Noncommutative Grassmannian of codimension two has coherent coordinate ring

TL;DR

This paper proves that the coordinate ring of a noncommutative Grassmannian of codimension two is coherent, using properties of a PBW-basis, which advances understanding of its algebraic structure.

Contribution

It demonstrates the coherence of the noncommutative Grassmannian's algebra specifically for codimension two, providing a new proof approach based on PBW-basis properties.

Findings

The algebra is coherent when codimension d=2.

Provides a new proof method distinct from recent approaches.

Establishes a t-structure on the derived category of the noncommutative Grassmannian.

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.