The Seifert-van Kampen theorem and generalized free products of $S$-algebras

In a Seifert-van Kampen situation a path-connected space $Z$ may be written as the union of two open path-connected subspaces $X$ and $Y$ along a common path-connected intersection $W$. The fundamental group of $Z$ is isomorphic to the colimit of the diagram of fundamental groups of the three subspaces. In case the maps of fundamental groups are all injective, the fundamental group of $Z$ is a classical free product with amalgamation, and the integral group ring of the fundamental group of $Z$ is also a free product with amalgamation in the category of rings. In this case relations among the $K$-theories of the group rings have been studied. Here we describe a generalization and stablization of this algebraic fact, where there are no injectivity hypotheses on the fundamental groups and where we work in the category of $S$-algebras. Some of the methods we use are classical and familiar, but the passage to $S$-algebras blends classical and new techniques. Our most important application is a description of the algebraic $K$-theory of the space $Z$ in terms of the algebraic $K$-theories of the other three spaces and the algebraic $K$-theory of spaces $\operatorname{Nil}$-term.