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Localizing Motives over Nonzero Connective E_2-Rings Are Not Compactly Generated

Author: Wei Yang · Added: August 14, 2026

Article date: August 14, 2026.

Abstract

We prove that $\operatorname{Mot}^{\mathrm{loc}}_R$ is not compactly generated for every nonzero connective $\mathbb{E}_2$ -ring spectrum $R$ , removing the nonzero-rationalization hypothesis from Ramzi’s theorem. The new input is the positive-characteristic case. For a perfect field $k$ of characteristic $p$ , we construct a $k$ -linear motive whose cyclotomic topological Hochschild homology is compact but not dualizable over $\operatorname{THH}(k)$ . The compact object comes from the one-weight polygonic cell, and its nondualizability is detected by relative Tate assembly. Ramzi’s universal $\operatorname{THH}$ -motive realizes the cell, while Efimov’s rigidity theorem converts the discrepancy into failure of compact generation. The general case follows by passage to a perfect residue field, together with Ramzi’s rational result.