The cellular Tate filtration is lax E2-monoidal, and the HKR filtrations of Antieau, Bhatt-Lurie, and Raksit on negative cyclic and periodic homology coincide.
Glasman, Day convolution for -categories , Mathematical Research Letters 23 (2017), no
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On the circle-equivariant cellular Tate filtration
The cellular Tate filtration is lax E2-monoidal, and the HKR filtrations of Antieau, Bhatt-Lurie, and Raksit on negative cyclic and periodic homology coincide.