let $r$ be a right artinian ring or a perfect commutative‎‎ring‎. ‎let $m$ be a noncosingular self-generator $sum$-lifting‎‎module‎. ‎then $m$ has a direct decomposition $m=oplus_{iin i} m_i$‎,‎where each $m_i$ is noetherian quasi-projective and each‎‎endomorphism ring $end(m_i)$ is local‎.

Proof. Consider the map g:A/I → C , a+I 7→ f (a). It is well defined: a+I = a′ +I implies a− a′ ∈ I implies f (a) = f (a′). The element a + I belongs to the kernel of g iff g(a + I) = f (a) = 0, i.e. a ∈ I , i.e. a + I = I is the zero element of A/I . Thus, ker(g) = 0. The image of g is g(A/I) = {f (a) : a ∈ A} = C . Thus, g is an isomorphism. The inverse morphism to g is given by f (a) 7→ a + I .

A free divisor D in C is linear if its module of logarithmic vector fields has a basis of global vector fields of degree 0. It is then defined by a homogeneous polynomial of degree n and its complement is an open orbit of an algebraic subgroup GD in Gln(C). The best known example is the normal crossing divisor. Many other such divisors arise, for instance, from quiver representations. We give a...

In this note, we outline the general development of a theory of symmetric homology of algebras, an analog of cyclic homology where the cyclic groups are replaced by symmetric groups. This theory is developed using the framework of crossed simplicial groups and the homological algebra of module-valued functors. The symmetric homology of group algebras is related to stable homotopy theory. Two sp...

We associate a diagrammatic monoidal category $\mathcal{H}\textit{eis}_k(A;z,t)$, which we call the quantum Frobenius Heisenberg category, to symmetric superalgebra $A$, central charge $k \in \mathbb{Z}$, and invertible parameters $z,t$ in some ground ring. When $A$ is trivial, i.e. it equals ring, these categories recover introduced our previous work, when $k$ zero they yield generalizations o...

Model verification is to check the correctness of the model implementation comparing with a model specification. The specification is modeling and description properties of a real system. The implementation is the ready to simulation model described from model specification. Our proposed frame work is system morphism based approach, which has various levels of morphism depending on system descr...

Let G be a simple, simply connected algebraic group defined and split over the prime field F p > 3. Let F be a finite dimensional rational G-module. As shown in [1] , for d suitably large, the Eilenberg-Mac Lane cohomology groups H*(G(Fpd), V) achieve a stable value H*en(G, V)9 the so-called generic cohomology of V. This generic cohomology can in turn be determined from the rational cohomology ...

Inspired by the work of Cherry, we introduce and study a new notion Brody hyperbolicity for rigid analytic varieties over non-archimedean field $K$ characteristic zero. We use this to show following algebraic statement: if projective variety admits non-constant morphism from an abelian variety, then so does any specialization it. As application result, that moduli space is $K$-analytically hype...