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Spectral sequences for cyclic homology
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We prove that for a homologically smooth and proper DG algebra over a field of characteristic 0, the Hodge-to-de Rham spectral sequence degenerates. This has been conjectured by M. Kontsevich and Y. Soibelman arXiv:math/0606241 and proved in arXiv:math/0611623 under a technical assumption. In this paper, the assumption is removed, and the argument is considerably simplified (in particular, it no longer uses Dold-Kan equivalence and simplicial methods). We also analyse the degeneration of the conjugate spectral sequence in positive caracteristic constructed in arXiv:1509.08784.
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$\mathrm{Mot}^{\mathrm{loc}}$ is not compactly generated
For any connective E2-ring R with R_Q ≠ 0, the ∞-category of R-linear localizing motives is not compactly generated.
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