Recollement of homotopy categories and Cohen-Macaulay modules

n -representation-finite algebras and twisted fractionally Calabi-Yau algebras

“…The derived categories of finite dimensional non-semisimple algebras are never CY, but they are often fractionally CY. Some recent meetings [Ba, Bi, P, T] as well as recent results including [D,DL,IIKNS,IKM,KST,Ke2,Ladk,Le,Mi,MY,Y] suggest that the fractionally CY property becomes more and more important in representation theory, singularity theory, commutative and non-commutative algebraic geometry. The aim of this short paper is to apply the fractionally CY property in the study of n-representation-finite algebras defined below.…”

The stable monomorphism category of a Frobenius category

“…The next observation is essentially due to Li and Zhang ([27, Theorem 1.1]; also see [18,Proposition 3.6]). Recall that for an artin algebra A, a (right) module over T 2 (A) is identified with a morphism of (right) A-modules; in fact, this yields an equivalence mod T 2 (A) ≃ Mor(mod A) of categories; see [3, Chapter III, Proposition 2.2].…”

“…We show that Mon(A) is a Frobenius exact category and then the stable category Mon(A) modulo projective objects is triangulated; it is called the stable monomorphism category of A. Recently this category is also studied by [18]. Note that the triangulated categories above are algebraical in the sense of Keller.…”

“…With a slightly modified proof as above, one can show that a similar result holds if the category A is an exact category. In this case, one replaces Mon(A) by the category of inflations in A; compare [19, section 5] and [18].…”