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Categorical smooth compactifications and generalized Hodge-to-de Rham degeneration

Alexander I. Efimov

TL;DR

The paper refutes Kontsevich's generalized Hodge-to-de Rham degeneration conjectures for both smooth and proper DG categories by constructing explicit counterexamples, including a minimal $A_{ abla ext{infty}}$-algebra of dimension $10$ with nonzero $ ext{str}( u_3)$. A nilpotence and factorization theorem for cohomology classes enables these constructions, and the work also yields a homotopically finitely presented DG category without a smooth categorical compactification, answering Toën's question negatively. Additionally, it produces a proper DG category that cannot be categorically resolved in the Kuznetsov–Lunts sense. Collectively, the results demonstrate fundamental limits of noncommutative resolution and degeneration phenomena beyond the classical smooth/proper regime, highlighting significant gaps between conjectural expectations and actual behavior in the noncommutative setting.

Abstract

We disprove two (unpublished) conjectures of Kontsevich which state generalized versions of categorical Hodge-to-de Rham degeneration for smooth and for proper DG categories (but not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 $μ_3$ 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 Toën. 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).

Categorical smooth compactifications and generalized Hodge-to-de Rham degeneration

TL;DR

The paper refutes Kontsevich's generalized Hodge-to-de Rham degeneration conjectures for both smooth and proper DG categories by constructing explicit counterexamples, including a minimal -algebra of dimension with nonzero . A nilpotence and factorization theorem for cohomology classes enables these constructions, and the work also yields a homotopically finitely presented DG category without a smooth categorical compactification, answering Toën's question negatively. Additionally, it produces a proper DG category that cannot be categorically resolved in the Kuznetsov–Lunts sense. Collectively, the results demonstrate fundamental limits of noncommutative resolution and degeneration phenomena beyond the classical smooth/proper regime, highlighting significant gaps between conjectural expectations and actual behavior in the noncommutative setting.

Abstract

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 -dimensional -algebra over a field of characteristic zero, for which the supertrace of 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 Toën. 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).

Paper Structure

This paper contains 8 sections, 14 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on DG categories and $A_{\infty}$-algebras
  3. DG categories
  4. $A_{\infty}$-algebras and $A_{\infty}$-(bi)modules
  5. Preliminaries on the Hochschild complex, pairings and copairings
  6. A counterexample to the generalized degeneration conjecture
  7. A counterexample to Conjecture \ref{['conj:degeneration_for_smooth_intro']}
  8. A counterexample to Conjecture \ref{['conj:degeneration_for_proper_intro']}

Key Result

Theorem 1

Ka Let $A$ be a smooth and proper DG algebra. Then the Hochschild-to-cyclic spectral sequence degenerates, so that we have an isomorphism $HP_{\bullet}(A)=HH_{\bullet}(A)((u)).$

Theorems & Definitions (45)

  • Theorem 1
  • Conjecture 2
  • Conjecture 3
  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Remark 1.7
  • ...and 35 more