TY - JOUR

T1 - On the obscure axiom for one-sided exact categories

AU - Henrard, Ruben

AU - van Roosmalen, Adam Christiaan

N1 - Publisher Copyright:
© 2024 Elsevier B.V.

PY - 2024/7

Y1 - 2024/7

N2 - 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.

AB - 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.

KW - Derived category

KW - Exact category

KW - Homological algebra

KW - Weak idempotent completion

UR - http://www.scopus.com/inward/record.url?scp=85186537995&partnerID=8YFLogxK

U2 - 10.1016/j.jpaa.2024.107635

DO - 10.1016/j.jpaa.2024.107635

M3 - Article

AN - SCOPUS:85186537995

SN - 0022-4049

VL - 228

JO - Journal of Pure and Applied Algebra

JF - Journal of Pure and Applied Algebra

IS - 7

M1 - 107635

ER -