Lower bounds for Betti numbers over fiber product rings

Abstract

The present paper deals with the investigation of the structure of general fiber product rings R × 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 × T S is bounded by a rational function. In addition, we give a description of depth(R × T S), which is an open problem in this theory. As a biproduct, using the characterization of the Betti numbers over R × T S obtained, we provide certain cases of the Cohen-Macaulayness of R × T S and, in particular, we show that R × T S is always non-regular. Some positive answers for the Buchsbaum-Eisenbud-Horrocks and Total rank conjectures over R × T S are also established.2010 Mathematics Subject Classification. 13D02.