Let T be a smooth proper dg-category over C. Write K_operatornametop(T) for its topological K-theory spectrum, operatornameHP(T/C) for its periodic cyclic homology spectrum, and operatornameCh^operatornametop for the topological Chern character. Blanc's lattice conjecture. The map operatornameCh^operatornametopwedge_SHC:K_operatornametop(T)wedge_SHC→operatornameHP(T/C) is an equivalence. In particular, for any i, pi_iK_operatornametop(T)otimes_ZCsimeq operatornameHP_i(T/C). This is the non-commutative analogue of the de Rham comparison between singular and de Rham cohomology; the source presents it as Blanc's prediction, and no resolution is supplied here.

Blanc's lattice conjecture for smooth proper dg-categories

From papers

Let $\mathcal{T}$ be a smooth proper dg-category over $\mathbb{C}$ . Write $K_{\operatorname{top}}(\mathcal{T})$ for its topological K-theory spectrum, $\operatorname{HP}(\mathcal{T}/\mathbb{C})$ for its periodic cyclic homology spectrum, and $\operatorname{Ch}^{\operatorname{top}}$ for the topological Chern character. Blanc's lattice conjecture. The map

\operatorname{Ch}^{\operatorname{top}}\wedge_{\mathbb{S}}H\mathbb{C}:K_{\operatorname{top}}(\mathcal{T})\wedge_{\mathbb{S}}H\mathbb{C}\to\operatorname{HP}(\mathcal{T}/\mathbb{C})

is an equivalence. In particular, for any $i$ ,

\pi_iK_{\operatorname{top}}(\mathcal{T})\otimes_{\mathbb{Z}}\mathbb{C}\simeq\operatorname{HP}_i(\mathcal{T}/\mathbb{C}).

This is the non-commutative analogue of the de Rham comparison between singular and de Rham cohomology; the source presents it as Blanc's prediction, and no resolution is supplied here.

Progress summary

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Blanc's lattice conjecture for smooth proper dg-categories
Let $\cC$ be a dg-category smooth and proper over $\C$ . Let $K_*^{\operatorname{top}}(\cC)_\C$ denote its complexified topological $K$ -theory, let $HP_*(\cC)$ denote its periodic cyclic homology, and let $\operatorname{ch}^{\operatorname{top}}$ be the induced topological Chern character map.

Blanc's lattice conjecture. If $\cC$ is smooth and proper over $\C$ , then
$K_*^{\operatorname{top}}(\cC)_\C \xrightarrow{\operatorname{ch}^{\operatorname{top}}} HP_*(\cC)$
is an isomorphism.

This is the noncommutative analogue of the comparison between topological $K$ -theory and periodic cyclic homology. The source states that the conjecture implies property (3) in the construction of a pre-Hodge structure; it is known for dg-categories arising from smooth proper complex varieties, while the general smooth proper case is not resolved here.
source: Michael K. Brown and Mark E. Walker, “The Hodge structure on the singularity category of a complex hypersurface”, arXiv:2407.09988 (2025).

Sources & referencesView supporting material

Primary source

Keiho Matsumoto, “Towards the p-adic Hodge theory for non-commutative algebraic varieties”, arXiv:2305.00292 (2024).

Additional references

2 papers in this index state this conjecture (2013–2023). The statement above is taken from the most recent of them; the others are arXiv:1307.6430.

1 week ago