$\aleph_m$-presented Gorenstein flat modules and Gorenstein dimensions of groups
Let $G$ be a group and $k$ be a commutative ring. The Gorenstein homological dimension $\text{Ghd}_k G$ and the Gorenstein cohomological dimension $\text{Gcd}_k G$ 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 $\text{Gcd}_k G\le\text{Ghd}_k G+m+1$ for every $m\ge0$ and every group $G$ of cardinality at most $\aleph_m$. When the supremum sfli$k$ of the flat dimensions of the injective $k$-modules is finite, this gives $\text{Ghd}_k G\le\text{Gcd}_k G\le\text{Ghd}_k G+1$ 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 $\aleph_m$-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.