论文标题
针对外部类别的完整同种学
Complete cohomology for extriangulated categories
论文作者
论文摘要
令$（\ Mathcal {C}，\ Mathbb {e}，\ Mathfrak {s}）$为一个外部缝制类别，具有适当的类$之一的$ \ Mathbb {e} $ - 三角形。在本文中，我们通过应用$ - 匹配$ -Projective Resolutions和$之一$（\ Mathcal {C}，\ Mathbb {E}，\ Mathfrak {s}）$。消失完整的共同体学检测有限$ξ$ - 标记维度和有限的$ξ$插入尺寸的对象。结果，我们获得了Wakamatsu倾斜猜想的有效性的一些标准，并给出了几乎Gorenstein代数为Gorenstein的必要和足够条件。此外，我们提供了一种通用技术，用于计算有限$ξ$ - $ \ MATHCAL {G} $投影维度的对象的完整共同体学。作为一个应用程序，给出了$（\ Mathcal {c}中的对象的$之一，$ξ$ -projective dimensions与$ξ$ - $ \ natercal {g} $投影尺寸。
Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an extriangulated category with a proper class $ξ$ of $\mathbb{E}$-triangles. In this paper, we study complete cohomology of objects in $(\mathcal{C},\mathbb{E},\mathfrak{s})$ by applying $ξ$-projective resolutions and $ξ$-injective coresolutions constructed in $(\mathcal{C},\mathbb{E},\mathfrak{s})$. Vanishing of complete cohomology detects objects with finite $ξ$-projective dimension and finite $ξ$-injective dimension. As a consequence, we obtain some criteria for the validity of the Wakamatsu Tilting Conjecture and give a necessary and sufficient condition for a virtually Gorenstein algebra to be Gorenstein. Moreover, we give a general technique for computing complete cohomology of objects with finite $ξ$-$\mathcal{G}$projective dimension. As an application, the relationships between $ξ$-projective dimensions and $ξ$-$\mathcal{G}$projective dimensions for objects in $(\mathcal{C},\mathbb{E},\mathfrak{s})$ are given.
