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The Happel functor and homologically well-graded Iwanaga-Gorenstein algebras. (English) Zbl 1468.18015

Summary: D. Happel [Prog. Math. 95, 389–404 (1991; Zbl 0759.16007)] constructed a fully faithful functor \(\mathcal{H} : \mathsf{D}^{\text{b}} \pmod{\Lambda} \to \underline{\bmod}^{\mathbb{Z}} \text{T}(\Lambda)\) 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 \(\bmod^{\mathbb{Z}} \text{T}(\Lambda)\). In this paper we generalize the Happel 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 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 \(\mathcal{H}\) gives an equivalence, then it provides a canonical tilting object \(\mathcal{H}(T)\) of \(\mathsf{CM}_{\_}^{\mathbb{Z}}A\). For some special classes of finitely graded IG-algebras, our tilting objects \(\mathcal{H}(T)\) coincide with tilting object constructed in previous works.

MSC:

18G80 Derived categories, triangulated categories
16E10 Homological dimension in associative algebras
16D90 Module categories in associative algebras
16G10 Representations of associative Artinian rings

Citations:

Zbl 0759.16007
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