Representations of Finite-Dimensional Algebras

Almost Split Sequences and Approximations

“…In the module category over an artin algebra, the existence of almost split sequences is derived from the Auslander-Reiten duality. In a general Hom-finite Krull-Schmidt exact category, Gabriel and Roiter showed that the existence of the Auslander-Reiten duality is necessary for the existence of almost split sequences; see [11], which is later proved to be sufficient by Lenzing and Zuazua in case the category is in addition Ext-finite; see [21]. On the other hand, it is natural to study when a subcategory of a category having almost split sequences has almost split sequences.…”

Differential graded categories

“…We will state the axioms these sequences have to satisfy using the terminology of [52]: The morphisms p are called deflations, the morphisms i inflations and the pairs (i, p) conflations. The axioms are: Ex0 The identity morphism of the zero object is a deflation.…”

“…e.g. [52] In the particular case where F = S −d Σ, the triangulated category H 0 (A/F Z ) is CalabiYau [91] of CY-dimension d (cf. [84]).…”

On a problem of Nazarova and Roiter

“…By kS we denote the spectroid whose objects are the elements of S, whose morphism-spaces kS(x, y) are one-dimensional with basis (y|x) if y ≥ x, or else are 0. The composition is such that (z|y) • (y|x) = (z|x) [5]. Each interval I of S gives rise to a module k I over kS such that k I (x) = 0 if x / ∈ I and k I (y) = k, k I (z|y) = 1 1 k if y, z ∈ I and y ≤ z [5].…”

“…The composition is such that (z|y) • (y|x) = (z|x) [5]. Each interval I of S gives rise to a module k I over kS such that k I (x) = 0 if x / ∈ I and k I (y) = k, k I (z|y) = 1 1 k if y, z ∈ I and y ≤ z [5]. We set L − i = k I if I = S − i and L + i = k I if I = S + i .…”