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

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.