Lefschetz trace formula
Solomon Lefschetz wanted to count the fixed point set of a continuous map.
Fix a ground field . Given a continuous map of topological space, its Lefschetz number is the alternating sum of the traces
\sum_i (-1)^i Tr (H^i(f) \colon H^i(X,k)\to H^i(X,k)
of the endomorphisms of the ordinary cohomology groups with coefficients in the ground field .
One sometimes also talks of the Lefschetz number of the induced endomorphism of the chain/cochain complexes, see algebraic Lefschetz formula.
For the identity map, the Lefschetz trace reduces to the Euler characteristic.
Lefschetz fixed point theorem
The Lefschetz fixed point theorem says that if the Lefschetz number is non-zero, then has at least one fixed point.
The existence of a Lefschetz formula holds more general in Weil cohomology theories (by definition) and hence notably in ℓ-adic étale cohomology. This fact serves to prove the Weil conjectures.
follows from existence of
(Milne, section 25)
For ordinary cohomology
The original article is
For étale cohomology
For étale cohomology of schemes:
For algebraic stacks:
- Kai Behrend, The Lefschetz trace formula for algebraic stacks, Invent. Math. 112, 1 (1993), 127-149, doi
Revised on November 22, 2013 05:48:20
by Urs Schreiber