Test elements in equal characteristic semianalytic algebras

We establish a series of results showing that the Jacobian ideal is contained in the test ideal.

After proving a new result in characteristic $p$ for complete rings over a field $K$, we extend this containment to equal characteristic 0 for affine, analytic, affine-analytic and semianalytic $K$-algebras.

We also prove a result on equiheight algebras essentially of finite type over excellent local rings.