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 new category, let's call it , by formally inverting all morphisms in the class:

Let's rephrase the conditions on :

• A graded homomorphism is a collection of maps .
• The kernel, , is a graded submodule of , with . The condition means that the degree-0 part of the map, , is not injective.
• The cokernel, , is a graded quotient module of . Its degree-0 part is . The condition means that is not surjective.

So, a morphism is in if and only if its degree-0 component is neither injective nor surjective.

2. The Key Insight: The Zero Morphism

Now, let's consider the zero morphism between two graded modules and .

• , so .
• , so .

According to your definition, the zero morphism belongs to if and only if and .

3. The Consequence of Inverting Zero Morphisms

In your new category , every morphism in becomes an isomorphism. This has a drastic consequence:

If is any graded module with , then becomes a zero object in .

Proof: Consider the zero endomorphism . Since , this morphism is in . Therefore, in the category , is an isomorphism. In any category with a zero object, a morphism is the zero morphism if and only if it factors through the zero object. If the zero endomorphism is an isomorphism, it means that the identity morphism must be the zero morphism (since ). If , then must be a zero object.

This means that your construction collapses all graded modules with a non-trivial degree-zero component into a single object, which acts as the zero object of the resulting category .

4. What Remains of the Category?

Let's partition the objects of into two classes:

As we've seen, all objects in become isomorphic to the zero object in .

Now, what about the objects in ? The subcategory is the full subcategory of graded modules whose degree-zero component is trivial. Let be objects in (so ), and let be any morphism between them.

• , so .
• . Therefore, the conditions for being in can never be met. No morphism between objects of is in .

What about morphisms between an object in and an object in ?

• Let with and . Then , so .
• Let with and . Then , so .

The localization procedure only inverts morphisms between objects in . The morphisms within , and between and , are not in . However, in the localized category , any map to or from an object in becomes a zero map, because all objects in are identified with the zero object.

The zero module is in (since ). This object becomes the zero object of . All objects from become isomorphic to this object. The morphisms between objects in are unaffected, as they are not inverted.

Conclusion: An Elegant Description

The category you have constructed is equivalent to the full subcategory of graded -modules whose degree-zero component is trivial.

Let . This is itself an abelian category. Your category is equivalent to .

The localization functor can be described as follows:

• On objects:
  • If , then .
  • If , then (the zero module).
• On morphisms :
  • If and , then .
  • Otherwise, (the zero morphism).

This functor has the universal property of your localization: it inverts all morphisms in and any other such functor factors uniquely through it.

In summary, your construction effectively "kills" the degree-zero information, resulting in the category of graded modules that were trivial at degree zero to begin with. This is a very clean and canonical description.

Side Note: This is interestingly different from the more standard construction of a Serre quotient. If one were to take the Serre subcategory and form the quotient category , a morphism would become an isomorphism if and only if its kernel and cokernel were in . This means and , i.e., is an isomorphism. Your construction inverts a nearly opposite class of maps.