Papers by Marta Pérez Rodríguez
In this work we present the results of an investigation to diagnose the issue of mathematical voc... more In this work we present the results of an investigation to diagnose the issue of mathematical vocations in Primary Education, with special emphasis on girls in the third stage of Primary Education. The study uses a mixed and parallel focus qualitative and quantitative approximation on a sample of 172 girls and 168 boys between 9 and 12 years old from eight educational centers in the municipalities of Vigo and Pontevedra, Spain. The instruments used for data collection were a 28-question questionnaire, interviews and discussion groups on the perception, knowledge and assessment of different aspects related to the question of mathematical vocations and the importance of mathematics in life. We found that individuals recognize the usefulness and importance of mathematics, but not its realization in professional activity or in daily life, associating mathematics mainly with calculation. Moreover, there is a clear difference in self-concept in boys and girls with respect to mathematics, ...
Indiana University Mathematics Journal, 2021
We show that if X is a pseudo-proper smooth noetherian formal scheme over a positive characterist... more We show that if X is a pseudo-proper smooth noetherian formal scheme over a positive characteristic p field k then its De Rham complex τ ≤p (F X/k * Ω • X/k) is decomposable. Along the way we establish the Cartier isomorphism Ω i X (p) /Y γ → H i (F X/Y * Ω • X/Y) associated to a map f : X → Y of positive characteristic p noetherian formal schemes where X (p) denotes the base change of X along the Frobenius morphism of Y and F X/Y denotes the relative Frobenius of X over Y. Contents 17 6. Decomposition at p 24 References 27
Journal of Pure and Applied Algebra, 2017
Let X be an Adams geometric stack. We show that D(A qc (X)), its derived category of quasi-cohere... more Let X be an Adams geometric stack. We show that D(A qc (X)), its derived category of quasi-coherent sheaves, satisfies the axioms of a stable homotopy category defined by Hovey, Palmieri and Strickland in [HPS]. Moreover we show how this structure relates to the derived category of comodules over a Hopf algebroid that determines X.
En este trabajo se aborda de forma sistematica el estudio de las conduciones infinitesimales de m... more En este trabajo se aborda de forma sistematica el estudio de las conduciones infinitesimales de morfismos de esquemas formales, Tratamos primero las condiciones de levantamiento (formalmente liso, formalmente no ramificado, formalmente etale) en el contexto de esquemas formales e introducimos las nociones de morfismo liso, no ramificado y etale si el morfismo es ademas de pseudo tipo finito. Esta condicion permite la caracterizacion de las condiciones infinitesimales en terminos diferenciales incluyendo las versiones usuales de los criterios jacobianos. Si ademas el morfismo es adico, se prueba que las condiciones infinitesimales estan determinadas por las condiciones infinitesimales de los morfismos de esquemas subyacentes. Para morfismos no necesariamente adicos se obtine una caracterizacion local de los morfismos lisos (no ramificados, etales) en terminos de compleciones (pseudo encajes cerrados, compleciones) y morfismos lisos adicos (etales adicos, etales adicos) que entendemos que esclarece completamente la estructura de los morfismos lisos (etales, no ramificados, respectivamente) de esquemas formales. Por otro lado, este estudio permite el desarrollo de una teoria de la deformacion formal para morfismos lisos de esquemas formales. La memoria se completa con aplicaciones de la teoria desarrollada. En concreto, estudiamos la cohomologia de De Rham en caracteristica positiva. Hemos demostrado el analogo del teorema de descomposicion para un esquema formal liso sobre un cuerpo de caracteristica positiva.
Journal of Algebra, 2016
Abstract We define the cotangent complex of a morphism f : X → Y of locally noetherian formal sch... more Abstract We define the cotangent complex of a morphism f : X → Y of locally noetherian formal schemes as an object in the derived category D − ( X ) through local homology. We discuss its basic properties and establish the basics results of a deformation theory, providing a characterization of smooth and etale morphisms. This leads to simpler lifting results depending on a differential module, for a class of non-smooth morphism of usual schemes. We also give descriptions of the cotangent complex in the case of regular closed immersions and complete intersection morphisms of formal schemes.
We make a systematic study of the infinitesimal lifting conditions of a pseudo finite type map of... more We make a systematic study of the infinitesimal lifting conditions of a pseudo finite type map of noetherian formal schemes. We recover the usual general properties in this context, and, more importantly, we uncover some new phenomena. We define a completion map of formal schemes as the one that arises canonically by performing the completion of a noetherian formal scheme along a subscheme, following the well-known pattern of ordinary schemes. These maps are etale in the sense of this work (but not adic). They allow us to give a local description of smooth morphisms. This morphisms can be factored locally as a completion map followed from a smooth adic morphism. The latter kind of morphisms can be factored locally as an etale adic morphism followed by a (formal) affine space. We also characterize etale adic morphisms giving an equivalence of categories between the category of etale adic formal schemes over a noetherian formal scheme (X, O_X) and the category of etale schemes over th...
Expositiones Mathematicae, 2015
A geometric stack is a quasi-compact and semi-separated algebraic stack. We prove that the quasi-... more A geometric stack is a quasi-compact and semi-separated algebraic stack. We prove that the quasi-coherent sheaves on the small flat topology, Cartesian presheaves on the underlying category, and comodules over a Hopf algebroid associated to a presentation of a geometric stack are equivalent categories. As a consequence, we show that the category of quasi-coherent sheaves on a geometric stack is a Grothendieck category. We also associate, in a 2-functorial way, to a 1-morphism of geometric stacks f : X → Y, an adjunction f * ⊣ f * for the corresponding categories of quasi-coherent sheaves that agrees with the classical one defined for schemes. This construction is described both geometrically in terms of the small flat site and algebraically in terms of comodules over the Hopf algebroid. CONTENTS 25 6. Properties and functoriality of quasi-coherent sheaves 33 7. Describing functoriality via comodules 40 8. Deligne-Mumford stacks and functoriality for the étale topology 45 References 49
Journal of Pure and Applied Algebra, 2009
We deepen our study on infinitesimal lifting properties of maps between locally noetherian formal... more We deepen our study on infinitesimal lifting properties of maps between locally noetherian formal schemes started in [3]. In this paper, we focus on properties which make sense specifically in the formal context. In this vein, we make a detailed study of the relationship between the infinitesimal lifting properties of a morphism of formal schemes and those of the corresponding maps of usual schemes associated to the directed systems that define the corresponding formal schemes. Among our main results, we obtain the characterization of completion morphisms as pseudo closed immersions that are flat. Also, the local structure of smooth andétale morphisms between locally noetherian formal schemes is described: the former factors locally as a completion morphism followed by a smooth adic morphism and the latter as a completion morphism followed by anétale adic morphism.
Journal of Pure and Applied Algebra, 2008
Applied Categorical Structures, 2009
In this paper, we prove that for a noetherian formal scheme X, its derived category of sheaves of... more In this paper, we prove that for a noetherian formal scheme X, its derived category of sheaves of modules with quasi-coherent torsion homologies Dqct(X) is generated by a single compact object. In an appendix we prove that the category of compact objects in Dqct(X) is skeletally small.
Advances in Mathematics, 2008
We prove in this paper that for a quasi-compact and semiseparated (non necessarily noetherian) sc... more We prove in this paper that for a quasi-compact and semiseparated (non necessarily noetherian) scheme X, the derived category of quasi-coherent sheaves over X, D(Aqc(X)), is a stable homotopy category in the sense of Hovey, Palmieri and Strickland, answering a question posed by Strickland. Moreover we show that it is unital and algebraic. We also prove that for a noetherian semi-separated formal scheme X, its derived category of sheaves of modules with quasi-coherent torsion homologies Dqct(X) is a stable homotopy category. It is algebraic but if the formal scheme is not a usual scheme, it is not unital, therefore its abstract nature differs essentially from that of the derived category Dqc(X) (which is equivalent to D(Aqc(X))) in the case of a usual scheme. 3 We will add a reference to [EGA I60] for the benefit of readers using the free www.numdam.org version. Unfortunately, in this case there is no equivalence though the results follow easily from [EGA I60, §1] together with the well-known fact that a sheaf F is quasi-coherent if and only if its restriction to an affine open set U = Spec(A) is such that F|U ∼ = f M for a certain A-module M. 4 See the previous footnote.
Communications in Algebra, 2007
This a first step to develop a theory of smooth,étale and unramified morphisms between noetherian... more This a first step to develop a theory of smooth,étale and unramified morphisms between noetherian formal schemes. Our main tool is the complete module of differentials, that is a coherent sheaf whenever the map of formal schemes is of pseudo finite type. Among our results we show that these infinitesimal properties of a map of usual schemes carry over into the completion with respect to suitable closed subsets. We characterize unramifiedness by the vanishing of the module of differentials. Also we see that a smooth morphism of noetherian formal schemes is flat and its module of differentials is locally free. The paper closes with a version of Zariski's Jacobian criterion.
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Papers by Marta Pérez Rodríguez