algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety commutative cancellative torsion-free monoid; monoid ring; \(K\)-homotopy invariance; multiplicative action; nilpotence conjecture; toric variety; local-global patching; Mayer-Vietoris sequence for singular varieties; excision; pyramidal descent; big Witt vectors; nil-\(K\)-theory 