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algebraic Lefschetz formula

Let (C,d)(C,d) be a nonnegative cochain complex of vector spaces over a field of (total) finite dimension dimC= p=0 dimC p<dim C = \sum_{p=0}^\infty dim C^p \lt \infty and f=(f p) p0:(C,d)(C,d)f = (f^p)_{p\geq 0} :(C,d)\to (C,d) an endomorphism of cochain complexes.

The algebraic Lefschetz formula is the statement

p0(1) ptr(f p:C pC p)= p0(1) ptr(H p(f):H p(C)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=idf = id is the Euler-Poincaré formula

p0(1) pdimC p= p0(1) pdimH p(C). \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)