On a theorem of Keller over a base ring

TL;DR

This work generalizes Keller's theorem, which identifies Hochschild and cyclic invariants of a quasi-compact separated scheme $X$ over a base field with those of the dg category ${\sf Perf}(X)$, to the relative setting over a base commutative ring $R$. It develops a relative theory of Hochschild and cyclic invariants using dg categories and Shukla-type resolutions, and proves that the trace map from $M({\sf Perf}(\mathfrak X))$ to $R\Gamma(\mathfrak X, M(\mathcal{O})^\sharp)$ is invertible in the derived category of mixed complexes $\mathcal{D}\mathrm{Mix}(R)$. The paper achieves this by adapting Keller's Mayer–Vietoris and localization arguments to the relative setting via semi-free resolutions, ensuring flatness and exactness in $\mathcal{D}\mathrm{Mix}(R)$. The main result shows canonical isomorphisms $HH_\bullet({\sf Perf}(\mathfrak X)/R)\cong HH_\bullet(\mathfrak X/R)$ and $HC_\bullet({\sf Perf}(\mathfrak X)/R)\cong HC_\bullet(\mathfrak X/R)$, extending the categorical-geometric nature of Hochschild and cyclic invariants to the relative context. This provides a robust framework for traced invariants in relative algebraic geometry and their potential applications to log-Hochschild theory and deformation contexts.

Abstract

Let $X$ be a quasi-compact separated scheme over a base field. Keller proved a theorem stating that the cyclic homology of $X$ is canonically isomorphic to the cyclic homology of the dg category ${\sf Perf}(X)$ consisting of perfect complexes over $X$. This theorem shows the categorical nature of the cyclic homology. In this note, we generalize Keller's theorem to allow $X$ be defined over a base commutative ring.

Theorems & Definitions (18)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Definition 4.1
• Definition 4.2
• Lemma 4.3