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 $\mathsf{gr}(A)$ be the category of finitely presented graded left $A$-modules and $\mathsf{fdim}(A)$ its full subcategory of finite dimensional modules. Let $\mathsf{qgr}(A)$ be the quotient category $\mathsf{gr}(A)/\mathsf{fdim}(A)$. We compute the Grothendieck group $K_0(\mathsf{qgr}(A))$. In particular, if the reciprocal of the Hilbert series of $A$, which is a polynomial, is irreducible, then $K_0(\mathsf{qgr}(A))\cong \mathbb{Z}[\theta ]\subset \mathbb{R}$ as ordered abelian groups where $\theta$ is the smallest positive real root of that polynomial. When ${\text{dim}}_k(V)=2$, $\mathsf{qgr}(A)$ 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 $\mathsf{qgr}(A)$ behaves as if it is the category of "coherent sheaves" on a non-commutative, non-noetherian, analogue of ${\mathbb{P}}^1$.

18 pages. Expanded introduction and small changes and corrections following the referee's suggestions