Category equivalences involving graded modules over path algebras of quivers

Let kQ be the path algebra of a quiver Q with its standard grading. We show that the category of graded kQ-modules modulo those that are the sum of their finite dimensional submodules, QGr(kQ), is equivalent to several other categories: the graded modules over a suitable Leavitt path algebra, the modules over a certain direct limit of finite dimensional multi-matrix algebras, QGr(kQ') where Q' is the quiver whose incidence matrix is the n^{th} power of that for Q, and others. A relation with a suitable Cuntz-Krieger algebra is established. All short exact sequences in the full subcategory of finitely presented objects in QGr(kQ), split so that subcategory can be given the structure of a triangulated category with suspension functor the Serre degree twist (-1); it is shown that this triangulated category is equivalent to the "singularity category" for the radical square zero algebra kQ/kQ_{\ge 2}.Comment: Several changes made as a result of the referee's report. Added Lemma 3.5 and Prop. 3.6 showing that O is a generato

Integral Non-commutative Spaces

This paper introduces a notion of integrality that is suitable for non-commutative varieties. It is compatible with the usual notion of integrality for schemes. The function field and generic point of a non-commutative integral space are also defined. These agree with the usual notion for noetherian schemes. It is shown that these notions behave as one would wish. For example, various non-commutative analogues of projective space are integral.Comment: 14 page

Degenerate 3-dimensional Sklyanin algebras are monomial algebras

The 3-dimensional Sklyanin algebras, S(a,b,c), over a field k, form a flat family parametrized by points (a,b,c) lying in P^2-D, the complement of a set D of 12 points in the projective plane, P^2. When (a,b,c) is in D the algebras having the same defining relations as the 3-dimensional Sklyanin algebras are said to be "degenerate". Chelsea Walton showed the degenerate 3-dimensional Sklyanin algebras do not have the same properties as the non-degenerate ones. Here we prove that a degenerate Sklyanin algebra is isomorphic to the free algebra on u,v,w, modulo either the relations u^2=v^2=w^2=0 or the relations uv=vw=wu=0. These monomial algebras are Zhang twists of each other. Therefore all degenerate Sklyanin algebras have the same category of graded modules. A number of properties of the degenerate Sklyanin algebras follow from this observation. We exhibit a quiver Q and an ultramatricial algebra R such that if S is a degenerate Sklyanin algebra then the categories QGr(S), QGr(kQ), and Mod(R), are equivalent; neither Q nor R depends on S. Here QGr(-) denotes the category of graded right modules modulo the full subcategory of graded modules that are the sum of their finite dimensional submodules