nLab
algebraic Lefschetz formula

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

The algebraic Lefschetz formula is the statement

$$\sum _{p\ge 0}(-1{)}^p\mathrm{tr}(f^p:C^p\to C^p)=\sum _{p\ge 0}(-1{)}^p\mathrm{tr}(H^p(f):H^p(C)\to H^p(C)).$$\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

$$\sum _{p\ge 0}(-1{)}^p\dim C^p=\sum _{p\ge 0}(-1{)}^p\dim H^p(C).$$\sum_{p\geq 0} (-1)^p dim C^p = \sum_{p\geq 0} (-1)^p dim H^p(C).