  This paper is concerned with the prime spectrum of a tensor product of
algebras over a field. It seeks necessary and sufficient conditions for such a
tensor product to have the S-property, strong S-property, and catenarity. Its
main results lead to new examples of stably strong S-rings and universally
catenarian rings. The work begins by investigating the minimal prime ideal
structure. Throughout, several results on polynomial rings are recovered, and
numerous examples are provided to illustrate the scope and sharpness of the
results.
