Abstract
It is well known that the decomposition of injective modules over noetherian rings is one of the most aesthetic and important results in commutative algebra. Our aim is to prove similar results for graded noetherian rings. In this paper, we will study the structure theorem for $gr$-injective modules over $gr$-noetherian $G$-graded commutative rings, give a definition of the $gr$-Bass numbers, and study their properties. We will show that every $gr$-injective module has an indecomposable decomposition. Let $R$ be a $gr$-noetherian graded ring and $M$ be a $gr$-finitely generated $R$-module, we will give a formula for expressing the Bass numbers using the functor $Ext$. We will define the section functor $\Gamma_{V}$ with support in a specialization-closed subset $V$ of $Spec^{gr}(R)$ and the abstract local cohomology functor. Finally, we will show that a left exact radical functor $F$ is of the form $\Gamma_V$ for a specialization-closed subset $V$.
Highlights
Considerable interest has been noted in rings and other algebraic structures equipped with grading
Paul Smith in his work [21] proved some category equivalences involving the quotient category QGr(kQ) := Gr(kQ)/F dim(kQ) of graded kQ-modules modulo those that are the sum of their finite-dimensional submodules
Kim in their work [3] show that if a graded submodule of a noetherian module cannot be written as a proper intersection of graded submodules, it cannot be written as a proper intersection of submodules at all
Summary
It is well known that the decomposition of injective modules over noetherian rings is one of the most aesthetic and important results in commutative algebra. We will study the structure theorem for gr-injective modules over gr-noetherian G-graded commutative rings, give a definition of the gr-Bass numbers, and study their properties. Let R be a gr-noetherian graded ring and M be a gr-finitely generated R-module, we will give a formula for expressing the Bass numbers using the functor Ext. We will define the section functor ΓV with support in a specialization-closed subset V of Specgr(R) and the abstract local cohomology functor.
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