[Paper Review] Non-commutative Hodge-to-de Rham degeneration via the method of Deligne-Illusie

D. Kaledin|arXiv (Cornell University)|Nov 21, 2006

Homotopy and Cohomology in Algebraic Topology15 references11 citations

TL;DR

This paper establishes the degeneration of the Hodge-to-de Rham spectral sequence for smooth, compact DG algebras over a field of characteristic zero using a non-commutative adaptation of the Deligne-Illusie method. By constructing a generalized Cartier map in the non-commutative setting and leveraging reduction to positive characteristic, the author proves that the spectral sequence degenerates at the first term, confirming a non-commutative version of the Kontsevich-Soibelman conjecture under finiteness conditions.

ABSTRACT

We use a version of the method of Deligne-Illusie to prove that the Hodge-to-de Rham, a.k.a. Hochschild-to-cyclic spectral sequence degenerates for a large class of associative, not necessariyl commutative DG algebras. This proves, under some assumption, a conjecture by Kontsevich and Soibelman made in math.RA/0606241. The approach is similar to my earlier paper math.AG/0511665, but the proof is more straightforward, and the underlying algebraic topology notions are explicitly described. The paper is independent of math.AG/0511665 and in a sense, supercedes it.

Motivation & Objective

• To extend the classical Hodge-to-de Rham degeneration theorem to the non-commutative setting using algebraic geometry techniques.
• To resolve the Kontsevich-Soibelman conjecture on non-commutative Hodge-to-de Rham degeneration under smoothness and compactness assumptions.
• To adapt the Deligne-Illusie method—originally relying on the Cartier isomorphism in positive characteristic—to non-commutative algebras by constructing a non-commutative Cartier map.
• To demonstrate that the Hodge-to-de Rham spectral sequence degenerates at the first term for smooth, compact DG algebras over characteristic zero fields.
• To bridge algebraic geometry and algebraic topology by making topological constructions (e.g., cyclotomic spectra) accessible via purely homological algebra, avoiding stable homotopy theory.

Proposed method

• The paper constructs a generalized Cartier map in the non-commutative setting, extending the classical Cartier isomorphism used in the Deligne-Illusie proof.
• It uses reduction modulo a prime to transfer the problem from characteristic zero to positive characteristic, where the Cartier map becomes effective.
• The method relies on the existence of a weak splitting of the complex $\overline{Q}_{\bullet}(\mathcal{A}) \to \mathcal{A}$, which allows the construction of a splitting of the Hodge-to-de Rham spectral sequence.
• The proof employs the periodic cyclic homology $HP_\bullet(A^{\bullet})$ as a key invariant, showing that $HH_\bullet(A^{\bullet})^{(1)}((u)) \cong HP_\bullet(A^{\bullet})$ via the generalized Cartier map.
• It applies Nakayama’s lemma to show that vanishing of differentials modulo all maximal ideals implies degeneration over the original ring.
• The argument is formalized using cosimplicial algebras and pro-objects to handle the limit behavior in reduction to positive characteristic.

Experimental results

Research questions

• RQ1 Does the Hodge-to-de Rham spectral sequence degenerate for smooth, compact DG algebras over a field of characteristic zero in the non-commutative setting?
• RQ2 Can the Deligne-Illusie method be adapted to non-commutative algebras despite the absence of a Frobenius map?
• RQ3 Is there a meaningful non-commutative analogue of the Cartier isomorphism that supports the degeneration argument?
• RQ4 How can the topological origin of the Cartier map (via cyclotomic spectra) be made accessible to algebraic geometers without stable homotopy theory?
• RQ5 Under what conditions does the spectral sequence $HH_\bullet(A^{\bullet})[u^{-1}] \Rightarrow HC_\bullet(A^{\bullet})$ degenerate at the first term?

Key findings

• The Hodge-to-de Rham spectral sequence $HH_\bullet(A^{\bullet})[u^{-1}] \Rightarrow HC_\bullet(A^{\bullet})$ degenerates at the first term for smooth, compact DG algebras over a field of characteristic zero.
• The generalized Cartier map induces an isomorphism $HH_\bullet(A^{\bullet})^{(1)}((u)) \cong HP_\bullet(A^{\bullet})$, which is essential for the degeneration proof.
• The spectral sequence degenerates modulo every maximal ideal of a suitable subring $R \subset K$, and by Nakayama’s lemma, this implies degeneration over the original algebra.
• The finite-dimensionality and vanishing of Hochschild and cyclic homology in high degrees ensure that the degeneration criterion from Deligne [D2] applies.
• The proof establishes the non-commutative version of the Kontsevich-Soibelman conjecture under the assumptions of smoothness and compactness of the DG algebra.
• The construction of the Cartier map in the non-commutative world is shown to be compatible with the topological framework of cyclotomic spectra, though the paper avoids explicit use of spectra.