2018/11/20 by Hiroyuki Minamoto, Minamoto, Hiroyuki, Kota Yamaura +1
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.1811.08036
openalex publication_date 2018/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Happel constructed a fully faithful functor H :Db(mod Λ) → \underlinemodℤ 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 \underlinemodℤ T(Λ). In this paper we generalize Happel's 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 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 H gives an equivalence, then it provides a canonical tilting object H(T) of \underlineCMℤ A. For some special classes of finitely graded IG algebras, our tilting objects H(T) coincide with tilting object constructed in previous works.