scholarly journals DERIVED NON-ARCHIMEDEAN ANALYTIC HILBERT SPACE

2020 ◽  
pp. 1-32
Author(s):  
Jorge António ◽  
Mauro Porta
Keyword(s):  
Hilbert Space ◽  
Short Paper ◽  
Formal Models ◽  
Analytic Geometry ◽  
Localization Theorem ◽  
Formal Scheme ◽  
Generic Fiber ◽  
Link Type ◽  
Perfect Complexes ◽  
Rigid Analytic Geometry

In this short paper, we combine the representability theorem introduced in [Porta and Yu, Representability theorem in derived analytic geometry, preprint, 2017, arXiv:1704.01683; Porta and Yu, Derived Hom spaces in rigid analytic geometry, preprint, 2018, arXiv:1801.07730] with the theory of derived formal models introduced in [António, $p$ -adic derived formal geometry and derived Raynaud localization theorem, preprint, 2018, arXiv:1805.03302] to prove the existence representability of the derived Hilbert space $\mathbf{R}\text{Hilb}(X)$ for a separated $k$ -analytic space $X$ . Such representability results rely on a localization theorem stating that if $\mathfrak{X}$ is a quasi-compact and quasi-separated formal scheme, then the $\infty$ -category $\text{Coh}^{-}(\mathfrak{X}^{\text{rig}})$ of almost perfect complexes over the generic fiber can be realized as a Verdier quotient of the $\infty$ -category $\text{Coh}^{-}(\mathfrak{X})$ . Along the way, we prove several results concerning the $\infty$ -categories of formal models for almost perfect modules on derived $k$ -analytic spaces.

scholarly journals Definable Transformation to Normal Crossings over Henselian Fields with Separated Analytic Structure

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 934
Author(s):  
Krzysztof Jan Nowak
Keyword(s):  
Quantifier Elimination ◽  
General Setting ◽  
Resolution Of Singularities ◽  
Analytic Structure ◽  
Analytic Geometry ◽  
Skolem Functions ◽  
Algebraic Properties ◽  
Rank One ◽  
Rigid Analytic Geometry ◽  
Definable Functions

We are concerned with rigid analytic geometry in the general setting of Henselian fields K with separated analytic structure, whose theory was developed by Cluckers–Lipshitz–Robinson. It unifies earlier work and approaches of numerous mathematicians. Separated analytic structures admit reasonable relative quantifier elimination in a suitable analytic language. However, the rings of global analytic functions with two kinds of variables seem not to have good algebraic properties such as Noetherianity or excellence. Therefore, the usual global resolution of singularities from rigid analytic geometry is no longer at our disposal. Our main purpose is to give a definable version of the canonical desingularization algorithm (the hypersurface case) due to Bierstone–Milman so that both of these powerful tools are available in the realm of non-Archimedean analytic geometry at the same time. It will be carried out within a category of definable, strong analytic manifolds and maps, which is more flexible than that of affinoid varieties and maps. Strong analytic objects are those definable ones that remain analytic over all fields elementarily equivalent to K. This condition may be regarded as a kind of symmetry imposed on ordinary analytic objects. The strong analytic category makes it possible to apply a model-theoretic compactness argument in the absence of the ordinary topological compactness. On the other hand, our closedness theorem enables application of resolution of singularities to topological problems involving the topology induced by valuation. Eventually, these three results will be applied to such issues as the existence of definable retractions or extending continuous definable functions. The established results remain valid for strictly convergent analytic structures, whose classical examples are complete, rank one valued fields with the Tate algebras of strictly convergent power series. The earlier techniques and approaches to the purely topological versions of those issues cannot be carried over to the definable settings because, among others, non-Archimedean geometry over non-locally compact fields suffers from lack of definable Skolem functions.

Perfect complexes on Deligne-Mumford stacks and applications

2009 ◽  
Vol 4 (3) ◽  
pp. 559-603 ◽  
Author(s):  
Amalendu Krishna
Keyword(s):  
Full Subcategory ◽  
Derived Categories ◽  
Derived Category ◽  
Triangulated Categories ◽  
Localization Theorem ◽  
Coherent Sheaves ◽  
Perfect Complexes ◽  
Tensor Triangulated Categories ◽  
Resolution Property

AbstractFor a tame Deligne-Mumford stack X with the resolution property, we show that the Cartan-Eilenberg resolutions of unbounded complexes of quasicoherent sheaves are K-injective resolutions. This allows us to realize the derived category of quasi-coherent sheaves on X as a reflexive full subcategory of the derived category of X-modules.We then use the results of Neeman and recent results of Kresch to establish the localization theorem of Thomason-Trobaugh for the K-theory of perfect complexes on stacks of above type which have coarse moduli schemes. As a byproduct, we get a generalization of Krause's result about the stable derived categories of schemes to such stacks.We prove Thomason's classification of thick triangulated tensor subcategories of D(perf / X). As the final application of our localization theorem, we show that the spectrum of D(perf / X) as defined by Balmer, is naturally isomorphic to the coarse moduli scheme of X, answering a question of Balmer for the tensor triangulated categories arising from Deligne-Mumford stacks.

Sangaku Optimization Problems: An Algebraic Approach

2018 ◽  
Vol 111 (5) ◽  
pp. 385-389
Author(s):  
David Schultz ◽  
Enrico Serpone
Keyword(s):  
Optimization Problems ◽  
Algebraic Approach ◽  
Formal Definition ◽  
Analytic Geometry ◽  
Edo Period ◽  
Link Type ◽  
Mathematical Problems ◽  
Unique Approach ◽  
Definition Of ◽  
Solution Techniques

During the Edo Period (1603-1867), Japan was isolated from the influence of western mathematics. Despite this isolation, Japanese mathematics, called Wasan, flourished, and a unique approach to present mathematical problems was developed. Painted wooden tablets called sangaku were hung on display at Shinto shrines and Buddhist temples for recreational enjoyment and religious offerings. More than 900 tablets have been discovered with problems developed by priests, samurai, farmers, and children. The vast majority of these problems were solved using analytic geometry and algebraic means, and the collection as a whole is frequently referred to as Japanese Temple Geometry. Within the collection of the sangaku, several optimization problems appear with answers included. However, the methods used to obtain those answers are absent. Because the work of Newton and Leibniz was unknown to the Japanese mathematicians of that time and no evidence exists of contemporaneous Japanese mathematicians having a formal definition of the derivative, their solution techniques to these problems remains unresolved (Fukagawa and Rothman 2008). To illustrate a possible noncalculus approach for the solution to sangaku optimization problems, we will examine two specific examples. To help readers visualize the two examples, Maple™ animations have been created by the authors and can be found at http://www.mesacc.edu/~davvu41111/Sangaku.htm.

On series whose rearrangements possess discrete sets of limit points

2014 ◽  
Vol 20 (1) ◽  
Author(s):  
Roman Wituła ◽  
Edyta Hetmaniok ◽  
Konrad Kaczmarek
Keyword(s):  
Hilbert Space ◽  
Short Paper ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space ◽  
Limit Points

Abstract.Aim of this short paper is to construct in any infinite-dimensional Hilbert space a series with terms tending to zero such that some of its rearrangements possess the discrete set of limit points.

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