Global dimension of cartesian squares

AbstractThe local rings of global dimension two are characterized. Such rings are either noetherian rings, valuation rings, or the so-called umbrella rings, A local ring A is an umbrella ring if and only if A contains a prime ideal P such that: 1.(a) Ap is a valuation ring of global dimension one or two,2.(b) AP is a regular local ring of global dimension two,3.(c) PAp = P,4.(d) A has only countably many principal prime ideals.These results are generalized by defining an F-ring to be a domain A containing a prime ideal Q such that AQ is a valuation ring and QAQ = Q. We show that A is the fiber-product of AQ and AQ over AQQAQ and determine the global dimension of A in terms of the global dimensions of AQ and AQ