$\aleph_m$-呈现的 Gorenstein 平坦模与群的 Gorenstein 维数
$\aleph_m$-presented Gorenstein flat modules and Gorenstein dimensions of groups
- University of Thessaly（塞萨利大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明可数群及基数至多 $\aleph_m$ 的群的 Gorenstein 上同调维数不超过同调维数加 $m+1$，推广 Bieri 不等式，并建立 $\aleph_m$-呈现 Gorenstein 平坦模的维数界。
AI中文摘要：
设 $G$ 为一个群，$k$ 为一个交换环。$G$ 在 $k$ 上的 Gorenstein 同调维数 $\text{Ghd}_k G$ 和 Gorenstein 上同调维数 $\text{Gcd}_k G$ 分别定义为平凡 $kG$-模 $k$ 的 Gorenstein 平坦维数和 Gorenstein 投射维数。我们证明：可数群 $G$ 的 Gorenstein 上同调维数至多比其 Gorenstein 同调维数大 1；更一般地，对每个 $m\ge0$ 以及每个基数至多为 $\aleph_m$ 的群 $G$，有 $\text{Gcd}_k G\le\text{Ghd}_k G+m+1$。当内射 $k$-模的平坦维数的上确界 sfli$k$ 有限时，这给出对每个可数群 $G$ 有 $\text{Ghd}_k G\le\text{Gcd}_k G\le\text{Ghd}_k G+1$，从而这两个维数同时有限。这些结果是 Bieri 关于可数群同调维数与上同调维数不等式的 Gorenstein 类比。它们基于一个具有独立兴趣的模论定理，该定理是 Jensen 和 Osofsky 关于平坦模投射维数结果的 Gorenstein 版本：在任意环上，每个 $\aleph_m$-呈现的 Gorenstein 平坦模的投射余解 Gorenstein 平坦维数至多为 $m+1$。附录刻画了 PGF-维数至多为 $n$ 的模恰为强 $n$-PGF 模的直和项。
英文摘要：
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.
