ℵ_m-presented Gorenstein flat modules and Gorenstein dimensions of groups

@misc{stergiopoulou2026ℵmpresented,
  title = {{$\aleph_m$-presented Gorenstein flat modules and Gorenstein dimensions of groups}},
  author = {Dimitra-Dionysia Stergiopoulou},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.24512},
  eprint = {2609.24512},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.24512},
}

TY  - UNPB
TI  - ℵ_m-presented Gorenstein flat modules and Gorenstein dimensions of groups
AU  - Dimitra-Dionysia Stergiopoulou
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.24512
UR  - https://arxiv.org/abs/2609.24512
AB  - Let G be a group and k be a commutative ring. The Gorenstein homological dimension Ghd_k G and the Gorenstein cohomological dimension Gcd_k G of G over k are defined as the Gorenstein flat and the Gorenstein projective dimension, respectively, of the trivial kG-module k. We prove that the Gorenstein cohomological dimension of a countable group G is at most one more than its Gorenstein homological dimension and, more generally, that Gcd_k G≤Ghd_k G+m+1 for every m≥0 and every group G of cardinality at most ℵ_m. When the supremum sflik of the flat dimensions of the injective k-modules is finite, this gives Ghd_k G≤Gcd_k G≤Ghd_k G+1 for every countable group G, so that the two dimensions are finite simultaneously. These results are Gorenstein analogues of Bieri's inequality for the homological and the cohomological dimension of countable groups. They rest on a module-theoretic theorem of independent interest, which is a Gorenstein version of results of Jensen and Osofsky on the projective dimension of flat modules: over an arbitrary ring, every ℵ_m-presented Gorenstein flat module has projectively coresolved Gorenstein flat dimension at most m+1. An appendix characterizes the modules of PGF-dimension at most n as the direct summands of the strongly n-PGF modules.
ER  -