Geodesic
Spectral Algebras and Non-commutative Hodge-to-de Rham Degeneration
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebra, number theory, and algebraic geometry, Tome 307 (2019), pp. 63-77

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We revisit the non-commutative Hodge-to-de Rham degeneration theorem of the first author and present its proof in a somewhat streamlined and improved form that explicitly uses spectral algebraic geometry. We also try to explain why topology is essential to the proof. 
@article{TM_2019_307_a2,
     author = {D. B. Kaledin and A. A. Konovalov and K. O. Magidson},
     title = {Spectral {Algebras} and {Non-commutative} {Hodge-to-de} {Rham} {Degeneration}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {63--77},
     publisher = {mathdoc},
     volume = {307},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2019_307_a2/}
}
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D. B. Kaledin; A. A. Konovalov; K. O. Magidson. Spectral Algebras and Non-commutative Hodge-to-de Rham Degeneration. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebra, number theory, and algebraic geometry, Tome 307 (2019), pp. 63-77. http://geodesic.mathdoc.fr/item/TM_2019_307_a2/