On the Poincare' Series of a Semigraded Module

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In his paper [10], M. Steurich introduced the notion of the semigraded local ring as a generalized concept of a power series ring over a field and in [9], we studied the homogeneous Golod homomorphism and some change of ring theorems about Poincare series for semigraded rings. In this paper, we investigate how the properties of inert modules introduced by J. Lescot [7] which are connected with Golod homomorphisms can be transfered to the semigraded case. Throughout the paper, all rings are commutative and Noetherian, and the symbol (R, m, k) stands for R is a local ring with maximal ideal rrt and residue field k. By definition, a local ring (R, m, k) is semigraded if (i) R=П^^∞__<i=0> R_i as an abelian group, (ii) R_iR_j⊆R_<i+j>. An R-module M is called semigraded if M satisfies the conditions: (i) M=П^^∞__<i=0> M_i, (ii) R_iM_j⊆M_<i+j>. For any semigraded (abbreviated by s.g.) R-modules M and N, K-homomorphism f:M→N is said to be homogeneous of degree d≧0 if(i)f(M_i)⊆N_<i+d> for all i, (ii)f(Σ^^∞__<i=0> xi)=Σ^^∞__<i=0>f(x_i) where x_i∈M_i. Since there exists a minimal free resolution for a finitely generated s.g. R-module M, this extends to a grading on the modules Tor^R_i (k, M) and we can define the Poincare series P^M_R(X, Y) of M as the power series in two variables X and Y: P^M_R(X, Y)=Σ_<i,j≧0> dim_k Tor^R_<i,j> (k,M)X^iY^j where Torfy^R_<i,j>(k, M) is the j-th homogeneous component of Tor^R_i (k, M). For the detail of the definitions and results the reader is referred to [9], [10]. Unless otherwise specified, we shall use the same notations and the same terminology which appeared in [9].
徳島大学の論文
1985-10-30

On the Poincare' Series of a Semigraded Module

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