On General fiber product rings, Poincaré series and their structure

Abstract

The present paper deals with the investigation of the structure of general fiber product rings $R{\times }_{T}S$, where $R$, $S$ and $T$ are local rings with common residue field. We show that the Poincaré series of any $R$-module over the fiber product ring $R{\times }_{T}S$ is bounded by a rational function. In addition, we give a description of $\mathrm{depth}(R{\times }_{T}S)$, which is an open problem in this theory. As a biproduct, using the characterization of the Betti numbers over $R{\times }_{T}S$ obtained, we provide certain cases of the Cohen-Macaulayness of $R{\times }_{T}S$ and, in particular, we show that $R{\times }_{T}S$ is always non-regular. Some positive answers for the Buchsbaum-Eisenbud-Horrocks and Total rank conjectures over $R{\times }_{T}S$ are also established.

Comments and suggestions are welcome