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Title: | Auslander's formula and correspondence for exact categories | Authors: | HENRARD, Ruben Sondre Kvamme VAN ROOSMALEN, Adam-Christiaan |
Issue Date: | 2020 | Publisher: | ACADEMIC PRESS INC ELSEVIER SCIENCE | Source: | ADVANCES IN MATHEMATICS (401) | Abstract: | The Auslander correspondence is a fundamental result in Auslander-Reiten theory. In this paper we introduce the category $\operatorname{mod_{\mathsf{adm}}}(\mathcal{E})$ of admissibly finitely presented functors and use it to give a version of Auslander correspondence for any exact category $\mathcal{E}$. An important ingredient in the proof is the localization theory of exact categories. We also investigate how properties of $\mathcal{E}$ are reflected in $\operatorname{mod_{\mathsf{adm}}}(\mathcal{E})$, for example being (weakly) idempotent complete or having enough projectives or injectives. Furthermore, we describe $\operatorname{mod_{\mathsf{adm}}}(\mathcal{E})$ as a subcategory of $\operatorname{mod}(\mathcal{E})$ when $\mathcal{E}$ is a resolving subcategory of an abelian category. This includes the category of Gorenstein projective modules and the category of maximal Cohen-Macaulay modules as special cases. Finally, we use $\operatorname{mod_{\mathsf{adm}}}(\mathcal{E})$ to give a bijection between exact structures on an idempotent complete additive category $\mathcal{C}$ and certain resolving subcategories of $\operatorname{mod}(\mathcal{C})$. | Keywords: | Mathematics - Representation Theory | Document URI: | http://hdl.handle.net/1942/32745 | Link to publication/dataset: | http://arxiv.org/abs/2011.15107v1 | ISSN: | 0001-8708 | e-ISSN: | 1090-2082 | DOI: | 10.1016/j.aim.2022.108296 | ISI #: | 000793102900002 | Category: | R2 | Type: | Research Report | Validations: | ecoom 2023 |
Appears in Collections: | Research publications |
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