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. 
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     title = {Structural theorem for $gr$-injective modules over $gr$-noetherian $G$-graded commutative rings and local cohomology functors},
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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/

[1] Beilinson A. A., Bernstein J., Deligne P., “Faisceaux pervers, 1”, Asterisque, 100, 1982, 5–171 | MR

[2] Bell J., Zhang J., “An isomorphism lemma for graded rings”, Proceedings of the American Mathematical Society, 145 (2017), 989–994 | DOI | MR | Zbl

[3] Chen J., Kim Y., “Graded-irreducible modules are irreducible”, Communications in Algebra, 45:5 (2017), 1907–1913 | DOI | MR | Zbl

[4] DellAmbrogio I., Stevenson G., “On the derived category of a graded commutative Noetherian ring”, Journal of Algebra, 373 (2013), 356–376 | DOI | MR | Zbl

[5] Dickson S. E., “A torsion theory for abelian categories”, Transactions of the American Mathematical Society, 121 (1966), 223–235 | DOI | MR | Zbl

[6] Foxby H. B., “Bounded complexes of flat modules”, Journal of Pure and Applied Algebra, 15:2 (1979), 149–172 | DOI | MR | Zbl

[7] Goldman O., “Rings and modules of quotients”, Journal of Algebra, 13:1 (1969), 10–47 | DOI | MR | Zbl

[8] Heinzer W., Roitman M., “The homogeneous spectrum of a graded commutative ring”, Proceedings of the American Mathematical Society, 130 (2002), 1573–1580 | DOI | MR | Zbl

[9] Hochster M., Local cohomology, Unpublished notes, 2011 https://www.math.lsa.umich.edu/h̃ochster/615W11/loc.pdf

[10] Kanda R., “Classifying Serre subcategories via atom spectrum”, Advances in Mathematics, 231 (2012), 1572–1588 | DOI | MR | Zbl

[11] Krause H., “The stable derived category of a Noetherian scheme”, Compositio Mathematica, 141 (2005), 1128–1162 | DOI | MR | Zbl

[12] Lamber J., Torsion theories, additive semantics, and rings of quotients, Springer, Berlin–Heidelberg, 2006 | DOI | MR

[13] Maranda J.-M., “Injective structures”, Transactions of the American Mathematical Society, 110 (1964), 98–135 | DOI | MR | Zbl

[14] Miyachi J., “Localization of triangulated categories and derived categories”, Journal of Algebra, 141 (1991), 463–483 | DOI | MR | Zbl

[15] Nastasescu C., Oystaeyen F., Graded ring theory, Elsevier, 2011 | MR

[16] Nastasescu C., Oystaeyen F., Methods of graded rings, Springer Science, 2004 | MR | Zbl

[17] Park C., Park M., “Integral closure of a graded Noetherian domain”, Journal of the Korean Mathematical Society, 48 (2011), 449–464 | DOI | MR | Zbl

[18] Popescu N., Abelian categories with applications to rings and modules, Academic Press, 1973 | MR | Zbl

[19] Ratliff L., Rush D., “Two notes on homogeneous prime ideals in graded Noetherian rings”, Journal of Algebra, 264 (2003), 211–230 | DOI | MR | Zbl

[20] Rush D., “Noetherian properties in monoid rings”, Journal of Pure and Applied Algebra, 185 (2003), 259–278 | DOI | MR | Zbl

[21] Smith S., “Category equivalences involving graded modules over path algebras of quivers”, Advances in Mathematics, 230 (2012), 1780–1810 | DOI | MR | Zbl

[22] Tarrio L. A., Lopez A. J., Salorio M. J. S., “Localization in categories of complexes and unbounded resolutions”, Canadian Journal of Mathematics, 52 (2000), 225–247 | DOI | MR | Zbl

[23] Yoshino Y., Yoshizawa T., “Abstract local cohomology functors”, Mathematical Journal of Okayama University, 53 (2011), 129–154 | MR | Zbl