Happel's functor and homologically well-graded Iwanaga-Gorenstein algebras
Happel's functor and homologically well-graded Iwanaga-Gorenstein algebras
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发表时间：
2018-11
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通讯作者：
H. Minamoto;K. Yamaura
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作者：
H. Minamoto;K. Yamaura
Happl构造了一个完全忠实的函子$\mathcal{H}：\mathsf{D}^{\mathrm{b}}(\Text{mod}\\lambda)\为有限维代数$\lambda$在{\Text{mod}}^{\bbb{Z}}\\Text{T}(\Lambda)$下划线。他还证明了当$\Text{gldim}\lambda<；\infty$时，这个函子$\mathcal{H}$给出了一个等价。因此，如果$\mathcal{H}$给出等价性，则它提供$\mathcal{H}(\lambda)$的规范倾斜对象$\下划线{\Text{mod}}^{\mathbb{Z}}\\Text{T}(\Lambda)$。在本文中，我们推广了Happl函子$\mathcal{H}$，当$\Text{T}(\Lambda)$替换为有限分次IG代数$A$时。我们研究这个函子什么时候是完全忠诚的或给出等价的。为此，我们引入了同调良好分次(HWG)IG-代数的概念，它可以被刻画为一个代数具有同调对称性，并且后验地保证该代数是IG。证明了HWG IG-代数正是Happl函子完全忠实的有限分次IG-代数类。我们还确定了Happl函子给出的等价类。作为我们结果的结果，我们看到如果$\mathcal{H}$给出一个等价，则它提供了一个标准倾斜对象$\mathcal{H}(T)$$\下划线{\Text{CM}}^{\bbb{Z}}A$。对于某些特殊的有限分次IG代数，我们得到的倾斜对象与前人构造的倾斜对象重合。
Happel constructed a fully faithful functor $\mathcal{H} :\mathsf{D}^{\mathrm{b}}(\text{mod} \ \Lambda) \to \underline{\text{mod}}^{\Bbb{Z}} \ \text{T}(\Lambda)$ for a finite dimensional algebra $\Lambda$. He also showed that this functor $\mathcal{H}$ gives an equivalence precisely when $\text{gldim } \Lambda < \infty$. Thus if $\mathcal{H}$ gives an equivalence, then it provides a canonical tilting object $\mathcal{H} (\Lambda)$ of $\underline{\text{mod}}^{\mathbb{Z}} \ \text{T}(\Lambda)$. In this paper we generalize Happel's functor $\mathcal{H}$ in the case where $\text{T}(\Lambda)$ is replaced with a finitely graded IG algebra $A$. We study when this functor is fully faithful or gives an equivalence. For this purpose we introduce the notion of homologically well-graded (hwg) IG-algebra, which can be characterized as an algebra posses a homological symmetry which, a posteriori, guarantee that the algebra is IG. We prove that hwg IG-algebras is precisely the class of finitely graded IG-algebras that Happel's functor is fully faithful. We also identify the class that Happel's functor gives an equivalence. As a consequence of our result, we see that if $\mathcal{H}$ gives an equivalence, then it provides a canonical tilting object $\mathcal{H}(T)$ of $\underline{\text{CM}}^{\Bbb{Z}} A$. For some special classes of finitely graded IG algebras, our tilting objects $\mathcal{H}(T)$ coincide with tilting object constructed in previous works.