We disprove two (unpublished) conjectures of Kontsevich which state generalized versions of categorical Hodge-to-de Rham degeneration for smooth and for proper DG categories (but not smooth and proper, in which case degeneration is proved by Kaledin \cite{Ka}). In particular, we show that there exists a minimal $10$-dimensional $A_{\infty}$-algebra over a field of characteristic zero, for which the supertrace of $\mu_3$ on the second argument is non-zero. As a byproduct, we obtain an example of a homotopically finitely presented DG category (over a field of characteristic zero) that does not have a smooth categorical compactification, giving a negative answer to a question of To\"en. This can be interpreted as a lack of resolution of singularities in the noncommutative setup. We also obtain an example of a proper DG category which does not admit a categorical resolution of singularities in the terminology of Kuznetsov and Lunts \cite{KL} (that is, it cannot be embedded into a smooth and proper DG category).

中文翻译：

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分类平滑紧缩和广义 Hodge-to-de Rham 退化

我们反驳了 Kontsevich 的两个（未发表的）猜想，这些猜想陈述了分类 Hodge-to-de Rham 退化的广义版本，用于平滑和适当的 DG 类别（但不是平滑和适当的，在这种情况下退化由 Kaledin \cite{Ka} 证明） . 特别地，我们证明在特征为零的域上存在一个最小的 $10$-维 $A_{\infty}$-代数，为此 $\mu_3$ 在第二个参数上的超迹是非零的。作为副产品，我们获得了一个同伦有限呈现的 DG 类别（在特征为零的域上）的示例，它没有平滑的分类紧化，对 To\"en 问题给出否定答案。这可以解释为在非对易设置中缺乏奇点的分辨率。