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We associate to each real algebraic variety a filtered chain complex, the weight complex, which is well-defined up to filtered quasi-isomorphism, and which induces on classical (compactly supported) homology with \(\mathbb Z _2\) coefficients an analog of the weight filtration for complex algebraic varieties. This complements our previous definition of the weight filtration of Borel–Moore homology.

The weight filtration for real algebraic varieties II: classical homology