Finite ambient algebras do not control their subalgebras

Exact statement

Finite generation of an ambient commutative algebra does not imply finite generation of each of its subalgebras.

StatusProved here by the explicit factorial subalgebra; applied to rule out an invalid proof step

External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

The control algebra is finitely generated, but the derivative algebra generated by the output can be a non-finitely-generated subalgebra.

Proof or evidence

The factorial construction realizes this obstruction inside the automata model: the finite algebra generated by the two controls contains an infinitely generated derivative algebra of the output.

Failure of the subalgebra argument →

Verification notes

The grading argument for the factorial example was rechecked and does not use a false Noetherianity claim for subalgebras.

Limitations

This does not rule out a different finite commutative over-algebra.

Open work

Keep finite over-algebras separate from the output's derivative-generated subalgebra in future arguments.