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arXiv · 1903.10861

Localizations of (one-sided) exact categories

Ruben Henrard·Adam-Christiaan van Roosmalen
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Abstract

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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Ruben Henrard, Adam-Christiaan van Roosmalen. 2020-06-18. Localizations of (one-sided) exact categories. https://arxiv.org/abs/1903.10861

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