On General fiber product rings, Poincar\'e series and their structure

The present paper deals with the investigation of the structure of general fiber product rings RTS, where R, S and T are local rings with common residue field. We show that the Poincar\'e series of any R-module over the fiber product ring RTS is bounded by a rational function. In addition, we give a description of depth (RTS), which is an open problem in this theory. As a biproduct, using the characterization of the Betti numbers over RTS obtained, we provide certain cases of the Cohen-Macaulayness of RTS and, in particular, we show that RTS is always non-regular. Some positive answers for the Buchsbaum-Eisenbud-Horrocks and Total rank conjectures over RTS are also established.