AI 中文总结
研究完美域上表示无限的有限维 \(k -\) 代数的奥斯特兰德 - 赖滕箭图，通过结合局部化多项式表示嵌入、半线性扭转及可分基变换等方法，证明其有无限多个连通分支，确立相关猜想。
AI 中文摘要
设 \(k\) 为完美域，\(A\) 为表示无限的有限维 \(k -\) 代数。我们证明 \(A\) 的奥斯特兰德 - 赖滕箭图有无限多个连通分支。这为完美域上的有限维代数确立了奥斯特兰德、赖滕和斯马洛关于阿廷代数的一个猜想。在代数闭域上，证明结合了局部化多项式表示嵌入与由域自同构诱导的半线性扭转。通过可分基变换从完美域过渡到其代数闭包：我们证明若 \(A\) 的奥斯特兰德 - 赖滕箭图只有有限多个分支，则标量扩张到代数闭包时同样成立。
英文摘要
Let $k$ be a perfect field and let $A$ be a representation-infinite finite-dimensional $k$-algebra. We prove that the Auslander--Reiten quiver of $A$ has infinitely many connected components. This establishes, for finite-dimensional algebras over perfect fields, a conjecture of Auslander, Reiten, and Smalø concerning Artin algebras. Over an algebraically closed field, the proof combines a localized polynomial representation embedding with semilinear twists induced by field automorphisms. The passage from a perfect field to its algebraic closure is obtained by separable base change: we prove that if the Auslander--Reiten quiver of $A$ has only finitely many components, then the same holds for the scalar extension to the algebraic closure.
Comments19 pages, comments welcome!
