Note
Multiplicativity of the Synthetic Tate Filtration
Article date: September 2026.
Abstract
We resolve the two monoidality conjectures for the synthetic cellular Tate filtration of Annala and Pstrągowski. The perfect-even module filtration has no unital lax monoidal structure for diagonal tensor products, but the synthetic Tate filtration admits a lax -monoidal structure compatible with realization. The positive construction uses change of rings for modules with even complex bordism homology and a local monoidal refinement on bounded-below modules with flat even complex bordism homology. We obtain an comparison of the Antieau, Bhatt-Lurie, and Raksit HKR bifiltrations and identify their double associated graded with a derived exterior-power Laurent algebra. We also identify neutral décalage with a Whitehead tower and prove a completed relative Künneth theorem for arbitrary inputs. The associative refinement reconstructs module categories. An explicit pair of filtered algebras has the same underlying filtration and filtration-zero algebra, but the Grothendieck groups of their compact module categories have different ranks. The coefficient -algebra admits no refinement.