Note
Cyclotomic Realization by Topological Hochschild Homology
Abstract
We study the essential image of topological Hochschild homology on small idempotent-complete stable -categories. We construct a genuine cyclotomic THH functor by Morita descent and compare it naturally with Nikolaus-Scholze THH. Every cyclotomic spectrum whose underlying spectrum is bounded below is realizable by THH. The proof combines corepresentability of integral TR with a retract construction for arbitrary coproducts of reduced polynomial motives and a convergent cellular tower. In the unbounded case there are two distinct failures of realization. An Anderson-dual example has no genuine cyclotomic refinement. A second example admits such a refinement but violates the uniform prime-wise estimate supplied by the integer May filtration on THH. Consequently the essential image of THH is strictly smaller than the essential image of genuine cyclotomic spectra in the Nikolaus-Scholze category, and the latter is itself a proper subcategory.