# nLab algebraic Lefschetz formula

Let $(C,d)$ be a nonnegative cochain complex of vector spaces over a field of (total) finite dimension $dim C = \sum_{p=0}^\infty dim C^p \lt \infty$ and $f = (f^p)_{p\geq 0} :(C,d)\to (C,d)$ an endomorphism of cochain complexes.

The algebraic Lefschetz formula is the statement

$\sum_{p\geq 0} (-1)^p tr (f^p :C^p\to C^p) = \sum_{p\geq 0} (-1)^p tr (H^p(f):H^p(C)\to H^p(C)).$

Its special case for $f = id$ is the Euler-Poincaré formula

$\sum_{p\geq 0} (-1)^p dim C^p = \sum_{p\geq 0} (-1)^p dim H^p(C).$
Created on March 9, 2010 14:34:55 by Zoran Škoda (161.53.130.104)