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Finitistic dimension and restricted flat dimension. (English) Zbl 1160.16003

The main result of this paper is the following statement: Theorem. Let \(R\) be a ring and \(T\) an \(R\)-module. Denote \(A=\text{End}_R(T)\).
(1) If \(T\) is selfsmall and coproduct-selforthogonal in the category of all \(R\)-modules, then the restricted flat dimension of \(T_A\) does not exceed the finitistic dimension of \(_AA\), which, in turn, is bounded by the restricted flat dimension of \(T_A\) plus the injective dimension of \(\text{Add}_RT\).
(2) If \(T\) is selforthogonal in the category of all finitely generated \(R\)-modules, then the restricted flat dimension of \(T_A\) does not exceed the finitistic dimension of \(_AA\), which, in turn, is bounded by the restricted flat dimension of \(T_A\) plus the injective dimension of \(_RT\).
(3) If \(T\) is selfsmall and coproduct-selforthogonal in the category of all \(R\)-modules and has finite projective dimension, then the restricted flat dimension of \(T_A\) does not exceed the finitistic dimension of \(_AA\), which, in turn, is bounded by the restricted flat dimension of \(T_A\) plus the finitistic dimension of \(_RR\).
(4) If \(T\) is selforthogonal in the category of all finitely generated \(R\)-modules and has finite projective dimension, then the finitistic dimension of \(_AA\) does not exceed the finitistic dimension of \(_RR\) plus the right flat dimension of \(T_A\).
This is used to disprove Foxby’s conjecture on restricted flat dimension [from H. Holm, J. Pure Appl. Algebra 189, No. 1-3, 167-193 (2004; Zbl 1050.16003)] and to give a partial answer to the question about the relation between the finitistic dimensions of \(_RR\) and that of \(_AA\) [from V. Mazorchuk, Algebra Discrete Math. 2004, No. 3, 77-88 (2004; Zbl 1067.16008)].

MSC:

16E10 Homological dimension in associative algebras
16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
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