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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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Intersection of Cycles
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Intersection of Cycles
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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The Lefschetz Fixed Point Theorem
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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The Lefschetz Fixed Point Theorem
Diagonal M M and graph f = {( p,f ( p)) | p M } of f .
xed point = intersection of and f
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The Lefschetz Fixed Point Theorem
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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The Lefschetz Fixed Point Theorem
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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A p -adic Cohomology of the Affine Line
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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A p -adic Cohomology of the Affine Line
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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A p -adic Cohomology of the Affine Line
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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A p -adic Cohomology of the Affine Line
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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A p -adic Cohomology of the Affine Line
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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A p -adic Cohomology of the Affine Line
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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