Exercise 1, § 4

Let Γ be a commutative group and H a graded ring of type Γ (A, II, p. 164). For γ ∈ Γ, we denote by H_γ the homogeneous component of degree γ of H. We suppose that H is generated by H_0 and a finite family (x_i)_{i∈I} of homogeneous elements of degree ≠ 0. We denote by γ_i the degree of x_i. For every commutative group K, we denote by K[[Γ]] the product group K^Γ and we designate by $\sum_{\gamma\in\Gamma} m_\gamma T^\gamma$ the element (m_γ) of K^Γ. We denote by K[Γ] the subgroup K^{(Γ)} of K[[Γ]]. We equip K[[Γ]] with its natural structure of module over the algebra Z[Γ] of the group Γ. Finally, let C be a hereditary set of classes of H_0-modules (A, VIII, § 10, no 1, def. 1) and K(C) the corresponding grothendieck group (loc. cit., no 2). For every H_0-module E of type C, we denote by [E] its class in K(C). a) Suppose that H_0 is noetherian, and let M be a finitely generated graded H-module such that for every γ ∈ Γ, the H_0-module M_γ is of type C. Now prove that the element $\prod_{i\in\Gamma} (1 - T^{\gamma_i}) \sum_{\gamma\in\Gamma} [M_\gamma] T^\gamma$ of K(C)[[Γ]] belongs to K(C)[Γ].

b) Suppose now that one has $\Gamma = Z^s$, $H_\gamma = 0$ and $M_\gamma = 0$ for $\gamma \notin N^s$, and that the $\gamma_i$ are basis vectors of $Z^s$; denoting by $a_k$ the number of $\gamma_i$ which are equal to the $k$-th basis vector, proving that one has

where $T = (T_1, ..., T_s)$, and $(m_\gamma)$ is a family with finite support. Deduce the existence of elements $c_b$, for $b \in Z^s$, of $K(C)$, zero except for a finite number of $b$ and such that, for $\gamma \in N^s$ sufficiently large, one has

where

the sum being taken over the elements $b$ of $Z^s$ satisfying $b_i < a_i$ for $1 \leq i \leq s$.