Let A be a commutative ring, B a commutative A-algebra and M a complex of B-modules. We begin by constructing the square SqB/A M , which is also a complex of B-modules. The squaring operation is a quadratic functor, and its construction requires differential graded (DG) algebras. If there exists an isomorphism ρ : M ≃ −→ SqB/A M then the pair (M,ρ) is called a rigid complex over B relative to A...

let $r$ be a commutative noetherian ring with non-zero identity, $fa$ an ideal of $r$, and $x$ an $r$--module. here, for fixed integers $s, t$ and a finite $fa$--torsion $r$--module $n$, we first study the membership of $ext^{s+t}_{r}(n, x)$ and $ext^{s}_{r}(n, h^{t}_{fa}(x))$ in the serre subcategories of the category of $r$--modules. then, we present some conditions which ensure the exi...

we introduce the notions of t-dual rickart and strongly t-dual rickart modules. we provide several characterizations and investigate properties of each of these concepts. it is shown that every free (resp. finitely generated free) $r$-module is t-dual rickart if and only if $overline{z}^2(r)$ is a  direct summand of $r$ and end$(overline{z}^2(r))$ is a semisimple (resp. regular) ring. it is sho...

Perovskite solar cells have shown considerable developments in the last decade, and commercial applications are drawing closer. In this article, we present a techno-economic study of perovskite PV technologies. We...

This paper proves Lee-Lee's conjecture that establishes a coincidence between the set of associated roots non-self-intersecting curves in an n-punctured disc and real Schur acyclic (valued) quivers with n vertices.

We introduce the notion of mutation on the set of n-cluster tilting subcategories in a triangulated category with Auslander-Reiten-Serre duality. Using this idea, we are able to obtain the complete classifications of rigid Cohen-Macaulay modules over certain Veronese subrings.

We study cluster tilting modules in mesh algebras of Dynkin type as defined [12], providing a new proof for their existence. Except G2, we show that these are precisely the maximal rigid modules, and they equivariant certain automorphism. further mutation, an example mutation abelian category which is not stably 2-Calabi-Yau, explicitly describe combinatorics.