Lower bounds for Betti numbers over fiber product rings

In this paper we study the behavior of the Betti numbers of a finitely generated R-module over the fiber product ring R×TS, with R, S and T being local rings with common residual field. We investigate the Buchsbaum-Eisenbud-Horrocks and Total Rank Conjectures over R×TS, and as a consequence, positive answers are given for a certain class of quotients of power series rings. A positive answer for the question posed by Jorgensen and Leuschke is also provided.