Poincaré series of fiber products and weak complete intersection ideals

We study the Poincaré series of modules over a fiber product of commutative local rings. We introduce the notion of a weak complete intersection ideal; these are the ideals with the property that every differential in their minimal free resolutions can be represented by a matrix whose entries are in the ideal itself. We show that many properties of the Poincaré series that are known to hold for a fiber product over the maximal ideal also hold for those over weak intersection ideals.