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 -