The stable class of the augmentation ideal. (English) Zbl 1094.20001
Let \(\mathbb{Z} G\) be the integral group ring of a group \(G\). If \(G\) is a finite group, then the author [F. E. A. Johnson, “Stable modules and the \(D(2)\)-problem” (Lond. Math. Soc. Lect. Note Ser. 301), CUP (2003; Zbl 1055.57002)] has studied a class \(\Omega_1(\mathbb{Z})\) of \(\mathbb{Z} G\)-modules that are stably equivalent to the augmentation ideal of \(\mathbb{Z} G\). In this paper, the author continues these investigations for finitely generated infinite groups \(G\) satisfying the condition \(\text{Ext}^1(\mathbb{Z},\mathbb{Z} G)=0\).
For a given group \(G\) let \(\mathbf{SF}\) be the tree of nonzero stably free \(\mathbb{Z} G\)-modules. The main result asserts that the correspondence \(J\to\operatorname{Hom}_{\mathbb{Z} G}(J,\mathbb{Z} G)\) defines a surjective level preserving mapping of trees \(\delta\colon\Omega_1(\mathbb{Z})\to\mathbf{SF}\). Moreover, \(\delta\) induces a one-to-one correspondence between the minimal level of \(\Omega_1(\mathbb{Z})\) and the isomorphism classes of stably free modules of rank 1.
For a given group \(G\) let \(\mathbf{SF}\) be the tree of nonzero stably free \(\mathbb{Z} G\)-modules. The main result asserts that the correspondence \(J\to\operatorname{Hom}_{\mathbb{Z} G}(J,\mathbb{Z} G)\) defines a surjective level preserving mapping of trees \(\delta\colon\Omega_1(\mathbb{Z})\to\mathbf{SF}\). Moreover, \(\delta\) induces a one-to-one correspondence between the minimal level of \(\Omega_1(\mathbb{Z})\) and the isomorphism classes of stably free modules of rank 1.
Reviewer: S. V. Mihovski (Plovdiv)
MSC:
20C07 | Group rings of infinite groups and their modules (group-theoretic aspects) |
16S34 | Group rings |
05C05 | Trees |
16E65 | Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.) |
19A13 | Stability for projective modules |
Citations:
Zbl 1055.57002References:
[2] | Johnson, F. E. A.: Stable Modules and the D(2)-Problem, LMS Lecture Notes In Mathematics, vol. 301, CUP (2003). · Zbl 1055.57002 |
[3] | Johnson, F. E. A.: Relation Modules and Duality, (preprint) University College London (2004). |
[4] | Kaplansky, I.: Fields and Rings, University of Chicago Press (1969). |
[5] | Mac Lane, S.: Homology, Springer-Verlag (1963). |
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