持久性范畴的 Gabriel 谱
Gabriel Spectrum of Persistence Categories
中文总结 AI 辅助
本文用纯拓扑方法确定了向量空间层范畴的 Gabriel 谱，证明其同胚于索伯化空间的 Skula 拓扑，并应用于持久性理论，得到持久模范畴的 Gabriel 谱的具体计算。
中文摘要 AI 辅助
我们以纯拓扑术语确定了向量空间层范畴的 Gabriel 谱。对于每个拓扑空间 $X$，我们证明了 ${\mathbf{Sh}}(X)$ 的 Gabriel 谱同胚于 ${\mathrm{Sk}}({\mathrm{Sob}}(X))$，其中 ${\mathrm{Sob}}(X)$ 表示 $X$ 的 sobrification（索伯化），而 ${\mathrm{Sk}}$ 表示取 Skula 拓扑。证明基于不可分解内射层的分类以及用 Skula 开子集对局部化子范畴的刻画。我们证明了 Gabriel 谱总是 Hausdorff 的、零维的且完全不连通的，并且它是紧的当且仅当 $X$ 是 Noether 的。这导致了持久性理论中出现的 Gabriel 谱的计算。特别地，${\mathbb R}^n$ 上持久模范畴的 Gabriel 谱同胚于 ${\mathbb R}^n$ 的理想空间，并赋予自然拓扑。
英文摘要
We determine the Gabriel spectrum of a category of sheaves of vector spaces in purely topological terms. For every topological space $X$, we prove that the Gabriel spectrum of ${\mathbf{Sh}}(X)$ is homeomorphic to ${\mathrm{Sk}}({\mathrm{Sob}}(X))$, where ${\mathrm{Sob}}(X)$ denotes the sobrification of $X$ and ${\mathrm{Sk}}$ indicates passage to the Skula topology. The proof is based on a classification of indecomposable injective sheaves and a characterization of localizing subcategories in terms of Skula-open subsets. We show that the Gabriel spectrum is always Hausdorff, zero-dimensional, and totally disconnected, and that it is compact if and only if $X$ is Noetherian. This leads to computations of Gabriel spectra arising in persistence theory. In particular, the Gabriel spectrum of the category of persistence modules over ${\mathbb R}^n$ is homeomorphic to the space of ideals of ${\mathbb R}^n$ with a natural topology.
