Theorem 43.15.5. Let $A$ be a Noetherian local ring. Let $I = (f_1, \ldots , f_ r) \subset A$ be an ideal of definition. Let $M$ be a finite $A$-module. Then

[Theorem 1 in part B of Chapter IV, Serre_algebre_locale]

**Proof.**
Let us change the Koszul complex $K_\bullet (f_1, \ldots , f_ r)$ into a cochain complex $K^\bullet $ by setting $K^ n = K_{-n}(f_1, \ldots , f_ r)$. Then $K^\bullet $ is sitting in degrees $-r, \ldots , 0$ and $H^ i(K^\bullet \otimes _ A M) = H_{-i}(K_\bullet (f_1, \ldots , f_ r) \otimes _ A M)$. The statement of the theorem makes sense as the modules $H^ i(K^\bullet \otimes M)$ are annihilated by $f_1, \ldots , f_ r$ (More on Algebra, Lemma 15.28.6) hence have finite length. Define a filtration on the complex $K^\bullet $ by setting

Since $f_ i I^ p \subset I^{p + 1}$ this is a filtration by subcomplexes. Thus we have a filtered complex and we obtain a spectral sequence, see Homology, Section 12.24. We have

Since $K^ n$ is finite free we have

Note that $\text{Gr}_ I(K^\bullet )$ is the Koszul complex over $\text{Gr}_ I(A)$ on the elements $\overline{f}_1, \ldots , \overline{f}_ r \in I/I^2$. A simple calculation (omitted) shows that the differential $d_0$ on $E_0$ agrees with the differential coming from the Koszul complex. Since $\text{Gr}_ I(M)$ is a finite $\text{Gr}_ I(A)$-module and since $\text{Gr}_ I(A)$ is Noetherian (as a quotient of $A/I[x_1, \ldots , x_ r]$ with $x_ i \mapsto \overline{f}_ i$), the cohomology module $E_1 = \bigoplus E_1^{p, q}$ is a finite $\text{Gr}_ I(A)$-module. However, as above $E_1$ is annihilated by $\overline{f}_1, \ldots , \overline{f}_ r$. We conclude $E_1$ has finite length. In particular we find that $\text{Gr}^ p_ F(K^\bullet \otimes M)$ is acyclic for $p \gg 0$.

Next, we check that the spectral sequence above converges using Homology, Lemma 12.24.10. The required equalities follow easily from the Artin-Rees lemma in the form stated in Algebra, Lemma 10.51.3. Thus we see that

because as we've seen above the length of $E_1$ is finite (of course this uses additivity of lengths). Pick $t$ so large that $\text{Gr}^ p_ F(K^\bullet \otimes M)$ is acyclic for $p \geq t$ (see above). Using additivity again we see that

This is equal to

by our choice of filtration above and the definition of $\chi _{I, M}$ in Algebra, Section 10.59. The lemma follows from Lemma 43.15.4 and the definition of $e_ I(M, r)$. $\square$

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