{c}
{title2=$\sum_k(-1)^k\operatorname{tr}(F_k)=\sum_k(-1)^k\operatorname{tr}(H_k(F))$}

For an endomorphism of a finite-dimensional bounded <chain complex> over a <field>, the alternating chain trace equals the alternating trace on <homology>. Split each chain space into boundaries, representatives for <homology>, and a complement mapping isomorphically to the preceding boundaries. The chain-map relation makes the traces on the first and third summands cancel in successive degrees, leaving the <homology> traces. Over a <field> the induced <cohomology> maps are dual and have the same traces. This identity turns the absence of diagonal simplex contributions into a vanishing <Lefschetz number>.


 Hopf trace identity (source code)