let $mathcal{x}$ be a class of $r$-modules‎. ‎in this paper‎, ‎we investigate ;$mathcal{x}$-injective (projective) and dg-$mathcal{x}$-injective (projective) complexes which are generalizations of injective (projective) and dg-injecti‎‎ve (projective) complexes‎. ‎we prove that some known results can be extended to the class of ;$mathcal{x}$-injective (projective) and dg-$mathcal{x}$-injective ...

For a monomial special multiserial algebra, which in general is of wild representation type, we construct a radical embedding into an algebra of finite representation type. As a consequence, we show that the representation dimension of monomial and self-injective special multiserial algebras is less than or equal to three.

Let [Formula: see text] be an extriangulated category with a proper class of text]-triangles. We study complete cohomology objects in by applying text]-projective resolutions and text]-injective coresolutions constructed text]. Vanishing detects finite dimension dimension. As consequence, we obtain some criteria for the validity Wakamatsu tilting conjecture give necessary sufficient condition v...

Alexander duality is made into a functor which extends the notion for monomial ideals to any finitely generated N-graded module. The functors associated with Alexander duality provide a duality on the level of free and injective resolutions, and numerous Bass and Betti number relations result as corollaries. A minimal injective resolution of a module M is equivalent to the injective resolution ...

Recently, Yekutieli introduced projective dimension, injective dimension and flat of DG-modules by generalizing the characterization ordinary modules vanishing Ext or Tor-groups. In this paper, we introduce a DG-version projective, resolution for over connective DG-algebra which are different from known resolutions. An important feature these resolutions is that, roughly speaking, “length” give...