论文标题
通过态度类别引起的外侧类别产生的阿贝尔商
Abelian quotients arising from extriangulated categories via morphism categories
论文作者
论文摘要
我们研究了通过形态类别引起的外侧类别产生的阿贝尔商，这是针对确切类别和三角形类别的统一治疗方法。令$（\ Mathcal {C}，\ Mathbb {e}，\ Mathfrak {s}）$是一个外部类别，具有足够的投影$ \ Mathcal {p} $和$ \ Mathcal {m} $ be be $ \ mathcal {C c} $ can $ canc $ cage $ cancal $ cage $ cage $ cage $ {p pp p pp p pp p pp pp p p \ \ pp \ pp \ pp p pp \ pp p pp \ pp \ pp \ pp \ pp \ pp \ pp \ pp \ pp calc。我们表明，$ \ Mathfrak {s} \ textup {-def}（\ Mathcal {m}）$的某些商类别，$ \ Mathfrak {s} $ - deflations $ f：m_ {1} \ rightArrow m_2 $ with $ m_1，m_2 $} $ nas as b in \ in \ in \ in \ n M.2 $。我们的主要定理有两个应用。如果$ \ MATHCAL {M} = \ MATHCAL {C} $，我们获得了某些理想的商类别$ \ Mathfrak {s} \ textup {-tri}（\ Mathcal {c}）/\ Mathcal {r} $ \ textup {mod - } \ mathcal {c}/[\ mathcal {p}] $，其中$ \ mathfrak {s} $ - tri $（\ mathcal {c}）$是所有$ \ mathfrak {s} $ - triangles的类别。如果$ \ MATHCAL {M} $是一个刚性的子类别，我们表明$ \ Mathcal {M} _ {l}/[\ Mathcal {M}] \ cong \ textup {mod-} $ \ MATHCAL {M} _ {l}/[ω\ MATHCAL {M}] \ CONC（\ textUp {mod-}（\ Mathcal {M}/[\ Mathcal {p} {p}） $ \ MATHCAL {m} _l $（resp。$ω\ Mathcal {m} $）是$ \ Mathcal {c} $的完整子类别， $ \ xymatrixRowSep {0.1pc} \ xymatrix {x \ ar [r]＆m_1 \ ar [r]＆m_2 \ ar@ar@ar@{ - >} [r]＆} \ xymatRixRowSep {0.1pc} \ xymatrix {x \ ar [r]＆p \ ar [r]＆m \ ar@ar@ar { - >} [r]＆}）$，带有$ m_1，m_1，m_2 \ in \ mathcal {m} $ p \ in \ Mathcal {p} $）。特别是，我们有$ \ MATHCAL {C}/[\ MATHCAL {M}] \ cong \ textup {mod-}（\ Mathcal {m}/[\ Mathcal {p}]）$ $ \ MATHCAL {C}/[ω\ MATHCAL {M}] \ CONC（\ textUp {mod-}（\ Mathcal {M}/[\ Mathcal {\ Mathcal {p}]）^{\ textUp {op}}}}}子类别。
We investigate abelian quotients arising from extriangulated categories via morphism categories, which is a unified treatment for both exact categories and triangulated categories. Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with enough projectives $\mathcal{P}$ and $\mathcal{M}$ be a full subcategory of $\mathcal{C}$ containing $\mathcal{P}$. We show that certain quotient category of $\mathfrak{s}\textup{-def}(\mathcal{M})$, the category of $\mathfrak{s}$-deflations $f:M_{1}\rightarrow M_2$ with $M_1,M_2\in\mathcal{M}$, is abelian. Our main theorem has two applications. If $\mathcal{M}=\mathcal{C}$, we obtain that certain ideal quotient category $\mathfrak{s}\textup{-tri}(\mathcal{C})/\mathcal{R}_2$ is equivalent to the category of finitely presented modules $\textup{mod-}\mathcal{C}/[\mathcal{P}]$, where $\mathfrak{s}$-tri$(\mathcal{C})$ is the category of all $\mathfrak{s}$-triangles. If $\mathcal{M}$ is a rigid subcategory, we show that $\mathcal{M}_{L}/[\mathcal{M}]\cong\textup{mod-}(\mathcal{M}/[\mathcal{P}])$ and $\mathcal{M}_{L}/[Ω\mathcal{M}]\cong(\textup{mod-}(\mathcal{M}/[\mathcal{P}])^{\textup{op}})^{\textup{op}}$, where $\mathcal{M}_L$ (resp. $Ω\mathcal{M}$) is the full subcategory of $\mathcal{C}$ of objects $X$ admitting an $\mathfrak{s}$-triangle $\xymatrixrowsep{0.1pc}\xymatrix{X\ar[r]&M_1\ar[r] & M_2\ar@{-->}[r]&} (\textup{resp.} \xymatrixrowsep{0.1pc}\xymatrix{X\ar[r]&P\ar[r] & M\ar@{-->}[r]&})$ with $M_1, M_2\in\mathcal{M}$ (resp. $M\in\mathcal{M}$ and $P\in\mathcal{P}$). In particular, we have $\mathcal{C}/[\mathcal{M}]\cong\textup{mod-}(\mathcal{M}/[\mathcal{P}])$ and $\mathcal{C}/[Ω\mathcal{M}]\cong(\textup{mod-}(\mathcal{M}/[\mathcal{P}])^{\textup{op}})^{\textup{op}}$ provided that $\mathcal{M}$ is a cluster-tilting subcategory.
