... ontoZj,iwith coefficient ±1, under the inductivehypothesis:(2.2.2)Xj+1→Xjhas the lifting property.Indeed, admitting this,Z•is then isomorphic to the mapping cone ofZ•→ ⊕iC(±id ... if the inverse image of a closed point is a closed variety forexample. The independence of the choice of (L, γ) and S is also nontrivial. Sowe have to use Grothendieck’s general theory using ... with the weightfiltration. Then the assertion follows by taking the image in Hr´et(X•,l)fr, andusing the E2-degeneration of the weight spectral sequence modulo torsion.3.5. Theorem....