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Let $mathcal{H}$ be a hereditary abelian category over a field $k$ with finite dimensional $operatorname{Hom}$ and $operatorname{Ext}$ spaces. It is proved that the bounded derived category $mathcal{D}^b(mathcal{H})$ has a silting object iff $mathcal{H}$ has a tilting object iff $mathcal{D}^b(mathcal{H})$ has a simple-minded collection with acyclic $operatorname{Ext}$-quiver. Along the way, we obtain a new proof for the fact that every presilting object of $mathcal{D}^b(mathcal{H})$ is a partial silting object. We also consider the question of complements for pre-simple-minded collections. In contrast to presilting objects, a pre-simple-minded collection $mathcal{R}$ of $mathcal{D}^b(mathcal{H})$ can be completed into a simple-minded collection iff the $operatorname{Ext}$-quiver of $mathcal{R}$ is acyclic.
We introduce pre-silting and silting subcategories in extriangulated categories and generalize the silting theory in triangulated categories. We prove that the silting reduction $mathcal B/({rm thick}mathcal W)$ of an extriangulated category $mathcal
Derived categories were invented by Grothendieck and Verdier around 1960, not very long after the old homological algebra (of derived functors between abelian categories) was established. This new homological algebra, of derived categories and derive
In this paper, we first provide an explicit procedure to glue complete hereditary cotorsion pairs along the recollement $(mathcal{A},mathcal{C},mathcal{B})$ of abelian categories with enough projective and injective objects. As a consequence, we inve
Each object of any abelian model category has a canonical resolution as described in this article. When the model structure is hereditary we show how morphism sets in the associated homotopy category may be realized as cohomology groups computed from
We consider the quotient of an exact or one-sided exact category $mathcal{E}$ by a so-called percolating subcategory $mathcal{A}$. For exact categories, such a quotient is constructed in two steps. Firstly, one localizes $mathcal{E}$ at a suitable cl