The Grothendieck group of non-commutative non-noetherian analogues of $\mathbb{P}^1$ and regular algebras of global dimension two

Abstract

Let $V$ be a finite-dimensional positively-graded vector space. Let $b \in V \otimes V$ be a homogeneous element whose rank is $\text{dim}(V)$. Let $A=TV/(b)$, the quotient of the tensor algebra $TV$ modulo the 2-sided ideal generated by $b$. Let ${\sf gr}(A)$ be the category of finitely presented graded left $A$-modules and ${\sf fdim}(A)$ its full subcategory of finite dimensional modules. Let ${\sf qgr}(A)$ be the quotient category ${\sf gr}(A)/{\sf fdim}(A)$. We compute the Grothendieck group $K_0({\sf qgr}(A))$. In particular, if the reciprocal of the Hilbert series of $A$, which is a polynomial, is irreducible, then $K_0({\sf qgr}(A)) \cong \mathbb{Z}[\theta] \subset \mathbb{R}$ as ordered abelian groups where $\theta$ is the smallest positive real root of that polynomial. When $\text{dim}_k(V)=2$, ${\sf qgr}(A)$ is equivalent to the category of coherent sheaves on the projective line, $\mathbb{P}^1$, or a stacky $\mathbb{P}^1$ if $V$ is not concentrated in degree 1. If $\text{dim}_k(V) \ge 3$, results of Piontkovskii and Minamoto suggest that ${\sf qgr}(A)$ behaves as if it is the category of "coherent sheaves" on a non-commutative, non-noetherian, analogue of $\mathbb{P}^1$.