# nLab algebraic Lefschetz formula

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

The algebraic Lefschetz formula is the statement

$\sum _{p\ge 0}\left(-1{\right)}^{p}\mathrm{tr}\left({f}^{p}:{C}^{p}\to {C}^{p}\right)=\sum _{p\ge 0}\left(-1{\right)}^{p}\mathrm{tr}\left({H}^{p}\left(f\right):{H}^{p}\left(C\right)\to {H}^{p}\left(C\right)\right).$\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 $f=\mathrm{id}$ is the Euler-Poincaré formula

$\sum _{p\ge 0}\left(-1{\right)}^{p}\mathrm{dim}{C}^{p}=\sum _{p\ge 0}\left(-1{\right)}^{p}\mathrm{dim}{H}^{p}\left(C\right).$\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)