Scheme theoretic tropicalization

In this paper, we introduce ordered blueprints and ordered blue schemes, which serve as a common language for the different approaches to tropicalizations and which enhances tropical varieties with a schematic structure. As an abstract concept, we consider a tropicalization as a moduli problem about extensions of a given valuation $v:k\to T$ between ordered blueprints $k$ and $T$. If $T$ is idempotent, then we show that a generalization of the Giansiracusa bend relation leads to a representing object for the tropicalization, and that it has yet another interpretation in terms of a base change along $v$. We call such a representing object a scheme theoretic tropicalization. This theory recovers and improves other approaches to tropicalizations as we explain with care in the second part of this text. The Berkovich analytification and the Kajiwara-Payne tropicalization appear as rational point sets of a scheme theoretic tropicalization. The same holds true for its generalization by Foster and Ranganathan to higher rank valuations. The scheme theoretic Giansiracusa tropicalization can be recovered from the scheme theoretic tropicalizations in our sense. We obtain an improvement due to the resulting blueprint structure, which is sufficient to remember the Maclagan-Rinc\'on weights. The Macpherson analytification has an interpretation in terms of a scheme theoretic tropicalization, and we give an alternative approach to Macpherson's construction of tropicalizations. The Thuillier analytification and the Ulirsch tropicalization are rational point sets of a scheme theoretic tropicalization. Our approach yields a generalization to any, possibly nontrivial, valuation $v:k\to T$ with idempotent $T$ and enhances the tropicalization with a schematic structure.Comment: 66 pages; for information about the changes in this version of the paper, please cf. the paragraph "Differences to previous versions" in the introductio

On MV - topologies

En este trabajo estamos interesados en un tipo particular de topología fuzzy llamada MV-topología, la cual usa operaciones MV-algebraicas para generar abiertos fuzzy. Estos espacios topológicos fuzzy permiten generalizaciones naturales de definiciones y resultados importantes de la topología clásica. En este sentido, desarrollamos algunos conceptos y resultados centrales, con el proprósito de extender los correspondientes de la topología clásica, y al mismo tiempo siguiendo la ruta de la bien conocida teoría de espacios topológicos fuzzy. Mostramos que las MV-topologías son un tipo de topología fuzzy que goza de muy "buen comportamiento" matemático, en el sentido de que la mayoría de definiciones y resultados clásicos de topología general encuentran una extensión o adaptación natural en este marco. Entre otros resultados, también extendemos el concepto de haz para el caso en el que el espacio base es un espacio MV-topológico, y mostramos una representación por "MV-haces" para una clase de MV-álgebras.DoctoradoDOCTOR(A) EN CIENCIAS - MATEMÁTICA

Sheaf representations of MV-algebras and lattice-ordered abelian groups via duality

We study representations of MV-algebras -- equivalently, unital lattice-ordered abelian groups -- through the lens of Stone-Priestley duality, using canonical extensions as an essential tool. Specifically, the theory of canonical extensions implies that the (Stone-Priestley) dual spaces of MV-algebras carry the structure of topological partial commutative ordered semigroups. We use this structure to obtain two different decompositions of such spaces, one indexed over the prime MV-spectrum, the other over the maximal MV-spectrum. These decompositions yield sheaf representations of MV-algebras, using a new and purely duality-theoretic result that relates certain sheaf representations of distributive lattices to decompositions of their dual spaces. Importantly, the proofs of the MV-algebraic representation theorems that we obtain in this way are distinguished from the existing work on this topic by the following features: (1) we use only basic algebraic facts about MV-algebras; (2) we show that the two aforementioned sheaf representations are special cases of a common result, with potential for generalizations; and (3) we show that these results are strongly related to the structure of the Stone-Priestley duals of MV-algebras. In addition, using our analysis of these decompositions, we prove that MV-algebras with isomorphic underlying lattices have homeomorphic maximal MV-spectra. This result is an MV-algebraic generalization of a classical theorem by Kaplansky stating that two compact Hausdorff spaces are homeomorphic if, and only if, the lattices of continuous [0, 1]-valued functions on the spaces are isomorphic.Comment: 36 pages, 1 tabl

Extended Riemannian Geometry II: Local Heterotic Double Field Theory

We continue our exploration of local Double Field Theory (DFT) in terms of symplectic graded manifolds carrying compatible derivations and study the case of heterotic DFT. We start by developing in detail the differential graded manifold that captures heterotic Generalized Geometry which leads to new observations on the generalized metric and its twists. We then give a symplectic pre-NQ-manifold that captures the symmetries and the geometry of local heterotic DFT. We derive a weakened form of the section condition, which arises algebraically from consistency of the symmetry Lie 2-algebra and its action on extended tensors. We also give appropriate notions of twists-which are required for global formulations-and of the torsion and Riemann tensors. Finally, we show how the observed $\alpha'$-corrections are interpreted naturally in our framework.Comment: v2: 30 pages, few more details added, typos fixed, published versio

Twisted equivariant K-theory with complex coefficients

Using a global version of the equivariant Chern character, we describe the complexified twisted equivariant K-theory of a space with a compact Lie group action in terms of fixed-point data. We apply this to the case of a compact Lie group acting on itself by conjugation, and relate the result to the Verlinde algebra and to the Kac numerator at q=1. Verlinde's formula is also discussed in this context.Comment: Final version, to appear in Topology. Exposition improved, rational homotopy calculation completely rewritte