AI 中文总结
针对任意Artin代数，通过用有限长度模的子范畴逼近其模范畴，结合挠类格、墙腔结构与图像群的性质，构造并证明了极小相容散射图的存在性，且复有限维代数情形下与Bridgeland稳定性散射图典范同构。
AI 中文摘要
针对任意Artin代数$A$，我们通过用长度至多为$\nell\in\mathbb{N}$的模构成的子范畴$(\mathrm{mod}\\,A)_\ell$逼近其模范畴$\mathrm{mod}\\,A$，构造了一个极小且相容的散射图。我们证明每个子范畴$(\operatorname{mod}A)_\ell$都具备性质良好的挠类格、有限的墙腔结构$\mathfrak{D}_\ell(A)$，以及带有自然范畴解释的伴随图像群$G_\ell(A)$。利用这些性质，我们为每个$\nell\in\mathbb{N}$构建了有限、极小且相容的散射图。通过取逆极限，我们确立了$A$的极小相容散射图的存在性。特别地，当$A$是$\mathbb{C}$上的有限维代数时，我们的逆极限构造与Bridgeland稳定性散射图典范同构。
英文摘要
For an arbitrary Artin algebra $A$, we construct a minimal and consistent scattering diagram by approximating its module category $\mathrm{mod}\,A$ using the subcategories $(\mathrm{mod}\,A)_\ell$ of modules of length at most $\ell \in \mathbb{N}$. We prove that each subcategory $(\operatorname{mod}A)_\ell$ possesses a well-behaved lattice of torsion classes, a finite wall-and-chamber structure $\mathfrak{D}_\ell(A)$ and an associated picture group $G_\ell(A)$ with a natural categorical interpretation. Using these properties, we build a finite, minimal, consistent scattering diagram for each $\ell \in \mathbb{N}$. Passing to the inverse limit, we establish the existence of a minimal consistent scattering diagram for $A$. In particular, when $A$ is a finite-dimensional algebra over $\mathbb{C}$, our inverse limit construction is canonically isomorphic to Bridgeland's stability scattering diagram.
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