Coherent Algebraic Sheaves by Jean-Pierre Serre, Piotr Achinger, Lukasz Krupa

By Jean-Pierre Serre, Piotr Achinger, Lukasz Krupa

Translation of: Jean-Pierre Serre, "Faisceaux Algebriques Coherents", The Annals of arithmetic, second Ser., Vol. sixty one, No. 2. (Mar., 1955), pp. 197--278
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Be a decreasing sequence of closed subsets of a space X satisfying (c); since all Yi satisfy (A), there exists for all i an ni such that Fm ∩ Yi = Fni ∩ Yi for m ≥ ni ; if n = Sup(ni ), we then have Fm = Fn for m ≥ n, which shows (c). A space X is said to be irreducible if it is not a union of two closed subspaces, distinct from X itself; or equivalently, if any two non-empty open subsets have a non-empty intersection. Any finite family of non-empty open subsets of X then has a non-empty intersection, and any open subset of X is also irreducible.

We verify by a direct calculation (keeping in mind that f is a cocycle) that we have d (g 0 ) = ι (τ f ), . . , d (g p ) = d (g p−1 ), . . d (g n−1 ) = (−1)n ι (f ) hence d(g 0 − g 1 + . . + (−1)n−1 g n−1 ) = ι (τ f ) − ι (f ), which shows that ι (τ f ) and ι (f ) are cohomologous. Proposition 5. Suppose that V is finer than U and that H q (Vs , F ) = 0 for all s and all q > 0. Then the homomorphism σ(V, U) : H n (U, F ) → H n (V, F ) is bijective for all n ≥ 0. If we apply Proposition 3, switching the roles of U and V, we see that ι : H n (V, F ) → H n (U, V; F ) is bijective.

Let G be a coherent algebraic sheaf on V which is zero outside W ; the annihilator of G does not necessarily contain J (W ) (in other words, G not always can be considered as an coherent algebraic sheaf on W ); all we can say is that it contains a power of J (W ). 40 Sheaves of fractional ideals Let V be an irreducible algebraic variety and let K(V ) denote the constant sheaf of rational functions on V (cf. n◦ 36); K(V ) is an algebraic sheaf which is not coherent if dim V > 0. An algebraic subsheaf F of K(V ) can be called a ,,sheaf of fractional ideals” since each Fx is a fractional ideal of Ox,V .

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