The category you described, where graded module morphisms with non-zero degree 0 kernel and non-zero degree 0 cokernel are declared isomorphisms, is equivalent to the full subcategory of graded -modules consisting of those with trivial degree 0 component (i.e., ).

Reasoning:

• Objects with non-trivial : For any graded module with , the zero morphism $0_...

What you are really trying to do is a localization of the category of graded R–modules at a class Σ of morphisms. Let me spell out two key points:

1. What your Σ contains. By definition Σ = { f: A → B in gr–R–Mod | ker(f)0 ≠ 0 and coker(f)0 ≠ 0 }. In particular, whenever A0 ≠ 0 and B0 ≠ 0, the zero map 0A,B: A → B lies in Σ: indeed ker(0)0 = A0 and coker(0)0 = B0.

So Σ contains all zero morphism...

Let 𝒢 = gr ­R–Mod be the category of (ℤ–)graded left R–modules with the usual degree-preserving morphisms.
Put

 Σ = {f : A → B in 𝒢 | (ker f)0 ≠ 0 and (coker f)0 ≠ 0}.

Form the localization
 Q : 𝒢 → 𝒢[Σ-1]
that formally inverts every morphism in Σ (or, in the language you used, “declares them to be isomorphisms’’).

The localized category can be described very explicitly:

Definit...

<think> We are given a category whose objects are graded modules over a graded ring R. The morphisms are the usual morphisms of graded modules, but we are quotienting by a certain set Σ, meaning that we declare all morphisms in Σ to be isomorphisms. The set Σ consists of those graded module morphisms f: A → B such that the kernel of f in degree 0 is non-zero and the cokernel of f in degree 0 is no...

Let R = ⨁_{i∈ℤ} R_i be a (left) graded ring and Gr R the abelian category of ℤ-graded left R–modules with degree–preserving homomorphisms.
Write

 Σ = { f : A → B in Gr R  |  (ker f)_0 = 0 and (coker f)_0 = 0 } .

(If you really meant “≠0’’ in the definition, replace it by “=0’’ – otherwise Σ would contain no map whose kernel or cokernel vanished in degree 0 and nothing that follows would be t...

Analyzing Graded-Module Morphisms with Non-Zero Kernel and Cokernel at Degree 0

To understand this category better, I'll analyze what it means to make morphisms in Σ into isomorphisms.

Understanding the Quotient Category

When you make morphisms in Σ isomorphisms, you're creating a localization of the category of graded R-modules. A morphism f: A → B is in Σ when:

• ker(f)₀ ≠ 0: Some elemen...

This is an interesting construction! You're looking at the quotient category of graded -modules by the class of morphisms that have both non-zero kernel and cokernel in degree 0.

Let me offer some perspectives on this category:

1. Localization Perspective

Your construction is a localization of the category of graded -modules at the class . This is well-defined if admit...

This is an excellent question. Your construction is a specific example of a categorical localization, and it turns out to have a very clean and elegant description.

Let's break down the construction to understand its effect.

1. Analyzing the Class of Morphisms

You start with the category of graded -modules, let's call it . You are forming a ...

Fix a graded‐ring R = ⊕ R_i and let Gr R be the abelian category of ℤ–graded R–modules with degree‐zero maps. In your question you form the “localization” of Gr R by declaring as isomorphisms exactly those f : M→N for which

 ker f and coker f both lie in
  T = { X∈Gr R : X_0=0 }.

Equivalently one inverts all f whose 0th–component
f_0 : M_0→N_0 is already an isomorphism. But this is nothi...