We consider $\,R-$modules as functors in the following way: if $\,M\,$ is a (left) $R$-module, le... more We consider $\,R-$modules as functors in the following way: if $\,M\,$ is a (left) $R$-module, let $\,\mathcal M\,$ be the functor of $\,\mathcal R-$modules defined by $\,\mathcal M(S) := S \otimes_R M\,$ for every $\,R-$algebra $\,S$. With the corresponding notion of dual functor, we prove that the natural morphism of functors $\,\mathcal M\to \mathcal M^{**}\,$ is an isomorphism.
Let R be an associative ring with unit. Given an R-module M , we can associate the following cova... more Let R be an associative ring with unit. Given an R-module M , we can associate the following covariant functor from the category of R-algebras to the category of abelian groups: S → M ⊗ R S. With the corresponding notion of dual functor, we prove that the natural morphism of functors M → M ∨∨ is an isomorphism. We prove several characterizations of the functors associated with flat modules, flat Mittag-Leffler modules and projective modules.
Finite modules, finitely presented modules and Mittag-Leffler modules are characterized by their ... more Finite modules, finitely presented modules and Mittag-Leffler modules are characterized by their behaviour by tensoring with direct products of modules. In this paper, we study and characterize the functors of modules that preserve direct products.
Let $R$ be an associative ring with unit. %It is natural to We consider $R$-modules as module fun... more Let $R$ be an associative ring with unit. %It is natural to We consider $R$-modules as module functors in the following way: if $M$ is a (left) $R$-module, let $\mathcal M$ be the functor of $\mathcal R$-modules defined by $\mathcal M(S) := S\otimes_R M$ for every $R$-algebra $S$. With the corresponding notion of dual functor, we prove that the natural morphism of functors $\mathcal M\to \mathcal M^{**}$ is an isomorphism.
The schematic finite spaces are those finite ringed spaces where a theory of quasicoherent module... more The schematic finite spaces are those finite ringed spaces where a theory of quasicoherent modules can be developed with minimal natural conditions. We give various characterizations of these spaces and their natural morphisms. We show that schematic finite spaces are strongly related to quasi-compact quasi-separated schemes.
We present a simple remark that assures that the invariant theory of certain real Lie groups coin... more We present a simple remark that assures that the invariant theory of certain real Lie groups coincides with that of the underlying affine, algebraic R-groups. In particular, this result applies to the non-compact orthogonal or symplectic Lie groups.
We consider $R$-modules as module functors in the following way: if $M$ is a (left) $R$-module, l... more We consider $R$-modules as module functors in the following way: if $M$ is a (left) $R$-module, let $\mathcal M$ be the functor of $\mathcal R$-modules defined by $\mathcal M(S) := S\otimes_R M$ for every $R$-algebra $S$. With the corresponding notion of dual functor, we prove that the natural morphism of functors $\mathcal M\to \mathcal M^{**}$ is an isomorphism. We give functorial characterizations of finitely generated projective modules, flat modules and flat Mittag-Leffler modules.
El presente texto esta concebido por el autor como el manual de la asignatura cuatrimestral Algeb... more El presente texto esta concebido por el autor como el manual de la asignatura cuatrimestral Algebra I, del tercer curso del Grado de Matematicas de la UEX. En este curso estudiamos la Teoria de Galois como elemento nucleador para introducir topicos fundamentales en Matematicas y Algebra como la Teoria de Grupos, la Teoria de Cuerpos y herramientas como el producto tensorial de modulos y algebras, y la toma de invariantes por la accion de un grupo.
We study the notion of affine ringed space, see its meaning in topological, differentiable and al... more We study the notion of affine ringed space, see its meaning in topological, differentiable and algebrogeometric contexts and show how to reduce the affineness of a ringed space to that of a ringed finite space. Then, we characterize schematic finite spaces and affine schematic spaces in terms of combinatorial data. Finally, we prove Serre's criterion of affineness for schematic finite spaces. This yields, in particular, Serre's criterion of affineness on schemes. 1. Introduction. Let (S, O S) be a ringed space and A = O S (S). We say that (S, O S) is an affine ringed space if it satisfies: (1) S is acyclic, i.e., H i (S, O S) = 0 for any i > 0. (2) The global sections functor {Quasi-coherent O S-modules} −→ {A-modules} M −→ Γ(S, M) is an equivalence. If (S, O S) is a quasi-compact and quasi-separated scheme, then S is affine in the above sense if and only if S is affine in the usual sense, i.e., S = Spec A. In the topological case, i.e., O S is the constant sheaf Z, S is affine if and only if S is homotopically trivial, see Proposition 2.8 for details. In the differentiable case, i.e., S is a separated differentiable manifold, or a differentiable space, and O S = C ∞ S is the sheaf of differentiable functions, S is affine if and only if S is compact, see Proposition 2.9 and Remark 2.10.
El presente texto esta concebido por el autor como el manual de la asignatura cuatrimestral Teori... more El presente texto esta concebido por el autor como el manual de la asignatura cuatrimestral Teoria de Numeros, del cuarto curso del Grado de Matematicas de la UEX. Este curso es una introduccion a la Teoria de Numeros y hacemos un especial enfasis en la relacion de esta teoria con la Teoria de Curvas Algebraicas. Suponemos que los alumnos han cursado antes un curso de Teoria de Galois (Algebra I) y un curso de Variedades Algebraicas (Algebra II). El manual esta divido en cuatro temas. En cada tema incluimos un cuestionario, una lista de problemas (con sus soluciones) y la biografia de un matematico relevante (en ingles).
Let R be a commutative ring. Roughly speaking, we prove that an R-module M is flat iff it is a di... more Let R be a commutative ring. Roughly speaking, we prove that an R-module M is flat iff it is a direct limit of R-module affine algebraic varieties, and M is a flat Mittag-Leffler module iff it is the union of its R-submodule affine algebraic varieties.
El presente manual esta concebido como texto de referencia para los estudiantes del Grado de Mate... more El presente manual esta concebido como texto de referencia para los estudiantes del Grado de Matematicas de la UEX, en las asignaturas de Algebra: Algebra Conmutativa, Algebra I, Algebra II Teoria de Numeros. Incluye diversos temas de Algebra y Geometria Algebraica para alumnos de master y doctorado, y sirve tambien como manual de apoyo a los profesores del area de Algebra.
espanolEl presente manual esta concebido por el autor como el manual de la asignatura cuatrimestr... more espanolEl presente manual esta concebido por el autor como el manual de la asignatura cuatrimestral Algebra Conmutativa, del segundo curso del Grado en Matematicas de la UEX. Introducimos estructuras basicas del Algebra como las de grupo, anillo y modulo, herramientas fundamentales como el cociente de un grupo por un subgrupo, cociente de un anillo por un ideal, cociente de un modulo por un submodulo y la localizacion de un anillo o un modulo por un sistema multiplicativo. El manual esta divido en cuatro temas. En cada tema incluimos un cuestionario, una lista de problemas (con sus soluciones) y la biografia de un matematico relevante (en ingles). EnglishThis manual is conceived by the author as the manual of the subject Commutative Algebra, of the second course of the Degree in Mathematics of the UEX. We introduce basic structures of Algebra such as group, ring and module, fundamental tools such as the quotient of a group to a subgroup, the quotient of a ring to an ideal, the quoti...
Proceedings of the American Mathematical Society, 2007
Let S be a locally noetherian scheme and R an N-graded O Salgebra of finite type. We say that X =... more Let S be a locally noetherian scheme and R an N-graded O Salgebra of finite type. We say that X = Spec R is a homogeneous variety over S. In this paper we prove that the functor HomHilb X/S : Locally noetherian S-schemes ; Sets T ; closed subschemes of X × S T flat and homogeneous over T is representable by an S-scheme that is a disjoint union of locally projective schemes over S. The proof is very simple, and it only makes use of the theory of graded modules and standard flatness criteria. From this, one obtains an elementary construction (which does not make use of cohomology) of the ordinary Hilbert scheme of a locally projective S-scheme.
Let G = Spec A be an affine functor of monoids. We prove that A * is the enveloping functor of al... more Let G = Spec A be an affine functor of monoids. We prove that A * is the enveloping functor of algebras of G and that the category of Gmodules is equivalent to the category of A * -modules. Moreover, we prove that the category of affine functors of monoids is anti-equivalent to the category of functors of affine bialgebras. Applications of these results include Cartier duality, neutral Tannakian duality for affine group schemes and the equivalence between formal groups and Lie algebras in characteristic zero. Finally, we also show how these results can be used to recover and generalize some aspects of the theory of the Reynolds operator.
In this paper we solve the problemof desingularization of an absolutely isolatedsingularity of a ... more In this paper we solve the problemof desingularization of an absolutely isolatedsingularity of a differential equation, including thedicritical case. As an application, we prove thefiniteness of the number of dicritical points in theblowing up tree of an absolutely isolated singularity.
The R-module functors that are essential for the development of the theory of the linear represen... more The R-module functors that are essential for the development of the theory of the linear representations of an affine R-group are the quasi-coherent R-modules and the R-module schemes. The aim of this paper is to study when a quasi-coherent R-module is an R-module scheme. We will prove that it is equivalent to giving a characterization of projective R-modules of finite type.
Let k → K be a finite field extension and let us consider the automorphism scheme Aut k K. We pro... more Let k → K be a finite field extension and let us consider the automorphism scheme Aut k K. We prove that Aut k K is a complete k-group, i.e., it has trivial centre and any automorphism is inner, except for separable extensions of degree 2 or 6. As a consequence, we obtain for finite field extensions K 1 , K 2 of k, not being separable of degree 2 or 6, the following equivalence:
We consider $\,R-$modules as functors in the following way: if $\,M\,$ is a (left) $R$-module, le... more We consider $\,R-$modules as functors in the following way: if $\,M\,$ is a (left) $R$-module, let $\,\mathcal M\,$ be the functor of $\,\mathcal R-$modules defined by $\,\mathcal M(S) := S \otimes_R M\,$ for every $\,R-$algebra $\,S$. With the corresponding notion of dual functor, we prove that the natural morphism of functors $\,\mathcal M\to \mathcal M^{**}\,$ is an isomorphism.
Let R be an associative ring with unit. Given an R-module M , we can associate the following cova... more Let R be an associative ring with unit. Given an R-module M , we can associate the following covariant functor from the category of R-algebras to the category of abelian groups: S → M ⊗ R S. With the corresponding notion of dual functor, we prove that the natural morphism of functors M → M ∨∨ is an isomorphism. We prove several characterizations of the functors associated with flat modules, flat Mittag-Leffler modules and projective modules.
Finite modules, finitely presented modules and Mittag-Leffler modules are characterized by their ... more Finite modules, finitely presented modules and Mittag-Leffler modules are characterized by their behaviour by tensoring with direct products of modules. In this paper, we study and characterize the functors of modules that preserve direct products.
Let $R$ be an associative ring with unit. %It is natural to We consider $R$-modules as module fun... more Let $R$ be an associative ring with unit. %It is natural to We consider $R$-modules as module functors in the following way: if $M$ is a (left) $R$-module, let $\mathcal M$ be the functor of $\mathcal R$-modules defined by $\mathcal M(S) := S\otimes_R M$ for every $R$-algebra $S$. With the corresponding notion of dual functor, we prove that the natural morphism of functors $\mathcal M\to \mathcal M^{**}$ is an isomorphism.
The schematic finite spaces are those finite ringed spaces where a theory of quasicoherent module... more The schematic finite spaces are those finite ringed spaces where a theory of quasicoherent modules can be developed with minimal natural conditions. We give various characterizations of these spaces and their natural morphisms. We show that schematic finite spaces are strongly related to quasi-compact quasi-separated schemes.
We present a simple remark that assures that the invariant theory of certain real Lie groups coin... more We present a simple remark that assures that the invariant theory of certain real Lie groups coincides with that of the underlying affine, algebraic R-groups. In particular, this result applies to the non-compact orthogonal or symplectic Lie groups.
We consider $R$-modules as module functors in the following way: if $M$ is a (left) $R$-module, l... more We consider $R$-modules as module functors in the following way: if $M$ is a (left) $R$-module, let $\mathcal M$ be the functor of $\mathcal R$-modules defined by $\mathcal M(S) := S\otimes_R M$ for every $R$-algebra $S$. With the corresponding notion of dual functor, we prove that the natural morphism of functors $\mathcal M\to \mathcal M^{**}$ is an isomorphism. We give functorial characterizations of finitely generated projective modules, flat modules and flat Mittag-Leffler modules.
El presente texto esta concebido por el autor como el manual de la asignatura cuatrimestral Algeb... more El presente texto esta concebido por el autor como el manual de la asignatura cuatrimestral Algebra I, del tercer curso del Grado de Matematicas de la UEX. En este curso estudiamos la Teoria de Galois como elemento nucleador para introducir topicos fundamentales en Matematicas y Algebra como la Teoria de Grupos, la Teoria de Cuerpos y herramientas como el producto tensorial de modulos y algebras, y la toma de invariantes por la accion de un grupo.
We study the notion of affine ringed space, see its meaning in topological, differentiable and al... more We study the notion of affine ringed space, see its meaning in topological, differentiable and algebrogeometric contexts and show how to reduce the affineness of a ringed space to that of a ringed finite space. Then, we characterize schematic finite spaces and affine schematic spaces in terms of combinatorial data. Finally, we prove Serre's criterion of affineness for schematic finite spaces. This yields, in particular, Serre's criterion of affineness on schemes. 1. Introduction. Let (S, O S) be a ringed space and A = O S (S). We say that (S, O S) is an affine ringed space if it satisfies: (1) S is acyclic, i.e., H i (S, O S) = 0 for any i > 0. (2) The global sections functor {Quasi-coherent O S-modules} −→ {A-modules} M −→ Γ(S, M) is an equivalence. If (S, O S) is a quasi-compact and quasi-separated scheme, then S is affine in the above sense if and only if S is affine in the usual sense, i.e., S = Spec A. In the topological case, i.e., O S is the constant sheaf Z, S is affine if and only if S is homotopically trivial, see Proposition 2.8 for details. In the differentiable case, i.e., S is a separated differentiable manifold, or a differentiable space, and O S = C ∞ S is the sheaf of differentiable functions, S is affine if and only if S is compact, see Proposition 2.9 and Remark 2.10.
El presente texto esta concebido por el autor como el manual de la asignatura cuatrimestral Teori... more El presente texto esta concebido por el autor como el manual de la asignatura cuatrimestral Teoria de Numeros, del cuarto curso del Grado de Matematicas de la UEX. Este curso es una introduccion a la Teoria de Numeros y hacemos un especial enfasis en la relacion de esta teoria con la Teoria de Curvas Algebraicas. Suponemos que los alumnos han cursado antes un curso de Teoria de Galois (Algebra I) y un curso de Variedades Algebraicas (Algebra II). El manual esta divido en cuatro temas. En cada tema incluimos un cuestionario, una lista de problemas (con sus soluciones) y la biografia de un matematico relevante (en ingles).
Let R be a commutative ring. Roughly speaking, we prove that an R-module M is flat iff it is a di... more Let R be a commutative ring. Roughly speaking, we prove that an R-module M is flat iff it is a direct limit of R-module affine algebraic varieties, and M is a flat Mittag-Leffler module iff it is the union of its R-submodule affine algebraic varieties.
El presente manual esta concebido como texto de referencia para los estudiantes del Grado de Mate... more El presente manual esta concebido como texto de referencia para los estudiantes del Grado de Matematicas de la UEX, en las asignaturas de Algebra: Algebra Conmutativa, Algebra I, Algebra II Teoria de Numeros. Incluye diversos temas de Algebra y Geometria Algebraica para alumnos de master y doctorado, y sirve tambien como manual de apoyo a los profesores del area de Algebra.
espanolEl presente manual esta concebido por el autor como el manual de la asignatura cuatrimestr... more espanolEl presente manual esta concebido por el autor como el manual de la asignatura cuatrimestral Algebra Conmutativa, del segundo curso del Grado en Matematicas de la UEX. Introducimos estructuras basicas del Algebra como las de grupo, anillo y modulo, herramientas fundamentales como el cociente de un grupo por un subgrupo, cociente de un anillo por un ideal, cociente de un modulo por un submodulo y la localizacion de un anillo o un modulo por un sistema multiplicativo. El manual esta divido en cuatro temas. En cada tema incluimos un cuestionario, una lista de problemas (con sus soluciones) y la biografia de un matematico relevante (en ingles). EnglishThis manual is conceived by the author as the manual of the subject Commutative Algebra, of the second course of the Degree in Mathematics of the UEX. We introduce basic structures of Algebra such as group, ring and module, fundamental tools such as the quotient of a group to a subgroup, the quotient of a ring to an ideal, the quoti...
Proceedings of the American Mathematical Society, 2007
Let S be a locally noetherian scheme and R an N-graded O Salgebra of finite type. We say that X =... more Let S be a locally noetherian scheme and R an N-graded O Salgebra of finite type. We say that X = Spec R is a homogeneous variety over S. In this paper we prove that the functor HomHilb X/S : Locally noetherian S-schemes ; Sets T ; closed subschemes of X × S T flat and homogeneous over T is representable by an S-scheme that is a disjoint union of locally projective schemes over S. The proof is very simple, and it only makes use of the theory of graded modules and standard flatness criteria. From this, one obtains an elementary construction (which does not make use of cohomology) of the ordinary Hilbert scheme of a locally projective S-scheme.
Let G = Spec A be an affine functor of monoids. We prove that A * is the enveloping functor of al... more Let G = Spec A be an affine functor of monoids. We prove that A * is the enveloping functor of algebras of G and that the category of Gmodules is equivalent to the category of A * -modules. Moreover, we prove that the category of affine functors of monoids is anti-equivalent to the category of functors of affine bialgebras. Applications of these results include Cartier duality, neutral Tannakian duality for affine group schemes and the equivalence between formal groups and Lie algebras in characteristic zero. Finally, we also show how these results can be used to recover and generalize some aspects of the theory of the Reynolds operator.
In this paper we solve the problemof desingularization of an absolutely isolatedsingularity of a ... more In this paper we solve the problemof desingularization of an absolutely isolatedsingularity of a differential equation, including thedicritical case. As an application, we prove thefiniteness of the number of dicritical points in theblowing up tree of an absolutely isolated singularity.
The R-module functors that are essential for the development of the theory of the linear represen... more The R-module functors that are essential for the development of the theory of the linear representations of an affine R-group are the quasi-coherent R-modules and the R-module schemes. The aim of this paper is to study when a quasi-coherent R-module is an R-module scheme. We will prove that it is equivalent to giving a characterization of projective R-modules of finite type.
Let k → K be a finite field extension and let us consider the automorphism scheme Aut k K. We pro... more Let k → K be a finite field extension and let us consider the automorphism scheme Aut k K. We prove that Aut k K is a complete k-group, i.e., it has trivial centre and any automorphism is inner, except for separable extensions of degree 2 or 6. As a consequence, we obtain for finite field extensions K 1 , K 2 of k, not being separable of degree 2 or 6, the following equivalence:
El presente manual se ha escrito como texto de referencia del curso de Álgebra Lineal I del Grado... more El presente manual se ha escrito como texto de referencia del curso de Álgebra Lineal I del Grado en Química de la Uex. Esta asignatura es obligatoria en los grados de Biología, Estadística, Física, Química, Matemáticas, etc., de la facultad de Ciencias. Tiene también muchos contenidos comunes con Matemáticas I de Ingeniería Industrial. Hemos procurado escribir un texto apropiado para todas ellas, de modo que el estudiante o profesor pueda escoger o resaltar aquellas partes que considere más convenientes. Así, un estudiante de matemáticas dará más importancia a las demostraciones, a la coherencia y desarrollo de la teoría y un estudiante de ingeniería a las aplicaciones.
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muchos contenidos comunes con Matemáticas I de Ingeniería Industrial.
Hemos procurado escribir un texto apropiado para todas ellas, de modo que el estudiante o profesor pueda escoger o resaltar aquellas partes que considere más convenientes.
Así, un estudiante de matemáticas dará más importancia a las demostraciones, a la coherencia y desarrollo de la teoría y un estudiante de ingeniería a las aplicaciones.