On certain homological invariants of groups. (English) Zbl 1241.20058
The authors investigate properties of the algebraic invariants \(\text{silf\,}\mathbb ZG\) and \(\text{sfli\,}\mathbb ZG\) and related invariants, where \(\mathbb ZG\) is the integral group ring of the group \(G\), \(\text{sfli\,}\mathbb ZG\) is the supremum of the flat lengths of injective \(\mathbb ZG\)-modules and \(\text{silf\,}\mathbb ZG\) is the supremum of the injective lengths of flat \(\mathbb ZG\)-modules.
One of their results is that for any group \(G\), \(\text{sfli\,}\mathbb ZG\) equals the supremum of the injective lengths of the projective \(\mathbb ZG\)-modules. Using this they show that \(\text{sfli\,}\mathbb ZG\) is less than or equal to \(\text{silf\,}\mathbb ZG\), for any group \(G\). They investigate cases for which \(\text{sfli\,}\mathbb ZG\) finite implies \(\text{silf\,}\mathbb ZG\) finite and they also show that the Gorenstein flat dimension of the trivial \(\mathbb ZG\)-module \(\mathbb Z\) is related to these algebraic invariants.
One of their results is that for any group \(G\), \(\text{sfli\,}\mathbb ZG\) equals the supremum of the injective lengths of the projective \(\mathbb ZG\)-modules. Using this they show that \(\text{sfli\,}\mathbb ZG\) is less than or equal to \(\text{silf\,}\mathbb ZG\), for any group \(G\). They investigate cases for which \(\text{sfli\,}\mathbb ZG\) finite implies \(\text{silf\,}\mathbb ZG\) finite and they also show that the Gorenstein flat dimension of the trivial \(\mathbb ZG\)-module \(\mathbb Z\) is related to these algebraic invariants.
Reviewer: Olympia Talelli (Athens)
MSC:
| 20J05 | Homological methods in group theory |
| 16E10 | Homological dimension in associative algebras |
| 16S34 | Group rings |
| 20C07 | Group rings of infinite groups and their modules (group-theoretic aspects) |
| 16E30 | Homological functors on modules (Tor, Ext, etc.) in associative algebras |
| 16D40 | Free, projective, and flat modules and ideals in associative algebras |
| 16D50 | Injective modules, self-injective associative rings |
| 18G05 | Projectives and injectives (category-theoretic aspects) |
Keywords:
homological dimensions; flat modules; covering classes; integral group rings; flat lengths; injective modules; injective lengths; projective modules; Gorenstein flat dimensionReferences:
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