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Title: | On the obscure axiom for one-sided exact categories | Authors: | HENRARD, Ruben VAN ROOSMALEN, Adam-Christiaan |
Issue Date: | 2024 | Publisher: | ELSEVIER | Source: | JOURNAL OF PURE AND APPLIED ALGEBRA, 228 (7) (Art N° 107635) | Abstract: | One-sided exact categories are obtained via a weakening of a Quillen exact category. Such one-sided exact categories are homologically similar to Quillen exact categories: a one-sided exact category E can be (essentially uniquely) embedded into its exact hull Eex; this embedding induces a derived equivalence Db(E) -> Db(Eex). Whereas it is well known that Quillen's obscure axioms are redundant for exact categories, some one-sided exact categories are known to not satisfy the corresponding obscure axiom. In fact, we show that the failure of the obscure axiom is controlled by the embedding of E into its exact hull Eex or, equivalently, into its derived category Db(E). We introduce three versions of the obscure axiom (these versions coincide when the category is weakly idempotent complete) and establish equivalent homological properties, such as the snake lemma and the nine lemma. We show that a one-sided exact category admits a closure under each of these obscure axioms, each of which preserves the bounded derived category up to triangle equivalence. (c) 2024 Elsevier B.V. All rights reserved. | Notes: | van Roosmalen, AC (corresponding author), Xian Jiaotong Liverpool Univ, Dept Pure Math, Suzhou 215123, Peoples R China. ruben.henrard@uhasselt.be; ac.vanroosmalen@xjtlu.edu.cn |
Keywords: | Exact category;Homological algebra;Derived category;Weak idempotent completion | Document URI: | http://hdl.handle.net/1942/42932 | ISSN: | 0022-4049 | e-ISSN: | 1873-1376 | DOI: | 10.1016/j.jpaa.2024.107635 | ISI #: | 001198977200001 | Rights: | 2024 Elsevier B.V. | Category: | A1 | Type: | Journal Contribution |
Appears in Collections: | Research publications |
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