Sheaves of Paragraded Rings
DOI:
https://doi.org/10.5644/SJM.07.2.02Keywords:
Category, paragraded ring, paragraded prime spectrum, localization, sheaves of paragraded ringsAbstract
In this paper we deal with sheaves of the category of paragraded rings with the same set of grades $\Delta,$ in short, the category of paragraded rings of type $\Delta,$ denoted by $R^P_\Delta.$ The definition of a sheaf of category $R^P_\Delta$ is the same as for any other category, but we are interested in the paragraded structure sheaf, as we named it here, and so, in some way we introduce the theory of paragraded structures into algebraic geometry. For this purpose we need to deal with homogeneous ideals, particularly, we need to introduce the prime spectrum of $R\in\mathrm{obj}(R^P_\Delta)$ and to analyze the localization of $R$.
2010 Mathematics Subject Classification. 16S60, 16S85, 16W99
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References
S. Awodey, Category Theory, Clarendon Press, Oxford 2006.
N. Bourbaki, Alge`bre, Chap. II, 3e ´edit. Paris, Hermann, 1962.
G. E. Bredon, Sheaf Theory, Springer-Verlag New York, 2nd ed. 1997.
E. Ili´c-Georgijevi´c, Some problems from the theory of paragraded structures, Ph. D. Thesis, University of East Sarajevo, 2011.
E. Ili´c-Georgijevi´c, On the categories of paragraded groups and modules of type ∆, (accepted in Sarajevo J. Math.).
E. Ili´c-Georgijevi´c, Some properties of paragraded modules of type ∆, (submitted).
E. Ili´c-Georgijevi´c, Tensor product of paragraded modules, (in preparation).
M. Krasner and M. Vukovi´c, Structures paragradu´ees (groupes, anneaux, modules), I, Proc. Japan Acad. Ser, A, 62 (9) (1986), 350–352.
M. Krasner and M. Vukovi´c, Structures paragradu´ees (groupes, anneaux, modules) II, Proc. Japan Acad. Ser, A, 62 (10) (1986), 389–391.
M. Krasner and M. Vukovi´c, Structures paragradu´ees (groupes, anneaux, modules) III, Proc. Japan Acad. Ser, A, 63 (1) (1987), 10–12.
M. Krasner and M. Vukovi´c, Structures paragradu´ees (groupes, anneaux, modules), Queen’s Papers in Pure and Applied Mathematics, No. 77, Queen’s University, Kingston, Ontario, Canada 1987, pp. 1–163.
S. Mac Lane, Categories for the Working Mathematician, Springer, 2nd ed. 1998.
C. Nˇastˇasescu and F. Van Oystaeyen, Methods of Graded Rings, Springer-Verlag, Berlin Heidelberg 2004.
V. Peri´c, Algebra I, IP Svjetlost, Sarajevo, 1991.
J. J. Rotman, An Introduction to Homological Algebra, Springer, 2nd ed. 2009.
B. R. Tennison, Sheaf Theory, Cambridge University Press, 1976.
M. Vukovi´c, Structures gradu´ees et paragradu´ees, Prepublication de l’Institut Fourier, Universit´e de Grenoble I, No. 536 (2001), pp. 1–40.
M. Vukovi´c, About Krasner’s and Vukovi´c’s paragraduations, International Congress of Mathematicians, Hyderabad, India, 19-27 August 2010.
