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

Abstract

Happel constructed a fully faithful functor $\mathcal{H}:{\mathsf{D}}^{\mathrm{b}}(\text{mod}\Lambda )\to {\underline{\text{mod}}}^{\mathbb{Z}}\text{T}(\Lambda )$ 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 ${\underline{\text{CM}}}^{\mathbb{Z}}A$. For some special classes of finitely graded IG algebras, our tilting objects $\mathcal{H}(T)$ coincide with tilting object constructed in previous works.

38 pages. v.3 expositions of graded modules and their complexes added

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Cited by in corpus (3)

• Tilting modules and dominant dimension with respect to injective modules
• On commutative differential graded algebras
• On finitely graded Iwanaga-Gorenstein algebras and the stable categories of their (graded) Cohen-Macaulay modules