algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety étale cohomology; values of L-functions; Grothendieck's purity conjecture; Riemann-Roch problem; Atiyah-Bott fixed point formula; Tate conjecture; construction of a motivic cohomology theory; algebraic K- theory; topological K-theory; algebraic objects; stable homotopy objects 