Stratification

Abstract

We continue the discussion of how an action of a group by automorphisms on a ring helps in analyzing the prime spectrum. If, sayHis a group acting on a ring R, then we can partition spec Rinto H-orbits. This partition is nice for theoretical purposes, but in practice it can be very difficult to deal with — there may be infinitely many H-orbits in spec R, and they can be hard to calculate. It proves useful to look for a coarser partition. Note that if Pand P’are prime ideals of Rwith the same H-orbit, then

$$ (P:H) = \cap \{ Q\left| {Q \in H.P} \right.\} = \cap \{ Q'\left| {Q' \in H.P} \right.'\} = (P':H). $$

Following this hint, we partition spec Rinto “H-strata”, where prime ideals Pand P’belong to the same H-stratum if and only if (P: H) = (P’: H).}