Let $G$ be a group with identity $e$ and $R$ be a commutative $G$-graded ring with nonzero unity $1$. In this article, we introduce the concept of graded $r$-ideals. A proper graded ideal $P$ of a graded ring $R$ is said to be graded $r$-ideal if whenever $a, bin h(R)$ such that $abin P$ and $Ann(a)={0}$, then $bin P$. We study and investigate the behavior of graded $r$-ideals to introduce ...

• In Section 4, we define locally homogeneous singularities. A space with locally homogeneous singularities (SLHS) is an analytic space X such that for all x ∈ X , the x-completion of a local ring OxX is isomorphic to an x-completion of associated graded ring (OxX)gr. We show that a complex variety is SLHS if and only if the underlying real analytic variety is SLHS. This allows us to define inv...

Let X ⊂ PK be Drinfeld’s upper half space over a finite extension K of Qp. We construct for every GLd+1-equivariant vector bundle F on PK , a GLd+1(K)equivariant filtration by closed subspaces on the K-Fréchet H0(X ,F). This gives rise by duality to a filtration by locally analytic GLd+1(K)-representations on the strong dual H0(X ,F)′. The graded pieces of this filtration are locally analytic i...

Let K[〈X〉] denote the large polynomial ring (in the sense of Halter-Koch [7]) on the set X = {x1, x2, x3, . . . } of indeterminates. For each integer n, there is a truncation homomorphism ρn : K[〈X〉] → K[x1, . . . , xn]. If I is a homogeneous ideal of K[〈X〉], then the N-graded Hilbert series of K[x1,...,xn] ρn(I) can be written as gn(t) (1−t) ; it was shown in [13] that if in addition I is what...

Let X ⊂ PdK be Drinfeld’s upper half space over a finite extension K of Qp. We construct for every GLd+1-equivariant vector bundle F on PdK , a GLd+1(K)equivariant filtration by closed subspaces on the K-Fréchet H(X ,F). This gives rise by duality to a filtration by locally analytic GLd+1(K)-representations on the strong dual H(X ,F). The graded pieces of this filtration are locally analytic in...