algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety \(G\)-scheme; 2-cocycle; semidirect product algebra; twisted group algebra; equivariant algebraic \(K\)-theory; twisted projective homogeneous scheme; full exceptional collection; equivariant motivic measure; noncommutative algebraic geometry; \(G\)-equivariant Chow motive; \(G\)-equivariant perfect complex; noncommutative Chow motives 