Exercise 6, § 4

Commutative Algebra, VIII, p. 90

Let H be a graded ring of type Z, such that H_n = \{ 0 \} for n < 0, that H_0 is a local artinian ring and that H is a finitely generated H_0-algebra. Let L. be a bounded complex of finitely generated graded free H-modules (cf. A, X, p. 56). Thus for each i, L_i is isomorphic to an H-module of the form $\bigoplus_{j=1}^{r_i} H(-n_{ij})$ , with $n_{ij} \in \mathbf{Z}$ , where H(−k) is the graded H-module such that H(−k)_n = H_{n−k}. For every finitely generated graded H-module M, denote by G_M \in \mathbf{Q}[T] the unique polynomial such that there exists an integer $n_1 \geq 0$ satisfying

G_M(n) = \sum_{i<n} \operatorname{long}_{H_0}(M_i), \quad \text{for } n \geq n_1.

Prove the relation

\sum_{i \geq 0} (-1)^i G_{H_i(L.)} = \sum_{k \geq 0} \frac{(-1)^k}{k!} \rho_k \frac{\partial^k G_H}{\partial T^k}

where $\rho_k = \sum_{i \geq 0} (-1)^i \sum_{j=1}^{r_i} n_{ij}$ , the $n_{ij}$ being the degrees associated with L_i, and where H_i(L.) denotes the homology of the complex L..

Solution

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