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)
[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
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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