Geodesic
Structural theorem for $gr$-injective modules over $gr$-noetherian $G$-graded commutative rings and local cohomology functors
Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 53 (2019), pp. 127-137

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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 $V$.
Keywords: graded commutative rings, $gr$-Bass numbers, local cohomology functors, derived categories, radical functors. 
@article{IIMI_2019_53_a10,
     author = {L. Lu},
     title = {Structural theorem for $gr$-injective modules over $gr$-noetherian $G$-graded commutative rings and local cohomology functors},
     journal = {Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta},
     pages = {127--137},
     publisher = {mathdoc},
     volume = {53},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IIMI_2019_53_a10/}
}
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L. Lu. Structural theorem for $gr$-injective modules over $gr$-noetherian $G$-graded commutative rings and local cohomology functors. Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 53 (2019), pp. 127-137. http://geodesic.mathdoc.fr/item/IIMI_2019_53_a10/