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Homological finiteness conditions for modules over strongly group-graded rings. (English) Zbl 0857.16007

This paper proves two theorems about modules over strongly \(G\)-graded \(k\)-algebras in case \(G\) is a group belonging to one of the classes \(H_1{\mathcal F}\) or \(H{\mathcal F}\) introduced by P. H. Kropholler [in J. Pure Appl. Algebra 90, No. 1, 55-67 (1993; Zbl 0816.20042)]. Note that the class \(H_1{\mathcal F}\) contains all groups of finite virtual cohomological dimension and, in particular, it contains all polycyclic-by-finite and all arithmetic groups. Theorem A. Let \(G\) be an \(H_1{\mathcal F}\)-group, let \(R\) be a strongly \(G\)-graded \(k\)-algebra and let \(M\) be an \(R\)-module. Then \(M\) has finite projective dimension if and only if its restriction to \(R_H\) has finite projective dimension for all finite subgroups \(H\) of \(G\).
The class \(H{\mathcal F}\) is larger and the result for groups in this class is more technical, requiring the notion of a semidiagonal action. Let \(M\) be an \(R\)-module and let \(V\) be a \(kG\)-module. Then \(M\otimes_kV\) can be made into an \(R\)-module via \((m\otimes v)r_g=mr_g\otimes vg\). Theorem B. Let \(G\) be an \(H{\mathcal F}\)-group, let \(R\) be a strongly \(G\)-graded \(k\)-algebra, let \(M\) be an \(R\)-module of type \(FP_\infty\) which has finite projective dimension over \(R_1\), and let \(V\) be a \(kG\)-module which is \(k\)-projective. Then \(M\otimes V\) has finite projective dimension as an \(R\)-module if and only if it has finite projective dimension over \(R_H\) for all finite subgroups \(H\) of \(G\).

MSC:

16E10 Homological dimension in associative algebras
16S35 Twisted and skew group rings, crossed products
16W50 Graded rings and modules (associative rings and algebras)
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
20J05 Homological methods in group theory

Citations:

Zbl 0816.20042
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References:

[1] Cornick, QMW, University of London (1994)
[2] Brown, Cohomology of groups (1982)
[3] DOI: 10.1016/0022-4049(92)90116-W · Zbl 0807.20044
[4] DOI: 10.1112/jlms/s2-44.1.47 · Zbl 0688.16010
[5] DOI: 10.1006/jabr.1994.1056 · Zbl 0803.16027
[6] DOI: 10.1007/BF01161413 · Zbl 0424.16001
[7] DOI: 10.1016/0166-8641(94)90081-7 · Zbl 0810.20040
[8] DOI: 10.1016/0022-4049(93)90136-H · Zbl 0816.20042
[9] Kaplansky, Fields and rings (1972) · Zbl 1001.16500
[10] DOI: 10.1016/0022-4049(92)90009-5 · Zbl 0767.55004
[11] DOI: 10.1016/0022-4049(91)90048-7 · Zbl 0741.20030
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