## 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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