Hopf Modules, Coinvariants, and the Fundamental Theorem — Examples

Compute the projection and inverse decomposition for V⊗kG and V⊗k[x]. Include a compatible module/comodule and a pair failing compatibility; the theorem cannot be invoked for arbitrary simultaneous structures.

These are planned worked examples and counterexamples for hopf-modules-coinvariants-and-the-fundamental-theorem, with empty item lists pending normal authoring and proof review. Each example must calculate from the stated hypotheses, and each counterexample must verify both its premises and the failed conclusion. This companion is a dependency leaf and supplies no theorem to another page.