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

2014/03/04 by Sisodia, Gautam, Smith, S. Paul
#FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1403.0640

Abstract

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 \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 K₀(\sf qgr(A)). In particular, if the reciprocal of the Hilbert series of A, which is a polynomial, is irreducible, then K₀(\sf qgr(A)) ≅ ℤ[θ] ⊂ ℝ as ordered abelian groups where θ is the smallest positive real root of that polynomial. When dim_k(V)=2, \sf 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 dim_k(V) ≥ 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 ℙ¹.

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