 )r |H i (X/Qq )
¢
i=0
• Let C be a projective, smooth curve of genus g over Fq
– S a set of m Fq -points and A coordinate ring of C \ S
dim H 1 (A/Qq ) = 2g + m − 1
26