The Stacks project

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

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

\[ e_ I(M, r) = \sum (-1)^ i\text{length}_ A H_ i(K_\bullet (f_1, \ldots , f_ r) \otimes _ A M) \]

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

\[ F^ p(K^ n \otimes _ A M) = I^{\max (0, p + n)}(K^ n \otimes _ A M),\quad p \in \mathbf{Z} \]

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

\[ E_0 = \bigoplus \nolimits _{p, q} E_0^{p, q} = \bigoplus \nolimits _{p, q} \text{gr}^ p(K^{p + q} \otimes _ A M) = \text{Gr}_ I(K^\bullet \otimes _ A M) \]

Since $K^ n$ is finite free we have

\[ \text{Gr}_ I(K^\bullet \otimes _ A M) = \text{Gr}_ I(K^\bullet ) \otimes _{\text{Gr}_ I(A)} \text{Gr}_ I(M) \]

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

\begin{align*} \sum (-1)^ i\text{length}_ A(H^ i(K^\bullet \otimes _ A M)) & = \sum (-1)^{p + q} \text{length}_ A(E_\infty ^{p, q}) \\ & = \sum (-1)^{p + q} \text{length}_ A(E_1^{p, q}) \end{align*}

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

\[ \sum (-1)^{p + q} \text{length}_ A(E_1^{p, q}) = \sum \nolimits _ n \sum \nolimits _{p \leq t} (-1)^ n \text{length}_ A(\text{gr}^ p(K^ n \otimes _ A M)) \]

This is equal to

\[ \sum \nolimits _{n = -r, \ldots , 0} (-1)^ n{r \choose |n|} \chi _{I, M}(t + n) \]

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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