remark - ivorra

In this note, we sho w that the proof of [EGA Th´ eor` eme III.3.2.1] can be sligh tly modified to av oid spectral sequenc es. The stateme nt of the theorem is as follows: Let Y be a locally Noetherian scheme and f : X −→ Y a proper morphism. For each coher ent O X -module F , the O Y -modules R q f ∗ (F ) ar e co her ent for q ≥ 0. The previous results [EGA Th´ eor` eme III.3.1.2, Corollaire III.3.1.3] reduce the problem to show the following fact: (1) For each irreducible closed subset Z of X , with generic po int z, there exists a coherent O X -module F such that F z = 0 and R q f ∗ (F ) is coherent for q ≥ 0. To prove (1), we can suppose that Z = X is integral, i.e., it suffices to prove: (2) If X is integral with generic point x, there exists a coherent O X -module F such that F x = 0 and R q f ∗ (F ) is coherent for q ≥ 0. To prove that (2) ⇒ (1), we consider Z as reduced (and hence integral) closed subscheme of X and take the corresponding closed immersion j : Z −→ X . Since f ◦ j is a proper morphism, we know that there exists a coherent O Z -module G such that G z = 0 and R q (f ◦ j ) ∗ (G ) is coherent for q ≥ 0. Now, the O X -module F = j ∗ (G ) is coherent by [L, 5.1.14 d] and it satisfies (1). The fact that R q f ∗ (F ) is coherent is a consequence of the equality R q (f ◦ j ) ∗ (G ) = R q f ∗ ( j ∗ (G )), which can be proved by using that j ∗ is an exact functor and the same argument we use for proving (3) below. Now let us prove (2): By Chow’s lemma, there exists a projecti ve and surjective morphism g : X −→ X such that f ◦ g : X −→ Y is also projective. Let O X (1) be a very ample sheaf on X with respect to g. By [H, III.8.8] there exists an integer n ≥ 1 such that F = g ∗ (O X (n)) is a coherent O X -module, the natur al map g ∗ g ∗ (O X (n)) −→ O X (n) is surjective and R q g ∗ (O X (n)) = 0 for q ≥ 1. The surjectivity of g ∗ F −→ F implies that F x = 0. Now it suffices to prove that (3) R q f ∗ (F ) = R q (f ◦ g) ∗ (O X (n)), since the O Y -module s on the right are coher ent by [H, II I.8.8] . (This is the only step of the proof where spectral sequences are used in [EGA].) We start from an injective resolution of O X (n). By definition of deriv ed functors, the fact that R q g ∗ (O X (n)) = 0 means that the sequence remains exact if we apply to it the functor g ∗ , and so, we get a resolution of F , which is obvi ous ly flasque. Hen ce, it can be use d to calcu late the right hand side of (3). (Se e [H, III.8.3 and III.1.2 A]. ) So, if we appl y the functor f ∗ and take the cohomology groups, we are calculating both sides of (3), and this proves (2). References [EGA] A. Grothe ndieck, J. Dieudon n´ e, ´ Ele ments de G´ eom´ et ri e Alg´ ebri que , Publ. Math. IHES, 4, 8, 11, 17, 20, 24, 28, 32, 1960–1967. [H] R. Hartshorne, Algebraic Geometry, Grad. Texts Math., 52, Springer, New York–Heidelberg–Berlin, 1977. [L] Q. Liu, Algebraic Geometry and Arithmetic Curves, Oxford Grad. Texts Mat., 6, Oxford University Press, Oxford, 2002.

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In this note, we show that the proof of [EGA Theoreme III.3.2.1] can beslightly modified to avoid spectral sequences. The statement of the theorem isas follows:

Let Y be a locally Noetherian scheme and f : X −→ Y a proper morphism.

For each coherent OX-module F , the OY -modules Rqf ∗(F ) are coherent for

q ≥ 0.

The previous results [EGA Theoreme III.3.1.2, Corollaire III.3.1.3] reducethe problem to show the following fact:

(1) For each irreducible closed subset Z of X , with generic point z, there

exists a coherent OX-module F such that F z = 0 and Rqf ∗(F ) is coherent for

q ≥ 0.

To prove (1), we can suppose that Z = X is integral, i.e., it suffices to prove:

(2) If X is integral with generic point x, there exists a coherent OX-module

F such that F x = 0 and Rqf ∗(F ) is coherent for q ≥ 0.

To prove that (2) ⇒ (1), we consider Z as reduced (and hence integral) closedsubscheme of X and take the corresponding closed immersion j : Z −→ X . Sincef ◦ j is a proper morphism, we know that there exists a coherent OZ -module G

such that G z = 0 and Rq(f ◦ j)∗(G ) is coherent for q ≥ 0. Now, the OX-moduleF = j∗(G ) is coherent by [L, 5.1.14 d] and it satisfies (1). The fact that Rqf ∗(F )is coherent is a consequence of the equality Rq(f ◦ j)∗(G ) = Rqf ∗( j∗(G )), whichcan be proved by using that j∗ is an exact functor and the same argument weuse for proving (3) below.

Now let us prove (2): By Chow’s lemma, there exists a projective and surjective morphism g : X −→ X such that f ◦ g : X −→ Y is also projective.Let OX (1) be a very ample sheaf on X with respect to g. By [H, III.8.8] thereexists an integer n ≥ 1 such that F = g∗(OX(n)) is a coherent OX-module, thenatural map g∗g∗(OX(n)) −→ OX(n) is surjective and Rqg∗(OX(n)) = 0 forq ≥ 1.

The surjectivity of g∗F −→ F implies that F x = 0. Now it suffices to provethat

(3) Rqf ∗(F ) = Rq(f ◦ g)∗(OX(n)),

since the OY -modules on the right are coherent by [H, III.8.8]. (This is the onlystep of the proof where spectral sequences are used in [EGA].)

We start from an injective resolution of OX (n). By definition of derivedfunctors, the fact that Rqg∗(OX(n)) = 0 means that the sequence remainsexact if we apply to it the functor g∗, and so, we get a resolution of F , whichis obviously flasque. Hence, it can be used to calculate the right hand sideof (3). (See [H, III.8.3 and III.1.2A].) So, if we apply the functor f ∗ and takethe cohomology groups, we are calculating both sides of (3), and this proves (2).

References

[EGA] A. Grothendieck, J. Dieudonne, ´ Elements de Geometrie Algebrique, Publ.Math. IHES, 4, 8, 11, 17, 20, 24, 28, 32, 1960–1967.

[H] R. Hartshorne, Algebraic Geometry, Grad. Texts Math., 52, Springer, NewYork–Heidelberg–Berlin, 1977.

[L] Q. Liu, Algebraic Geometry and Arithmetic Curves, Oxford Grad. TextsMat., 6, Oxford University Press, Oxford, 2002.