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Category equivalences involving graded modules over weighted path algebras and weighted monomial algebras

Authors
  • Holdaway, Cody
  • Sisodia, Gautam1, 2
  • 1 Department of Mathematics
  • 2 Univ. Washington
Type
Published Article
Journal
Journal of Algebra
Publication Date
Jan 01, 2014
Volume
405
Pages
75–91
Identifiers
DOI: 10.1016/j.jalgebra.2013.12.032
Source
Elsevier
Keywords
License
Unknown

Abstract

Let k be a field, Q a finite directed graph, and kQ its path algebra. Make kQ an N-graded algebra by assigning each arrow a positive degree. Let I be an ideal in kQ generated by a finite number of paths and write A=kQ/I. Let QGrA denote the quotient of the category of graded right A-modules modulo the Serre subcategory consisting of those graded modules that are the sum of their finite dimensional submodules. This paper shows there is a finite directed graph Q′ with all its arrows placed in degree 1 and an equivalence of categories QGrA≡QGrkQ′. A result of Smith now implies that QGrA≡ModS, the category of right modules over an ultramatricial, hence von Neumann regular, algebra S.

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