A $t$-structure on the $\infty$-category of mixed graded modules
Emanuele Pavia
TL;DR
The paper develops a model-independent framework for the ∞-category of mixed graded $k$-modules and endows it with a left and right complete t-structure, the mixed graded Postnikov t-structure. It then establishes a precise link to Beilinson’s filtered modules by proving that the mixed graded category is the left completion of the Beilinson t-structure on filtered modules, realized via a t-exact embedding and a left adjoint that recovers the mixed graded category. The central result identifies $\varepsilon\operatorname{-}{\operatorname{Mod}}^{{\operatorname{gr}}}_{k}$ with $\widehat{\operatorname{Mod}}^{\operatorname{fil}}_{k}$ and provides a strongly monoidal equivalence, thereby unifying mixed graded and Beilinson filtrations in a cohesive framework. This construction yields a robust reference for mixed graded structures in derived geometry and deformation theory, with the heart of the t-structure corresponding to chain complexes when $k$ is discrete.
Abstract
In this work, we shall study in a purely model-independent fashion the $\infty$-category of mixed graded modules over a ring of characteristic $0$, and collect some basic results about its main formal properties. Finally, we shall endow such $\infty$-category with a both left and right complete accessible $t$-structure, showing how this identifies the $\infty$-category of mixed graded modules with the left completion of the Beilinson $t$-structure on the \infinity-category of filtered modules. Most of the content of this paper is already available in literature, and it serves mainly as a reference for future work.