Big Finitistic Dimensions for Categories of Quiver Representations

Document Type : Original Scientific Paper

Authors

1 Department of Mathematics, Isfahan University of Technology, Isfahan, I. R. Iran

2 Department of Pure Mathematics, Faculty of Mathematics and Computer Sciences, Shahid Chamran University, Ahvaz, Iran

10.22052/mir.2021.240439.1273

Abstract

Assume that A is a Grothendieck category and R is the category of all A-representations of a given quiver Q. If Q is left rooted and A has a projective generator, we prove that the big finitistic flat (resp. projective) dimension FFD(A) (resp. FPD(A)) of A is finite if and only if the big finitistic flat (resp. projective) dimension of R is finite. When A is the Grothendieck category of left modules over a unitary ring R, we prove that if FPD(R) < +∞ then any representation of Q of finite flat dimension has finite projective dimension. Moreover, if R is n-perfect then we show that FFD(R) < +∞  if and only if FPD(R) < +.
 

Keywords


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