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The Lefschetz Fixed Point Theorem

Frederik [email protected]

University of Bristol

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Homology

Chain complex K is a sequence {C n , n }n Z of Abelian groups

n 1 C n 1

n C n n +1 C n +1

n +2

and boundary maps (homomorphisms) such that n n +1 = 0 .

Since n n +1 = 0 one has Im n +1 Ker n and

H n (K ) := Ker n / Im n +1

is the n-th homology group of K.

Example: singular homology.

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Singular Homology

Let S n (X ) be free abelian group with basis singular n-simplices

S n (X ) = {

n | n = 0 nitely many }

By linearity n : S n (X ) S n 1(X ) and n n +1 = 0.

Element c S n (X ) is n-cycle if n (c) = 0.

Element d S n (X ) is n-boundary if d = (e) for e S n +1 (X ).

n-th singular homology group

H n (K ) := Ker n / Im n +1

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Singular Homology

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Cohomology

Cochain complex is a sequence {C n , dn }n Z of Abelian groups

dn 2 C n 1

dn 1 C ndn C n +1

dn +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.

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Algebraic de Rham Cohomology

X smooth, affine variety over K of char 0 with coordinate ring

A := K [x1 , . . . , x n ]/ (f 1 , . . . , f m )

Module of Kahler differentials 1A/K generated by dg with g A

1A/K = ( A dx1 + + A dxn )/ (m

i=1

A(f ix1

dx1 + +f ix n

dxn )) .

iA/K =i 1A/K and di :

iA/K

i+1A/K exterior diff.

Since di+1 di = 0 we get the de Rham complex A/K

0 A d0 1A/Kd1 2A/K

d2 3A/K

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i-th de Rham cohomology group of is dened as

H iDR (A/K ) := Ker di / Im di 1

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Intersection of Cycles

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Let A and B two cycles that intersect tranversely at point p.

The intersection number of A and B is

#( A B ) = p A B

p(A B )

Intersection index p(A B ) { 1, +1 } depends on orientation.

#( A B ) only depends on homology classes of A and B !

General: intersection number denes pairing

H k (M, Z ) H n k (M, Z ) Z

Poincare : for any k-cycle A on M there is closed ( n k)-form A

#( A B ) = B A12

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Let M be compact oriented manifold of dimension n andf : M M an endomorphism.

The Lefschetz number of f is dened as

L(f ) =

n

i=0( 1)

iTrace( f |H

iDR (M )) .

A point p M is called a xed point of f is

f ( p) = p

Question : what is # { p M | f ( p) = p}?

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Diagonal M M and graph f = {( p,f ( p)) | p M } of f .

xed point = intersection of and f

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If f has only nondegenerate xed points then

#( f )M M =f ( p)= p

f ( p)

The Lefschetz Fixed Point Formula

f ( p)= p f ( p) = L(f ) = i ( 1)i Trace( f |H iDR (M ))

Proof:#( f )M M =

f

Poincare dual of homology class of diagonal.

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Corollary 1: # { p M : f ( p) = p} | L(f )| .

Corollary 2: If L(f ) = 0 , then f has a xed point .

Theorem: for analytic cycles V and W of compact complex

manifold meeting transversally p(V W ) = +1 . Lefschetz Fixed Point Theorem : Let M be a compact complex

analytic manifold and f : M M an analytic map. Assume thatf only has isolated nondegenerate xed points then

# { p M | f ( p) = p} = L(f ) = i ( 1)i Trace( f |H iDR (M ))

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A p -adic Cohomology of the Affine Line

Frobenius F : F p F p : x x p then x F p iff F (x) = x.

Consider C : xy 1 = 0 with coordinate ring A = F p[x, 1/x ], then

N r = # C (F pr ) = # xed points of F r

= pr 1

Construct de Rham cohomology in characteristic p?

Only possible to compute N r (mod p). 1(A) := A dx/ (d A) is innite dimensional.

xk dx with k 1 (mod p) cannot be integrated.

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p -adic numbers

p-adic norm | | p of r = 0 Q is

|r | p = p , r = p u/v, , u, v Z , p | u, p | v.

Field of p-adic numbers Q p is completion of Q w.r.t. | | p,

ma i pi , ai { 0, 1, . . . , p 1}, m Z .

p-adic integers Z p is the ring with | | p 1 or m 0.

Unique maximal ideal M = {x Q p | |x | p < 1} = pZ p andZ p/M = F p.

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First attempt : lift situation to Z p and try again?

Consider two lifts to Z p

A1 = Z p[x, 1/x ] and A2 = Z p[x, 1/ (x(1 + px))]

A1 and A2 are not isomorphic; both x and 1 + px invertible in A2 .

H 1DR (A1 / Q p) =dxx and H

1DR (A2 / Q p) =

dxx ,

dx1+ px .

Frobenius does not always lift:

Example: A = F 3[x]/ (x2 2) and A = Z 3[x]/ (x2 2)

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Second attempt : use p-adic completion .

A1 = A2 = {

i Z i x i Z p[[x, 1/x ]] | lim

| i | + i = 0 }

However: H 1DR (A / Q p) is again innite dimensional!

i pi x p

i 1is in A but integral i x

pi is not.

Convergence property lost in integration.

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Third attempt : consider the dagger ring or weak completion

A = {i Z

i x i Z p[[x, 1/x ]] | R > 0 , R : v p( i ) |i | + }

Note: A1 is isomorphic to A2 , since 1 + px invertible in A

1 .

11 + px

=

i =0

( 1)i pi x i

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Monsky-Washnitzer := de Rham cohomology of A Q p

H 1(A/ Q p) = ( A Q p)dx/ (d(A Q p)) and clearly for k = 1

xk dx = d(xk +1

k + 1)

Conclusion: H 1(A/ Q p) has basis dxx

Lifting Frobenius F to A: innitely many possibilities

F (x) x p + pA

Examples: F 1(x) = x p or F 2(x) = x p + p

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Action of F 1 on basisdxx is given by

F 1dxx

=d(F 1(x))

F 1(x)=

d(x p)x p

= pdxx

Action of F 2 on basis dxx is given by

F 2dxx

=d(F 2(x))

F 2(x)=

d(x p + p)x p + p

=px p 1

x p + pdx =

p1 + px p

dxx

Power series expansion: (1 + px p) 1 = i=0 ( 1)i pi x ip A

F 2 dxx= pdx

x+ d

i =1

( 1)i+1 pi 1i

x ip

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Action of F 1 and F 2 are equal on H 1(A/ Q p)!

F (dxx

) = pdxx

F 1dxx

=1 p

dxx

Lefschetz Trace formula applied to C gives

# C (F pr ) = Trace ( pF 1)r |H 0(C/ Q p) Trace ( pF 1)r |H 1(C/ Q p)

Conclusion:

# C (F pr ) = pr 1

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Monsky-Washnitzer cohomology

X smooth affine variety over F q with coordinate ring A. Exists A := Z q[x1 , . . . , x n ]/ (f 1 , . . . , f m ) with A Z q F q = A

Dagger ring or weak completion A is dened

A

:=Z

q x1 , . . . , x n

/ (f 1 , . . . , f m )with Z q x1 , . . . , x n overconvergent power series

I

a I xI Z q[[x1 , . . . , x n ]] | lim inf | I |

v p( I )|I |

> 0

M-W cohomology is the de Rham cohomology of A Q q .

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Monsky-Washnitzer cohomology

Denition 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/ Q q) H i (B/ Q q) only depends on G.

Cohomology groups H i (A/ Q q) are nite dimensional .

Lefschetz trace formula : for X of dimension d

N r =d

i =0

( 1)i Tr (qd F 1)r |H i (X/ Q q)

Let C be a projective, smooth curve of genus g over F q

S a set of m F q-points and A coordinate ring of C \ S

dim H 1(A/ Q q) = 2 g + m 1

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