The non-commutative scheme having a free algebra as a homogeneous coordinate ring

Abstract: Let k be a field and TV the tensor algebra on a k-vector space V of dimension n>1. This paper proves that the quotient category QGr(TV) := Gr(TV)/Fdim of graded TV-modules modulo those that are unions of finite dimensional modules is equivalent to the category of modules over the direct limit of matrix algebras, M_n(k)^{\otimes r}. QGr(TV) is viewed as the category of "quasi-coherent sheaves" on the non-commutative scheme Proj(TV). The subcategory qgr(TV) consisting of the finitely presented objects is viewed as the category of coherent sheaves on Proj(TV). We show qgr(TV) has no indecomposable objects, no noetherian objects, and no simple objects. Moreover, every short exact sequence in qgr(TV) splits. The equivalence of categories result can be interpreted as saying that Proj(TV) is an "affine non-commutative scheme".