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

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 $\mathrm{\Lambda }$. He also showed that this functor $\mathcal{H}$ gives an equivalence precisely when $\text{gldim }\mathrm{\Lambda }<\mathrm{\infty }$. Thus if $\mathcal{H}$ gives an equivalence, then it provides a canonical tilting object $\mathcal{H}(\mathrm{\Lambda })$ of $\underline{$\text{mod}$}^{$\mathbb{Z}$}
$\text{T}$(\Lambda)$. In this paper we generalize Happel’s functor $\mathcal{H}$ in the case where $\text{T}(\mathrm{\Lambda })$ 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 ${\underset{‾}{\text{CM}}}^{\mathbb{Z}}A$. For some special classes of finitely graded IG algebras, our tilting objects $\mathcal{H}(T)$ coincide with tilting object constructed in previous works.