$\boldsymbol{n}$-Recollements and upper recollements of derived module categories

Abstract

For Artin algebras,we characterize $1$-recollements and $2$-recollements of their unbounded derived categories in terms ofupper (respectively lower) recollements of certain subcategories.As a result, we clarify the relationship betweenupper (respectively lower) recollements of derived categoriesand the finiteness of global (respectively finiteness) dimension of algebras,which generalizes a result of Yin and Gao (2016).Let $A$, $B$, and $C$ be algebras over a field $k$, and$\mathcal{D}A$admit a $3$-recollement relative to $\mathcal{D}B$ and $\mathcal{D}C$.We prove that $A$ is finite-dimensionalif and only if so are $B$ and $C$.