The Grothendieck group of non-commutative non-noetherian analogues of P1 and regular algebras of global dimension two

Let V be a finite-dimensional positively-graded vector space. Let b∈V⊗V be a homogeneous element whose rank is dim⁡(V). Let A=TV/(b), the quotient of the tensor algebra TV modulo the 2-sided ideal generated by b. Let gr(A) be the category of finitely presented graded left A-modules and fdim(A) its full subcategory of finite dimensional modules. Let qgr(A) be the quotient category gr(A)/fdim(A). We compute the Grothendieck group K0(qgr(A)). In particular, if the reciprocal of the Hilbert series of A, which is a polynomial, is irreducible, then K0(qgr(A))≅Z[θ]⊂R as ordered abelian groups where θ is the smallest positive real root of that polynomial. When dimk⁡(V)=2, qgr(A) is equivalent to the category of coherent sheaves on the projective line, P1, or a stacky P1 if V is not concentrated in degree 1. If dimk⁡(V)≥3, results of Piontkovski and Minamoto suggest that qgr(A) behaves as if it is the category of “coherent sheaves” on a non-commutative, non-noetherian analogue of P1.