= Double complex
{title2=$C_{p,q}$}
A double complex has modules $C_{p,q}$ and differentials lowering either index, each squaring to zero and anticommuting with the other. The total <chain complex> has term $\bigoplus_{p+q=i}C_{p,q}$ and differential the sum of the two differentials. For the tensor product of two <chain complexes>, a sign $(-1)^p$ on the second differential ensures anticommutation. If the double complex is in the first quadrant and one direction has <homology> only in degree zero, its total <homology> is computed by the surviving degree-zero complex. Applying this twice to two <free resolutions> proves the balanced calculation of the <Tor functor>.
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