  Test ideals are an important concept in tight closure theory and their
behavior via flat base change can be very difficult to understand. Our paper
presents results regarding this behavior under flat maps with reasonably nice
(but far from smooth) fibers. This involves analyzing, in depth, a special type
of ideal of test elements, called the CS test ideal. Besides providing new
results, the paper also contains extensions of a theorem by G. Lyubeznik and K.
E. Smith on the completely stable test ideal and of theorems by F. Enescu and,
independently, M. Hashimoto on the behavior of F-rationality under flat base
change.
