The Happel functor and homologically well-graded Iwanaga-Gorenstein algebras
The Happel functor and homologically well-graded Iwanaga-Gorenstein algebras
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DOI：
10.1016/j.jalgebra.2020.08.021
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发表时间：
2020-09
影响因子：
0.9
通讯作者：
H. Minamoto;K. Yamaura
中科院分区：
文献类型：
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作者：
H. Minamoto;K. Yamaura
Happl构造了一个有限维代数Λ)→的完全忠实函子H：Db(modΛmod_Z T(Λ).他还证明了这个函子H在Λ<；∞时给出了一个精确等价。因此，如果H给出等价，则它提供mod Z T(Λ)的正则倾斜对象H(Λ)。在这篇文章中，我们推广了Happl函子H，当T(Λ)被有限分次IG-代数A代替时，我们研究了这个函子是完全忠实的还是等价的。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。作为我们结果的一个结果，我们看到如果H给出等价，则它提供了CM_ZA的一个正则倾斜对象H(T)。对于某些特殊的有限分次IG-代数，我们的倾斜对象H(T)与以前工作中构造的倾斜对象重合。
Happel constructed a fully faithful functor H: D b (mod Λ)→ mod _ Z T (Λ) for a finite dimensional algebra Λ. He also showed that this functor H gives an equivalence precisely when gldim Λ<∞. Thus if H gives an equivalence, then it provides a canonical tilting object H (Λ) of mod Z T (Λ). In this paper we generalize the Happel functor H in the case where T (Λ) is replaced with a finitely graded IG-algebra A. We study when this functor is fully faithful or is 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 the Happel functor is fully faithful. We also identify the class that the Happel functor gives an equivalence. As a consequence of our result, we see that if H gives an equivalence, then it provides a canonical tilting object H (T) of CM _ Z A. For some special classes of finitely graded IG-algebras, our tilting objects H (T) coincide with tilting object constructed in previous works.