Happel's functor and homologically well-graded Iwanaga-Gorenstein algebras

Abstract

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.