Let $$𝕂$$ be an algebraically closed field. Consider a finite dimensional monomial relations algebra $$\Lambda =𝕂\Gamma /\phantom{𝕂\Gamma I}I$$ of finite global dimension, where Γ is a quiver and I an admissible ideal generated by a set of paths from the path algebra $$𝕂\Gamma$$ . There are many modules over Λ which may be represented graphically by a tree with respect to a top element, of which the indecomposable projectives are the most natural example. These trees possess branches which correspond to right subpaths of cycles in the quiver. A pattern...

Let $A$ be a standard Koszul standardly stratified algebra and $X$ an $A$-module. The paper investigates conditions which imply that the module ${\mathrm{Ext}}_A^{*}\left(X\right)$ over the Yoneda extension algebra $A^{*}$ is filtered by standard modules. In particular, we prove that the Yoneda extension algebra of $A$ is also standardly stratified. This is a generalization of similar results on quasi-hereditary and on graded standardly stratified algebras.

Let $𝒲$ be a self-orthogonal class of left $R$-modules. We introduce a class of modules, which is called strongly $𝒲$-Gorenstein modules, and give some equivalent characterizations of them. Many important classes of modules are included in these modules. It is proved that the class of strongly $𝒲$-Gorenstein modules is closed under finite direct sums. We also give some sufficient conditions under which the property of strongly $𝒲$-Gorenstein module can be inherited by its submodules and quotient modules....

We show that there is a one-to-one correspondence between basic cotilting complexes and certain contravariantly finite subcategories of the bounded derived category of an artin algebra. This is a triangulated version of a result by Auslander and Reiten. We use this to find an existence criterion for complements to exceptional complexes.