Proposition 11

Suppose A left noetherian, and let C be a bounded and flat complex of left a-modules such that H(C) is a finitely generated a-module. Let a and b be two integers such that a \leq b and H_n(C) = 0 for n < a, C_n = 0 for n > b. There exists a complex P of left a-modules such that P_n is projective and finitely generated for each n, and P_n = 0 for n \notin [a, b], and a homologism u : P \to C. Moreover, for every complex E of right a-modules, the homomorphism

By X, p. 53, prop. 7, there exists a complex (L, d) such that L_n is free and finitely generated for each n, and zero when n < a, and a homologism f : L \to C. Let P be the quotient complex L/L', where L'_n = 0 for n < b, L'_n = L_n for n > b, L'_b = B_b(L). Since C_n = 0 for n > b, f(L') = 0, so f factors through a morphism of complexes u : P \to C.

Since $f$ is a homologism, one has $H(\operatorname{Con}(f))=0$, whence an exact sequence

One then has an exact sequence

This shows on the one hand that the cone of $u$ is of zero homology, so that $u$ is a homologism, on the other hand that the module $P_b$ is flat (X, p. 76, cor. 2) ; since $P_b$ is finitely generated as a quotient of $L_{b+1}$, it is projective (X, p. 13, cor.). The pair $(P,u)$ then satisfies the required condition. The last assertion follows from X, p. 79, cor. 5.

\* Example. — Let A be a commutative noetherian ring, X a proper and flat A-scheme, $\mathscr F$ a coherent $\mathscr C_X$-module, flat over A. There exists a bounded complex P consisting of finitely generated projective A-modules such that for every A-module M, $H(X,\mathscr F\otimes_A M)$ identifies naturally with $H(P\otimes_A M)$. Indeed, let $\mathscr U$ be a covering of X by a finite number of affine open sets, $\mathscr C(\mathscr U,\mathscr F)$ the associated Čech complex. One shows that $H^i(\mathscr C(\mathscr U,\mathscr F))$ is isomorphic to the A-module $H^i(X,\mathscr F)$, and that the latter is finitely generated; moreover, for every A-module M, the complex $\mathscr C(\mathscr U,\mathscr F)\otimes_A M$ is isomorphic to $\mathscr C(\mathscr U,\mathscr F\otimes_A M)$. Applying Prop. 11 to the complex $\mathscr C(\mathscr U,\mathscr F)$ (which is bounded), one obtains a complex P that answers the question.

For every point $y$ of Spec (A), let $\kappa(y)$ denote the residue field of A at $y$, $X_y=X\otimes_A\kappa(y)$ the fibre of X over $y$, $\mathscr F_y=\mathscr F\otimes_A\kappa(y)$, and set $h^p(y)=\dim_{\kappa(y)}H^p(X_y,\mathscr F_y)$ for $p\geq 0$.

One easily deduces the following results from the existence of the complex P :

(i) the function $h^p$ is upper semi-continuous on Spec (A) ;

(ii) the function $\displaystyle\sum_{p\geq 0}(-1)^p h^p$ is locally constant on Spec (A). \*