具有任意系数和结合光滑性的导出表示方案
Derived representation schemes with arbitrary coefficients and associative smoothness
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中文总结 AI 辅助
研究通过任意有限维代数系数的表示同调探讨结合光滑性，证明有限生成形式光滑代数高阶同调消失，任意系数检验更强，对有限维代数而言其表示同调完全刻画形式光滑性，为相关理念提供证据。
中文摘要 AI 辅助
我们通过在任意有限维代数中具有系数的表示同调研究结合光滑性。证明了具有任意有限维系数的有限生成形式光滑代数的高阶表示同调消失。还表明允许任意系数产生更强的光滑性检验：合适的非矩阵系数能在矩阵系数无法检测时检测出非光滑性。这些结果引发了关于任意系数的表示同调是否能根据导出的孔采维奇 - 罗森伯格原理给出结合光滑性的同伦特征的问题。对于有限维代数，证明了任意系数的表示同调完全刻画形式光滑性，为这一理念提供了支持证据。
英文摘要
We study associative smoothness through representation homology with coefficients in arbitrary finite-dimensional algebras. We prove that the higher representation homology of a finitely generated formally smooth algebra with arbitrary finite-dimensional coefficients vanishes. We further show that allowing arbitrary coefficients yields a strictly stronger smoothness test: suitable non-matrix coefficients detect nonsmoothness in situations where matrix coefficients do not. These results raise the question of whether representation homology with arbitrary coefficients furnishes a homotopical characterization of associative smoothness in the spirit of the derived Kontsevich-Rosenberg principle. For finite-dimensional algebras, we prove that representation homology with arbitrary coefficients completely characterizes formal smoothness, providing supporting evidence for this philosophy.
