Search

We introduce higher dimensional analogues of the Nakayama algebras from the viewpoint of Iyama's higher Auslander–Reiten theory. More precisely, for each Nakayama algebra A and each positive integer d , we construct a finite dimensional algebra A (d) having a distinguished d -cluster-tilting A (d) -module whose endomorphism algebra is a higher dimensional analogue of the Auslander algebra of A. We also construct higher dimensional analogues of the mesh category of type ZA ∞ and the tubes. [ABSTRACT FROM AUTHOR]

Derived equivalences and t-structures are closely related. We use realisation functors associated to t-structures in triangulated categories to establish a derived Morita theory for abelian categories with a projective generator or an injective cogenerator. For this purpose we develop a theory of (non-compact, or large) tilting and cotilting objects that generalises the preceding notions in the literature. Within the scope of derived Morita theory for rings we show that, under some assumptions, the realisation functor is a derived tensor product. This fact allows us to approach a problem by Rickard on the shape of derived equivalences. Finally, we apply the techniques of this new derived Morita theory to show that a recollement of derived categories is a derived version of a recollement of abelian categories if and only if there are tilting or cotilting t-structures glueing to a tilting or a cotilting t-structure. As a further application, we answer a question by Xi on a standard form for recollements of derived module categories for finite dimensional hereditary algebras. [ABSTRACT FROM AUTHOR]

For a fixed finite dimensional algebra A, we study representation embeddings of the form $$mod(B)\rightarrow mod(A)$$ . Such an embedding is called homological, if it induces an isomorphism on all Ext-groups and weakly homological, if only Ext $$^1$$ is preserved. In case A is a preprojective algebra of Dynkin type, we give an explicit classification of all weakly homological and homological embeddings. Furthermore, we show that for self-injective algebras a classification of homological embeddings becomes accessible once these algebras fulfil the Tachikawa conjecture. [ABSTRACT FROM AUTHOR]