Noncommutative Grassmannian of codimension two has coherent coordinate ring

Abstract

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