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 is an element of V circle times 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 K-o(qgr(A)). In particular, if the reciprocal of the Hilbert series of A, which is a polynomial, is irreducible, then K-o(qgr(A)) congruent to Z[theta] subset of R as ordered abelian groups where theta is the smallest positive real root of that polynomial. When dim(k)(V) = 2, qgr(A) is equivalent to the category of coherent sheaves on the projective line, P-1, or a stacky P-1 if V is not concentrated in degree 1. If dim(k)(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 P-1. (C) 2014 Elsevier Inc. All rights reserved.