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Finitistic dimension and restricted injective dimension. (English) Zbl 1363.18010

Let \(R\) be a ring and \(T\) a left \(R\)-module with \(A := \text{End}_{R}(T)\). Then \(T\) can be viewed as a right \(A\)-module. The restricted injective dimension, \(\text{rid}\,(T_{A})\), of \(T_{A}\) is defined as the supremum of the natural numbers \(m\) such that \(\text{Ext}^{m}_{A}(Q, T_{A}) \neq 0\) for some finitely generated projective \(A\)-module \(Q\). The paper under review establishes a connection between the restricted injective dimension of \(T_{A}\) and the right finitistic dimension of \(A\).
Before stating the main result, we recall that \(T\) is said to be self-orthogonal if \(\text{Ext}^{i}(T,T)=0\) for all \(i\geq 1\). Furthermore, \(\text{findim}(_{R}T)\) is defined as the supremum of the lengths of coresolutions of \(_{R}T\) by finitely generated modules from \(\text{add}_{R}T\). The following is the main result of the paper.
Theorem. In the above notation, assume that \(T\) is self-orthogonal as a left \(R\)-module. Then \[ \text{rid}\,(T_{A}) \leq \text{findim}(A_{A}) \leq \text{findim}(_{R}T) + \text{rid}\,(T_{A}). \] Several consequences of this result provide more transparent inequalities (and equalities) between the restricted injective dimension and the finitistic dimension.

MSC:

18G20 Homological dimension (category-theoretic aspects)
18G10 Resolutions; derived functors (category-theoretic aspects)
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References:

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