Coherent cochain complexes and Beilinson t-structures, with an appendix by Achim Krause

TL;DR

This paper explores the concept of coherent cochain complexes in stable $mbda$-categories, establishing their equivalence to complete filtered objects, and investigates their relation to Beilinson t-structures, Toda brackets, and spectral sequences.

Contribution

It introduces the notion of coherent cochain complexes in stable $mbda$-categories and demonstrates their equivalence to complete filtered objects, providing new insights into Beilinson t-structures.

Findings

Equivalence between coherent cochain complexes and complete filtered objects.

Interpretation of Beilinson t-structure via this equivalence.

Construction of spectral sequences from coherent cochain complexes.

Abstract

We define and study coherent cochain complexes in arbitrary stable $\infty$-categories, following Joyal. Our main result is that the $\infty$-category of coherent cochain complexes in a stable $\infty$-category $\mathscr C$ is equivalent to the $\infty$-category of complete filtered objects in $\mathscr C$. We then show how the Beilinson t-structure can be interpreted in light of such equivalence, and analyze its behavior in the presence of symmetric monoidal structures. We also examine the relationship between the notion of (higher) Toda brackets and coherent cochain complexes. Finally, we prove how every coherent cochain complex gives rise to a spectral sequence and illustrate some examples.