Let $${\mathcal {X}}$$ X be a contravariantly finite resolving subcategory of $${\mathrm{{mod\text{- }}}}\varLambda $$ mod - Λ , the category of finitely generated right $$\varLambda $$ Λ -modules. We associate to $${\mathcal {X}}$$ X the subcategory $${\mathcal {S}}_{{\mathcal {X}}}(\varLambda )$$ S X ( Λ ) of the morphism category $$\mathrm{H}(\varLambda )$$ H ( Λ ) consisting of all monomorphisms $$(A{\mathop {\rightarrow }\limits ^{f}}B)$$ ( A → f B ) with A , B and $$\text {Cok} \ f$$ Cok f in $${\mathcal {X}}$$ X . Since $${\mathcal {S}}_{{\mathcal {X}}}(\varLambda )$$ S X ( Λ ) is closed under extensions it inherits naturally an exact structure from $$\mathrm{H}(\varLambda )$$ H ( Λ ) . We will define two other different exact structures other than the canonical one on $${\mathcal {S}}_{{\mathcal {X}}}(\varLambda )$$ S X ( Λ ) , and completely classify the indecomposable projective (resp. injective) objects in the corresponding exact categories. Enhancing $${\mathcal {S}}_{{\mathcal {X}}}(\varLambda )$$ S X ( Λ ) with the new exact structure provides a framework to construct a triangle functor. Let $${\mathrm{{mod\text{- }}}}{\underline{{\mathcal {X}}}}$$ mod - X ̲ denote the category of finitely presented functors over the stable category $${\underline{{\mathcal {X}}}}$$ X ̲ . We then use the triangle functor to show a triangle equivalence between the bounded derived category $${\mathbb {D}}^{\mathrm{b}}({\mathrm{{mod\text{- }}}}{\underline{{\mathcal {X}}}})$$ D b ( mod - X ̲ ) and a Verdier quotient of the bounded derived category of the associated exact category on $${\mathcal {S}}_{{\mathcal {X}}}(\varLambda )$$ S X ( Λ ) . Similar consideration is also given for the singularity category of $${\mathrm{{mod\text{- }}}}{\underline{{\mathcal {X}}}}$$ mod - X ̲ .

中文翻译：

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单态范畴上的不同精确结构

令 $${\mathcal {X}}$$ X 是 $${\mathrm{{mod\text{- }}}}\varLambda $$ mod - Λ 的逆变有限解析子范畴，有限生成权的范畴$$\varLambda $$ Λ -modules。我们将 $${\mathcal {X}}$$ X 关联到子范畴 $${\mathcal {S}}_{{\mathcal {X}}}(\varLambda )$$ SX ( Λ ) $$\mathrm{H}(\varLambda )$$ H ( Λ ) 由所有单态 $$(A{\mathop {\rightarrow }\limits ^{f}}B)$$ ( A → f B ) 组成A , B 和 $$\text {Cok} \ f$$ Cok f in $${\mathcal {X}}$$ X 。由于 $${\mathcal {S}}}_{{\mathcal {X}}}(\varLambda )$$ SX ( Λ ) 在扩展下是封闭的，它自然继承了 $$\mathrm{H}(\ varLambda )$$ H ( Λ ) 。我们将在 $${\mathcal {S}}_{{\mathcal {X}}}(\varLambda )$$ SX ( Λ ) 上定义除规范结构之外的另外两种不同的精确结构，并将不可分解的射影（或单射）对象完全分类在相应的确切类别中。用新的精确结构增强 $${\mathcal {S}}_{{\mathcal {X}}}(\varLambda )$$ SX ( Λ ) 提供了一个构建三角函子的框架。令 $${\mathrm{{mod\text{- }}}}{\underline{{\mathcal {X}}}}$$ mod - X ̲ 表示在稳定范畴 $${ 上的有限呈现函子的范畴\underline{{\mathcal {X}}}}$$ X ̲ . 然后我们使用三角形函子来展示有界派生范畴 $${\mathbb {D}}^{\mathrm{b}}({\mathrm{{mod\text{- }}}}{\下划线{{\mathcal {X}}}})$$ D b ( mod - X ̲ ) 和 $${\mathcal {S}}}_{{\数学 {X}}}(\varLambda )$$ SX ( Λ ) 。