A view toward the geometry of normalizing sequences in finitely semi-graded algebras
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Rubiano Suarez, Andrés Alejandro
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En 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.
In 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.
In 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.