The purpose of the present paper is to investigate cohomology of differential Lie algebra morphisms. First, we discuss representations of differential Lie algebra morphisms. Moreover, we construct cohomology of differential Lie algebra morphisms. As applications of our cohomology, we study formal deformations and abelian extensions of differential ] Lie algebra morphisms.

Effective codescent morphisms of ternary rings are characterized. To this end, it is shown that, the variety of ternary rings satisfies the strong amalgamation property, and, moreover, in this variety, the elements of amalgamated free products have unique normal forms. In view of the fact that the category of ternary rings contains the category of commutative associative unitary rings as a full subcategory, the class of effective codescent morphisms in the latter category (which, according to the well-known Joyal-Tierney's criterion, are precisely monomorphisms $R\rightarrowtail S$ which are pure as monomorphisms of $R$-modules) is compared with that of morphisms between commutative associative unitary rings which are effective codescent in the category of ternary rings. It turns out that the former class is contained in the latter one, but does not coincide with it.

In the domain of Riemannian Geometry, we explore $*$-$\eta$-Ricci-Yamabe soliton on a Riemannian manifold $(G^3, g)$. Initially, we establish that if the metric $g$ of $G^3$ constitutes a $*$-$\eta$-Ricci-Yamabe soliton, then $G^3$ is necessarily Einstein, when the soliton vector field $V$ is contact. Additionally, we investigated that the Riemannian manifold $(G^3, g),$ accommodates a gradient almost $*$-$\eta$-Ricci-Yamabe soliton, concluding that it must be Einstein with a consistent scalar curvature $r=-6.$ The associated functions of the $*$-$\eta$-Ricci soliton are characterized by $\alpha=1$ and $\beta=0.$

The purpose of this paper is to give, in the context of enriched category theory, some equivalent conditions characterising simultaneously reflective and coreflective full subcategories of the category of functors on a given small category. In particular, it is proved that such subcategories are (equivalent to) functor categories.

Let Ln for a positive integer n denote the stable homotopy category of v −1 n BP -local spectra at a prime number p. Then, M. Hopkins defines the Picard group of Ln as a collection of isomorphism classes of invertible spectra, whose exotic summand Pic 0 (Ln) is studied by several authors. In this paper, we study the summand for n with n 2 ≤ 2p + 2. For n 2 ≤ 2p − 2, it consists of invertible spectra whose K(n)-localization is the K(n)-local sphere. In particular, X is an exotic invertible spectrum of Ln if and only if X ∧ M J is isomorphic to a v −1 n BP -localization of the generalized Moore spectrum M J for an invarinat regular ideal J of length n. For n with 2p−2 < n 2 ≤ 2p+2, we consider the cases for (p, n) = (5, 3) and (7, 4) . In these cases, we characterize them by the Smith-Toda spectra V (n − 1). For this sake, we show that L 3 V (2) at the prime five and L 4 V (3) at the prime seven are ring spectra.