Direct limits of Jaffard domains and S-domains

It is proved under mild assumptions that the class of Jaffard domains and the class of S-domains are each stable under direct limit. New examples of Jaffard domains obtained thereby include the factorial domain of Fujita, and Nagata rings in arbitrarily many indeterminates over a Jaffard domain. New examples of S-domains are the polynomial rings in arbitrari ly many indeterminates over any domain. Also, any locally finite-dimensional directed union of universally catenarian going-down domains is itself a universally catenarian going-down domain. However, many related types of rings (such as [stably] strong S-domains or [universally] catenarian domains) are not preserved by direct l imit. Numerous examples i l lustrate the need for various hypotheses, the failure of various converses, etc., as well as the sharpness of bounds that we give for the dimension and the valuative dimension of a direct l imit.