In this paper, we first define the notion of finitely-cosmall quotient (singly-cosmall quotient) morphisms. Then give a characterization new concept. We show that an epimorphism p:Y→U is if and only for any right R-module Z morphism g:Z→Y such pg finitely-copartial isomorphism (singly-copartial isomorphism) from to Y with codomain U finitely (singly) split epimorphism. also investigate relation...

A coherent F -module on X is a coherent sheafM on X, together with a Frobenius action onM, that is, a morphism of sheaves of OX-modules FM : M→ F∗(M). In other words, FM is a morphism of sheaves of Fq-vector spaces OX → OX such that FM(am) = aFM(m) for every a ∈ OX(U) and m ∈ M(U), where U is any open subset of X. As above, since FM is Fq-linear, it follows that it induces Fq-linear maps on coh...

We consider a symmetric monoidal closed category V = (V ,⊗, I, [−,−]) together with a regular injective object Q such that the functor [−, Q] : V → V op is comonadic and prove that in such a category, as in the monoidal category of abelian groups, a morphism of commutative monoids is an effective descent morphism for modules if and only if it is a pure monomorphism. Examples of this kind of mon...

Let f : X → Y be a smooth morphism of complex analytic manifolds, and let F be an R-constructible sheaf on Y . LetM be a coherent DX -module. We prove that the microsupport of RHomDX (M, tHom(f −1F,OX)) is contained in the 1-microcharacteristic variety of M along V = X × Y T Y . As a consequence, we obtain a similar estimate for the microsupport of the complex of solutions of M in the sheaf of ...

We extend the ordinary concept of theory morphism in institutions to extra theory morphisms. Extra theory morphism map theories belonging to different institutions across institution morphisms. We investigate the basic mathematical properties of extra theory morphisms supporting the semantics of logical multiparadigm languages, especially structuring specifications (module systems) á la OBJ-Cle...

This a first step to develop a theory of smooth, étale and unramified morphisms between noetherian formal schemes. Our main tool is the complete module of differentials, that is a coherent sheaf whenever the map of formal schemes is of pseudo finite type. Among our results we show that these infinitesimal properties of a map of usual schemes carry over into the completion with respect to suitab...

let r be a commutative ring with non-zero identity and m be a unital r-module. then the concept of quasi-secondary submodules of m is introduced and some results concerning this class of submodules is obtained

Definition 1. Let S be a graded ring, set X = ProjS and letM a graded S-module. We define a sheaf of modulesM ̃ on X as follows. For each p ∈ ProjS we have the local ring S(p) and the S(p)module M(p) (GRM,Definition 4). Let Γ(U,M ̃) be the set of all functions s : U −→ ∐p∈U M(p) with s(p) ∈M(p) for each p, which are locally fractions. That is, for every p ∈ U there is an open neighborhood p ∈ V ⊆...

We start by showing that (1) is injective. Take an u ∈ Hom (A,B)⊗ Zl that maps to zero in HomΓ (Tl(A), Tl(B)). Write u = ∑∞ j=0 l uj , uj ∈ Hom(A,B), and [u]n for ∑n j=0 l uj . Note that since Hom (A,B) is a Z-module [u]n is in Hom (A,B). Now (since u maps to zero) [u]n is the zero morphism A[l ] → B[l], so it kills the ltorsion. As it is well known, this implies the existence of a certain ψn ∈...