N i l -closed Noetherian sub-algebras of H * ( W ) and their centres

For 𝒢 some groupoid whose objects are the sub-vector spaces of a 𝔽 p -vector space W , we define H * ( W ) 𝒢 a n i l -closed, noetherian, unstable sub-algebra of H * ( W ) over the Steenrod algebra. The application on the appropriate ordered set of groupoids, that maps 𝒢 to H * ( W ) 𝒢 defines an isomorphism of posets to the set of noetherian, n i l -closed, unstable sub-algebras of H * ( W ) of transcendence degree dim ( W ) , ordered by inclusion. Since any noetherian and integral unstable algebra of transcendence degree dim ( W ) admits an injection into H * ( W ) , any such n i l -closed unstable algebra is isomorphic to some H * ( W ) 𝒢 . We prove that 𝒢 encodes the centre, in the sense of Heard, of H * ( W ) 𝒢 . Also, there is a H * ( C ) -comodule structure on K that is associated with the centre of K . For K = H