Generalized Macaulay-Northcott Modules and Tor-Groups

Abstract:

      Let $(S, \leq)$ be a strictly totally ordered monoid which is also artinian, and $R$ a right noetherian ring. Assume that $M$ is a finitely generated right $R$-module and $N$ is a left $R$-module. Denote by $[[M^{S, \leq}]]$ and $[N^{S, \leq}]$ the module of generalized power series over $M$, and the generalized Macaulay-Northcott module over $N$, respectively. Then we show that there exists an isomorphism of Abelian groups: $$\Tor_i^{[[R^{S, \leq}]]}([[M^{S, \leq}]], [N^{S,\leq}])\cong \bigoplus_{s\in S}\Tor_i^R(M, N).$$