A view toward the geometry of normalizing sequences in finitely semi-graded algebras

dc.contributorReyes Villamil, Milton Armando
dc.contributorSAC2
dc.creatorRubiano Suarez, Andrés Alejandro
dc.date.accessioned2020-09-11T16:34:36Z
dc.date.available2020-09-11T16:34:36Z
dc.date.created2020-09-11T16:34:36Z
dc.date.issued2020-05-15
dc.identifierhttps://repositorio.unal.edu.co/handle/unal/78449
dc.description.abstractEn este trabajo vemos el comportamiento de la sucesión normalizadora en álgebras semi-graduadas. Primero, definimos la sucesión regular y el complejo de Koszul en el caso conmutativo. Usando la variedad de ideales máximos, llegamos a una geometría entre las subvariedades y las sucesiones regulares en álgebras graduadas. Luego, pasamos al caso no conmutativo. Definimos la sucesión normalizadora y vemos que está relacionada con la altura de un ideal. Luego, vemos que la sucesión normalizadora aparece en álgebras de Clifford torcidas graduadas. Además, definimos los módulos punto derechos y el zero locus. Con estas definiciones, vemos la relación entre el zero locus con las sucesiones normalizadoras en álgebras graduadas. Presentamos el contexto de álgebras finitamente semi-graduadas y la geometría de la sucesión normalizadora en este caso. Así, definimos el concepto de anillo finitamente semi-graduado y álgebra finitamente semi-graduada. Vemos que las extensiones PBW torcidas son anillos finitamente semi-graduados. También, vemos la aparición de la sucesión normalizadoras en un tipo de álgebra semi-graduada. Para esto, consideramos la teoría del álgebra envolvente universal de un álgebra de Lie. Definimos un álgebra completamente solucionable, un anillo de fracciones y llegamos a algunas propiedades en las que aparece la sucesión normalizadora. Finalmente, vemos cómo los módulos de puntos pueden parametrizarse con un cierto esquema en el caso de álgebras graduadas. Con la ayuda de esto, llegamos al objetivo principal de este trabajo, que es ver la geometría de las sucesiones normalizadoras en ciertas álgebras finitamente semi-graduadas. Aquí, encontramos una geometría de las sucesiones normalizadoras en las extensiones PBW torcidas graduadas. Luego, se deja una vía para continuar investigando las geometría de las sucesiones normalizadoras en objetos semigraduados más generales.
dc.description.abstractIn this work, we see the behavior of normalizing sequence in semi-graded algebras. First, we define regular sequence and the Koszul complex in commutative case. Using the variety of maximal ideals a geometry is reached between the sub-varieties and regular sequences in graded algebras. Then, we turn to non-commutative case. The normalizing sequence is defined and we see that it appears related to the height of an ideal. Then we see that the normalizing sequence appears in graded skew Clifford algebras. Also, we define the right point modules and the zero locus. With these definitions, we consider the relation between the zero locus with the normalizing sequences in graded algebras. We present the context of finitely semi-graded algebras and geometry of normalizing sequence in this case. Thus, we define the concept of finitely semi-graded ring and finitely semi-graded algebra. We note that skew PBW extension are finitely semi-graded rings. Also, we see the appearance of normalizing sequence in a type of semi-graded algebra. For this, we consider the theory of the universal enveloping algebra of a Lie algebra. We define a completely solvable algebra, a ring of fractions and we arrive at some properties in which the normalizing sequence appears. Finally, we see how point modules can be parameterized with a certain scheme in the case of graded algebras. With the help of this, we get to the main purpose of this work, which is to see the geometry of the normalizing sequences in certain finitely semi-graded algebras. Here, we find a geometry of the normalizing sequences in graded skew PBW extensions. Then, a way is left to continue investigating the geometry of normalizing sequences in more general semi-graded objects.
dc.languageeng
dc.publisherBogotá - Ciencias - Maestría en Ciencias - Matemáticas
dc.publisherDepartamento de Matemáticas
dc.publisherUniversidad Nacional de Colombia - Sede Bogotá
dc.relationV. Artamonov, O. Lezama, and W. Fajardo. Extended modules and Ore extensions. Commun. Math. Stat., 4(2):189–202, 2016.
dc.relationV. Artamonov. Derivations of Skew PBW Extensions. Commun. Math. Stat., 3(4):449–457, 2015.
dc.relationK. Ajitabh, S. Smith, and J. Zhang. Auslander-Gorenstein rings. Comm. Algebra, 26(7):2159–2180, 1998.
dc.relationM. Artin, J. Tate, and M. Van den Bergh. Some Algebras associated to Automorphisms of Elliptic Curves. In The Grothendieck Festschrift, pages 33–85. Springer, 2007.
dc.relationA. Bell and K. Goodearl. Uniform rank over differential operator rings and Poincaré-Birkhoff-Witt extensions. Pac. J. Math., 131(1):13–37, 1988.
dc.relationG. Bellamy, D. Rogalski, T. Schedler, J. Stafford, and M. Wemyss. Noncommutative Algebraic Geometry, volume 64. Cambridge University Press, 2016.
dc.relationP. Cohn. A Remark on the Birkhoff-Witt Theorem. J. Lond. Math. Soc., s1-38(1):197–203, 01 1963.
dc.relationT. Cassidy and M. Vancliff. Generalizations of Graded Clifford Algebras and of Complete Intersections. J. Lond. Math. Soc., 81(1):91–112, 2010.
dc.relationT. Cassidy and M. Vancliff. Corrigendum: Generalizations of Graded Clifford Algebras and of Complete Intersections. J. Lond. Math. Soc., 90(2):631–636, 2014.
dc.relationB. Doran, P. Flajolet, M. Ismail, TY Lam, and E. Lutwak. Categorical foundations: special topics in order, topology, algebra, and sheaf theory, volume 97. Cambridge University Press, 2004.
dc.relationV. Futorny and S. Ovsienko. Kostant’s theorem for special filtered algebras. Bull. Lond. Math. Soc., 37(2):187–199, 2005.
dc.relationC. Gallego and O. Lezama. Gröbner Bases for Ideals of -PBW Extensions. Comm. Algebra, 39(1):50–75, 2010.
dc.relationG. Greuel and G. Pfister. A Singular Introduction to Commutative Algebra. Springer Science & Business Media, 2012.
dc.relationK. Goodearl and R. Warfield Jr. An Introduction to Noncommutative Noetherian Rings, volume 61. Cambridge University Press, 2004.
dc.relationR. Hartshorne. Algebraic Geometry, volume 52. Springer Science & Business Media, 2013.
dc.relationP. Higgins. Baer Invariants and the Birkhoff-Witt Theorem. J. Algebra, 11(4):469–482, 1969.
dc.relationT. Hungerford. Algebra. 1974. Grad. Texts in Math, 1974.
dc.relationA. Kirillov. An Introduction to Lie Groups and Lie Algebras, volume 113. Cambridge University Press, 2008.
dc.relationA. Knapp. Lie Groups Beyond an Introduction, volume 140. Springer Science & Business Media, 2013.
dc.relationA. Knutson, T. Tao, et al. Puzzles and (equivariant) cohomology of Grassmannians. Duke Math. J., 119(2):221–260, 2003.
dc.relationO. Lezama, J. Acosta, and A. Reyes. Prime Ideals of Skew PBW Extensions. Rev. Un. Mat. Argentina, 56(2):39–55, 2015.
dc.relationT. Leinster. Basic Category Theory, volume 143. Cambridge University Press, 2014.
dc.relationT. Levasseur. Some properties of non-commutative regular graded rings. Glasg. Math. J., 34(3):277–300, 1992.
dc.relationO. Lezama. Cuadernos de álgebra, Volumen 2: Anillos y Módulos, Categorías, álgebra Homológica, álgebra no Conmutativa, Geometría Algebraica.
dc.relationO. Lezama. Some Homological Properties of Skew PBW Extensions arising in Non-Commutative Algebraic Geometry. Discuss. Math. Gen. Algebra Appl., 37(1):45–57, 2017.
dc.relationO. Lezama. Some open problems in the context of skew PBW extensions and semi-graded rings. arXiv preprint arXiv:1908.04880, 2019.
dc.relationO. Lezama. Computation of Point Modules of Finitely Semi-Graded Rings. Comm. Algebra, 48(2):866–878, 2020.
dc.relationO. Lezama, W. Fajardo, C. Gallego, A. Reyes, H. Suárez, and H. Venegas. Skew PBW Extensions, Ring and Module Theoretic Properties, Matrix and Gröebner Methods and Applications. To be published by Springer.
dc.relationO. Lezama and J. Gómez. Koszulity and Point Modules of Finitely Semi- Graded Rings and Algebras. Symmetry, 11(7):881, 2019.
dc.relationO. Lezama and E. Latorre. Non-Commutative Algebraic Geometry of Semi- Graded Rings. Internat. J. Algebra Comput., 27(04):361–389, 2017.
dc.relationO. Lezama and H. Venegas. Gelfand-Kirillov Dimension for Rings. São Paulo J. Math.Sci., 14(1):207–222, 2020.
dc.relationP. Morandi. Examples of Localization. Class notes, 1998.
dc.relationJ. McConnell, J. Robson, and L. Small. Noncommutative Noetherian Rings, volume 30. American Mathematical Society, 2001.
dc.relationB. Nathanson. An elementary proof for the Krull dimension of a polynomial ring. Amer. Math. Monthly, 125(7):623–637, 2018.
dc.relationA. Niño and A. Reyes. Some Ring Theoretical Properties of Skew Poincaré- Birkhoff-Witt Extensions. Bol. Mat., 24(2):131–148, 2017.
dc.relationP. Nuss. L’homologie cyclique des algèbres enveloppantes des algèbres de lie de dimension trois. J. Pure Appl. Algebra, 73(1):39–71, 1991.
dc.relationM. Nafari and M. Vancliff. Graded Skew Clifford Algebras that are twists of Graded Clifford Algebras. Comm. Algebra, 43(2):719–725, 2015.
dc.relationO. Ore. Theory of non-commutative polynomials. Ann. of Math., pages 480– 508, 1933.
dc.relationA. Reyes. Gelfand-Kirillov Dimension of Skew PBW Extensions. Rev. Colombiana Mat., 47(1):95–111, 2013.
dc.relationA. Reyes. Ring and Module Theoretical Properties of Skew PBW Extensions. PhD, National University of Colombia, Bogotá, Colombia, 2013.
dc.relationA. Reyes. Uniform Dimension over Skew PBW Extensions. Rev. Colombiana Mat., 48(1):79–96, 2014.
dc.relationA. Rosenberg. Noncommutative Algebraic Geometry and Representations of Quantized Algebras, volume 330. Springer Science & Business Media, 2013.
dc.relationJ. Rotman. An Introduction to Homological Algebra. Springer Science & Business Media, 2008.
dc.relationA. Rubiano and A. Reyes. On Graded Skew Clifford Algebras and Graded Skew PBW Extensions. preprint.
dc.relationA. Reyes and H. Suárez. Some Remarks about the Cyclic Homology of Skew PBW Extensions. Ciencia en Desarrollo, 7(2):99–107, 2016.
dc.relationA. Reyes and H. Suárez. Bases for Quantum Algebras and Skew Poincaré- Birkhoff-Witt Extensions. Momento, 1(54):54–75, 2017.
dc.relationA. Reyes and H. Suárez. Enveloping algebra and skew Calabi-Yau algebras over skew Poincaré–Birkhoff–Witt extensions. Far East J. Math. Sci. (FJMS), 102(2):373–397, 2017.
dc.relationH. Samelson. Notes on Lie Algebras. Springer Science & Business Media, 2012.
dc.relationH. Suárez, O. Lezama, and A. Reyes. Calabi-Yau property for graded skew PBW extensions. Rev. Colombiana Mat., 51(2):221–239, 2017.
dc.relationH. Suárez and A. Reyes. Koszulity for Skew PBW Extensions over Fields. JP J. Algebra Number Theory Appl., 39(2):181–203, 2017.
dc.relationH. Suárez and M. Reyes. A generalized Koszul property for skew PBW extensions. Far East J. Math. Sci. (FJMS), 101(2):301–320, 2017.
dc.relationH. Suárez. Koszulity for Graded Skew PBW Extensions. Comm. Algebra, 45(10):4569–4580, 2017.
dc.relationB. Shelton and M. Vancliff. Schemes of Line Modules I. J. Lond. Math. Soc., 65(3):575–590, 2002.
dc.relationM. Vancliff. The Interplay of Algebra and Geometry in the Setting of Regular Algebras. Commut. Algebra and Noncommut. Algebra. Geom., 6:371–390, 2015.
dc.relationM. Vancliff. On the Notion of Complete Intersection Outside the Setting of Skew Polynomial Rings. Comm. Algebra, 43(2):460–470, 2015.
dc.relationP. Veerapen. Graded Clifford Algebras and Graded Skew Clifford Algebras and Their Role in the Classification of Artin–Schelter Regular Algebras. Adv. Appl. Clifford Algebr., 27(3):2855–2871, 2017.
dc.relationM. Vancliff and K. Van Rompay. Embedding a Quantum Nonsingular Quadric in a Quantum P3. J. Algebra, 195(1):93–129, 1997.
dc.relationC. Weibel. An Introduction to Homological Algebra. Number 38. Cambridge University Press, 1995.
dc.relationJ. Zhang. A note on GK dimension of skew polynomial extensions. Proc. Amer. Math. Soc., 125(2):363–373, 1997.
dc.rightsAtribución-NoComercial 4.0 Internacional
dc.rightsAcceso abierto
dc.rightshttp://creativecommons.org/licenses/by-nc/4.0/
dc.rightsinfo:eu-repo/semantics/openAccess
dc.rightsDerechos reservados - Universidad Nacional de Colombia
dc.titleA view toward the geometry of normalizing sequences in finitely semi-graded algebras
dc.typeOtro


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