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تسجيل مستخدم جديدLocalizations of (one-sided) exact categories
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In this paper, we introduce quotients of exact categories by percolating subcategories. This approach extends earlier localization theories by Cardenas and Schlichting for exact categories, allowing new examples. Let $mathcal{A}$ be a percolating subcategory of an exact category $mathcal{E}$, the quotient $mathcal{E} {/mkern-6mu/} mathcal{A}$ is constructed in two steps. In the first step, we associate a set $S_mathcal{A} subseteq operatorname{Mor}(mathcal{E})$ to $mathcal{A}$ and consider the localization $mathcal{E}[S^{-1}_mathcal{A}]$. In general, $mathcal{E}[S_mathcal{A}^{-1}]$ need not be an exact category, but will be a one-sided exact category. In the second step, we take the exact hull $mathcal{E} {/mkern-6mu/} mathcal{A}$ of $mathcal{E}[S_mathcal{E}^{-1}]$. The composition $mathcal{E} rightarrow mathcal{E}[S_mathcal{A}^{-1}] rightarrow mathcal{E} {/mkern-6mu/} mathcal{A}$ satisfies the 2-universal property of a quotient in the 2-category of exact categories. We formulate our results in slightly more generality, allowing to start from a one-sided exact category. Additionally, we consider a type of percolating subcategories which guarantee that the morphisms of the set $S_mathcal{A}$ are admissible. In upcoming work, we show that these localizations induce Verdier localizations on the level of the bounded derived category.
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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
ass $S_mathcal{A} subseteq operatorname{Mor}(mathcal{E})$ 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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