algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety reconstruction theorem; abelian category; noncommutative algebraic geometry; quasi-coherent sheaves; automorphism class group; quasi-separated scheme; the spectrum of $\mathcal A$; equivalence of groupoids; automorphism class group; derived category of coherent sheaves; tensor triangulated category of perfect complexes; Tannaka duality 