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By Robert J. Walker

This advent to algebraic geometry examines how the more moderen summary ideas relate to standard analytical and geometrical difficulties. The presentation is saved as straightforward as attainable, because the textual content can be utilized both for a starting path or for self-study.

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Example text

5), π et π ˜ sont des ´equivalences de S-descente cohomologiques. Passons `a f˜. 3), il suffit de prouver que les morphismes d’espaces simpliciaux obtenus en regardant chaque ligne d’indice p ≥ 0 sont des ´equivalences. Ce morphisme est la premi`ere projection ϕp : (Y• )p+1 → (Y• )p o` u le produit est pris sur Y. 2), (Y• )p+1 et (Y• )p sont leur propre n + 1-cosquelettes. Par ailleurs, comme Y et Y co¨ıncident en degr´e ≤ n, le morphisme sqn (φp ) est un isomorphisme. 1), la section diagonale de ϕp = cosq (ϕp ) est un inverse `a homotopie pr`es.

Supposons que tp : Xp → Yp soit un isomorphisme pour p ≤ n. Alors, pour tout Faisceau F de SF , et at induit un isomorphisme ∼ τ

2, les fl`eches cosqn+1 (X)p → cosqn (X)p sont des ´equivalences de descente cohomologique pour tout p. 1.

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