Subtitle: Several examples in search of a theory

An infinite dimensional algebra may depend essentially on some parameters and yet be absolutely rigid in the classical deformation theory. However, the variability may be captured by the cohomolgy of an associated diagram of algebras. This is illustrated with several examples, the function field of a sphere with four marked points, that of the torus with one marked point, and the first Weyl algebra. While the theory is incomplete, these show that to understand the deformation of algebras one must consider not just that of single algebras but of diagrams even if at the start one was concerned with the deformation of but a single specific algebra.