论文标题
在单方面的晦涩的公理上
On the obscure axiom for one-sided exact categories
论文作者
论文摘要
单方面的精确类别是通过削弱Quillen精确类别获得的。这种单方面的精确类别在同源上与Quillen的精确类别相似：单方面的类别$ \ Mathcal {E} $可以（本质上是唯一的）嵌入其精确的船体$ {\ Mathcal {\ Mathcal {e}}}^{\ textrm {extrm {ex}} $;这嵌入诱导派生的等价$ \ textbf {d}^b（\ Mathcal {e}）\ to \ textbf {d}^b（{\ Mathcal {e}}}}}^{\ textrm {extrm {extrm {ex}}}）$。 众所周知，Quillen的晦涩公理对于精确的类别是多余的，但已知某些单面精确类别不满足相应的晦涩的公理。实际上，我们表明，晦涩的公理的故障是由$ \ mathcal {e} $的嵌入到其精确的船体$ {\ Mathcal {e}}}^{\ textrm {extrm {extrm {ex}} $中控制的。 在本文中，我们介绍了三个版本的晦涩的公理（这些版本是在类别弱化完成时重合的），并建立等效的同源性能，例如蛇的引理和九个引理。我们表明，单方面的类别承认在每个晦涩的公理下的封闭中，每个公理都保留了有限的派生类别，直至三角等效。
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 $\mathcal{E}$ can be (essentially uniquely) embedded into its exact hull ${\mathcal{E}}^{\textrm{ex}}$; this embedding induces a derived equivalence $\textbf{D}^b(\mathcal{E}) \to \textbf{D}^b({\mathcal{E}}^{\textrm{ex}})$. 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 $\mathcal{E}$ into its exact hull ${\mathcal{E}}^{\textrm{ex}}.$ In this paper, 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.
