### Algebraically constructible chains

We construct for a real algebraic variety (or more generally for a scheme essentially of finite type over a field of characteristic $0$) complexes of algebraically and $k$- algebraically constructible chains. We study their functoriality and compute their homologies for affine and projective spaces. Then we show that the lagrangian algebraically constructible cycles of the cotangent bundle are exactly the characteristic cycles of the algebraically constructible functions.