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Register a new userA reduction approach to silting objects for derived categories of hereditary categories
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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.
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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 $mathcal B$ with respect to a pre-silting subcategory $mathcal W$ can be realized as a certain subfactor category of $mathcal B$. This generalizes the result by Iyama-Yang. In particular, for a Gorenstein algebra, we get the relative version of the description of the singularity category due to Happel and Chen-Zhang by this reduction.
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 derived functors between them, provides a significantly richer and more flexible machinery than the old homological algebra. For instance, the important concepts of dualizing complex and tilting complex do not exist in the old homological algebra. This paper is an edited version of the notes for a two-lecture minicourse given at MSRI in January 2013. Sections 1-5 are about the general theory of derived categories, and the material is taken from my manuscript A Course on Derived Categories (available online). Sections 6-9 are on more specialized topics, leaning towards noncommutative algebraic geometry.
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 investigate how to establish recollements of triangulated categories from recollements of abelian categories by using the theory of exact model structures. Finally, we give applications to contraderived categories, projective stable derived categories and stable categories of Gorenstein injective modules over an upper triangular matrix ring.
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 these resolutions. We also give an alternative description of the morphism sets in terms of Yoneda Ext groups.
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 class $S_mathcal{A} subseteq operatorname{Mor}(mathcal{E})$ of morphisms. The localization $mathcal{E}[S_mathcal{A}^{-1}]$ need not be an exact category, but will be a one-sided exact category. Secondly, one constructs the exact hull $mathcal{E}{/mkern-6mu/} mathcal{A}$ of $mathcal{E}[S_mathcal{A}^{-1}]$ and shows that this satisfies the 2-universal property of a quotient amongst exact categories. In this paper, we show that this quotient $mathcal{E} to mathcal{E} {/mkern-6mu/} mathcal{A}$ induces a Verdier localization $mathbf{D}^b(mathcal{E}) to mathbf{D}^b(mathcal{E} {/mkern-6mu/} mathcal{A})$ of bounded derived categories. Specifically, (i) we study the derived category of a one-sided exact category, (ii) we show that the localization $mathcal{E} to mathcal{E}[S_mathcal{A}^{-1}]$ induces a Verdier quotient $mathbf{D}^b(mathcal{E}) to mathbf{D}^b(mathcal{E}[S^{-1}_mathcal{A}])$, and (iii) we show that the natural embedding of a one-sided exact category $mathcal{F}$ into its exact hull $overline{mathcal{F}}$ lifts to a derived equivalence $mathbf{D}^b(mathcal{F}) to mathbf{D}^b(overline{mathcal{F}})$. We furthermore show that the Verdier localization is compatible with several enhancements of the bounded derived category, so that the above Verdier localization can be used in the study of localizing invariants, such as non-connective $K$-theory.
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