Recollements from generalized tilting

TL;DR

This paper constructs recollements in derived categories from generalized tilting objects, providing a unified framework that applies to dg categories, tilting modules, and categories of simple modules, extending known results.

Contribution

It introduces a method to produce recollements from generalized tilting, encompassing dg categories, tilting modules, and simple module categories, unifying various cases.

Findings

Recollements are constructed from standard lifts of subcategories in derived categories.

The framework applies to dg categories, tilting modules, and simple module categories.

The results extend known recollement constructions to broader contexts.

Abstract

Let $\ca$ be a small dg category over a field $k$ and let $\cu$ be a small full subcategory of the derived category $\cd\ca$ which generate all free dg $\ca$-modules. Let $(\cb,X)$ be a standard lift of $\cu$. We show that there is a recollement such that its middle term is $\cd\cb$, its right term is $\cd\ca$, and the three functors on its right side are constructed from $X$. This applies to the pair $(A,T)$, where $A$ is a $k$-algebra and $T$ is a good $n$-tilting module, and we obtain a result of Bazzoni--Mantese--Tonolo. This also applies to the pair $(\ca,\cu)$, where $\ca$ is an augmented dg category and $\cu$ is the category of `simple' modules, e.g. $\ca$ is a finite-dimensional algebra or the Kontsevich--Soibelman $A_\infty$-category associated to a quiver with potential.