Graded algebras and their Euler-Poincaré series (Q1089389)

Let k be a field, R a connected graded k-algebra and M a graded R-module, both R and M finitely generated as k-modules in each degree. Let [M] be the Hilbert series of M, i.e., \([M]=\sum (\dim_ kM_ i)t^ i\quad and\) let \([H_*(R,M)]\) be \(\sum (-1)^ i\dim_ k(Tor_ i^ R(k,M))t^ i.\quad Then\) it is well known that \([M]=[R]\cdot [H_*(R,M)]\). This result is generalized in this article to the case where k is a commutative Noetherian regular ring. Also a result by Lemaire on the relation between the Hilbert series of the components of a fibre product is generalized in the same way. This also has a dual form concerning free products with amalgamation.