Noncommutative ampleness for multiple divisors

The twisted homogeneous coordinate ring is one of the basic constructions of the noncommutative projective geometry of Artin, Van den Bergh, and others. Chan generalized this construction to the multi-homogeneous case, using a concept of right ampleness for a finite collection of invertible sheaves and automorphisms of a projective scheme. From this he derives that certain multi-homogeneous rings, such as tensor products of twisted homogeneous coordinate rings, are right noetherian. We show that right and left ampleness are equivalent and that there is a simple criterion for such ampleness. Thus we find under natural hypotheses that multi-homogeneous coordinate rings are noetherian and have integer GK-dimension.Comment: 11 pages, LaTeX, minor corrections, to appear in J. Algebr

D-modules on rigid analytic spaces

We give an overview of the theory of \wideparen{\mathcal{D}}-modules on rigid analytic spaces and its applications to admissible locally analytic representations of $p$-adic Lie groups.Comment: to appear in the Proceedings of the IC

Noncommutative Grassmannian of codimension two has coherent coordinate ring

A noncommutative Grassmannian NGr(m, n) is introduced by Efimov, Luntz, and Orlov in `Deformation theory of objects in homotopy and derived categories III: Abelian categories' as a noncommutative algebra associated to an exceptional collection of n-m+1 coherent sheaves on P^n. It is a graded Calabi--Yau Z-algebra of dimension n-m+1. We show that this algebra is coherent provided that the codimension d = n-m of the Grassmannian is two. According to op. cit., this gives a t-structure on the derived category of the coherent sheaves on the noncommutative Grassmannian. The proof is quite different from the recent proofs of the coherence of some graded 3-dimensional Calabi--Yau algebras and is based on properties of a PBW-basis of the algebra.Comment: Unfortunately, an error appeared in the Groebner basis calculation. I am grateful to Alexander Efimov who have pointed this out. The algebra menioned in the title consideration is indeed PBW, but the proof of the restricted processing property fails. So, it is still an open problem is this algebra coherent or no