Proof of `Stabilized ranks and the radical quotient` (1st statement)
What this proof establishes
no assumptions
Assuming the claims on the left, the claim on the right holds — checked by the archive's pipeline. Proof code is not displayed here.
Description
Flatness makes the low-degree representatives surject onto the radical quotient. Transfer the selector shifts through this surjection. Their pairing identities prove well-definedness, self-adjointness, idempotence, and commutativity using nondegeneracy of the quotient pairing.