Kontsevich's proper DG algebra degeneration conjecture

Let BB be a proper DG algebra over a field cmathrmkcmathrm k. The Hochschild homology pairing is a map HH(B)HH(Bop)kHH_{\bullet}(B)\otimes HH_{\bullet}(B^{op})\to \mathrm k, and let δ:HC(Bop)HH(Bop)[1]\delta:HC_{\bullet}(B^{op})\to HH_{\bullet}(B^{op})[-1] be the connecting map. Kontsevich's proper degeneration conjecture. The composition map

(HH(B)HC(Bop))[1]idδHH(B)HH(Bop)k(HH_{\bullet}(B)\otimes HC_{\bullet}(B^{op}))[1]\xrightarrow{\operatorname{id}\otimes\delta}HH_{\bullet}(B)\otimes HH_{\bullet}(B^{op})\to\mathrm k

is zero. This is a proposed extension of Hodge-to-de Rham degeneration from smooth and proper DG algebras to proper DG algebras; the paper disproves it by constructing a counterexample.

Sources & referencesView supporting material

Primary source

Alexander I. Efimov, “Categorical smooth compactifications and generalized Hodge-to-de Rham degeneration”, arXiv:1805.09283 (2018).

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