Recursive Sequences and Faithfully Flat Extensions

This note arises from an attempt to give some model-theoretic interpretation of the concept of flatness. It is well-known that a necessary condition for a ring morphism A → B to be faithfully flat, is that any linear system of equations with coefficients from A which has a solution over B, must have already a solution over A. In fact, if we strenghten this condition to any solution over B comes from solutions over A by base change, then this becomes also a sufficient condition for being faithfully flat. However, whereas the first (necessary) condition is reminiscent of the model-theoretic notion of existentially closedness, the second seems to have no model-theoretic counterpart. Recall that a subring A of a ring B is said to be existentially closed in B, if any (not necessarily linear) system of equations with coefficients from A which is solvable over B is already solvable over A. This is the relative version of this concept, the absolute version reads: a ring A in a class of rings K is called existentially closed or generic for that class, if for any overring B ∈ K we have that A is existentially closed in B. So, paraphrasing this notion, one could say that if A ⊂ B is faithfully flat, then A is existentially closed in B with respect to linear equations. But as already observed, this is not a sufficient condition to guarantee faithful flatness. I will present a property of rings which is a consequence of faithfully flatness, but presumably stronger than existentially closedness for linear equations. The key definition is that of a (linear) recursive sequence (xn)n over a ring A, as a sequence satisfying some fixed linear relation over A among t consecutive terms. We will show that if A → B is faithfully flat and (xn)n is a sequence of elements in A satisfying a linear recursion relation with coefficients in B, then it already satisfies such a recursion relation (of the same length) with coefficients in A. As there is a strong connection between recursive sequences and rational power series, we obtain the following corollary. Assume, moreover, that A and B are normal domains, then any power series over A which is rational (meaning that it can be written as a quotient of two polynomials) over B, is already rational over A. Any direct attempt, however, to prove this corollary just using faithfully flatness seems to fail as far as I can tell.