Kontsevich's proper DG algebra degeneration conjecture

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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]→id⁡⊗δHH∙(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.

References

Primary source

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

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