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The Zariski spectrum as a formal geometry

Peter Schuster
2008 Theoretical Computer Science  
We next define the category of formal geometries, a natural abstraction from that of locally ringed spaces.  ...  We choose formal topology to deal in a basic manner with the Zariski spectra of commutative rings and their structure sheaves.  ...  The anonymous referees' critique was most helpful for bringing this paper into its final form.  ... 
doi:10.1016/j.tcs.2008.06.030 fatcat:nddz2pkt2ngwdmmocqritef3bm

On valuation spectra

Manfred Knebusch
1998 Banach Center Publications  
For the purpose of real geometry it is appropriate to replace Spev A by the real valuation spectrum Sperv A which is a sort of fibre product of Spev A with the real spectrum Sper A over the Zariski spectrum  ...  Indeed, this amounts to an extension of Zariski's approach in the thirties and forties to a truly "algebraic" geometry. Zariski used valuations more or less as "ideal points" of algebraic varieties.  ... 
doi:10.4064/-44-1-147-148 fatcat:nrkcllfrvffvbnvmye76ff2h3e

Derived noncommutative Zariski immersion and an equivalent reformulation of Friedlander-Milnor conjecture [article]

Ilias Amrani
2016 arXiv   pre-print
We il- lustrate the importance of such notion by reformulating the Friedlander-Milnor conjecture in terms of formal noncommutative Zariski immersions.  ...  This paper is based on the language developed by Dwyer, Greenless and Iyendar.  ...  Noncommutative Zariski immersion The notion of derived formal Zariski immersion shows up in the context of E ∞ring spectrum [9] seeking for a formulation of the Zariski topology in the setting of derived  ... 
arXiv:1603.01010v1 fatcat:mjvvbmjaazdjbhups6bmwxdqgy

"Brave New" Algebraic Geometry and global derived moduli spaces of ring spectra [article]

Bertrand Toen, Gabriele Vezzosi
2004 arXiv   pre-print
in Algebraic Geometry), and finally show how to define global moduli spaces of associative ring spectra structures and a moduli space related to topological modular forms as geometric S-stacks.  ...  We develop homotopical algebraic geometry (see math.AG/0207028) in the special context where the base symmetric monoidal model category is the category S of spectra, i.e. what might be called, after Waldhausen  ...  Any (formal) Zariski open covering of a commutative S-algebra is a (formal)étale covering.  ... 
arXiv:math/0309145v2 fatcat:th7ixp3nwfck5l6m7ubsvbgj5y

Formal Zariski topology: Positivity and points

Peter Schuster
2006 Annals of Pure and Applied Logic  
The topic of this article is the formal topology abstracted from the Zariski spectrum of a commutative ring.  ...  We further show that, constructively, the formal Zariski topology cannot have enough points.  ...  Last but not least, the author is grateful to Laura Crosilla for taking on also his portion of the joint organisation of a workshop while [54] was written.  ... 
doi:10.1016/j.apal.2005.05.026 fatcat:x2q72vfhhfeb5cb6mn43zepupu

The geometric semantics of algebraic quantum mechanics

John Alexander Cruz Morales, Boris Zilber
2015 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences  
In this paper we will present an ongoing project which aims to use model theory as a suitable mathematical setting for studying the formalism of quantum mechanics.  ...  We will argue that this approach provides a geometric semantics for such formalism by means of establishing a (non-commutative) duality between certain algebraic and geometric objects.  ...  Zariski geometries Zariski geometries were introduced in [3] as a generalization of Zariski topologies, a well-known concept in algebraic geometry, in order to study the hierarchy of stable structures  ... 
doi:10.1098/rsta.2014.0245 pmid:26124252 fatcat:ohukkcsmgne5zodvkqgaurvmbi

Page 4684 of Mathematical Reviews Vol. , Issue 80M [page]

1980 Mathematical Reviews  
The real Zariski spectrum of a ring A is defined to be the subspace of Spec(A) consisting of real prime ideals. It is proved to be compact.  ...  Let me describe some of the background ideas which lead to the definition and investigation of the real étale spectrum, as carried out in this section.  ... 

The Basic Zariski Topology

Davide Rinaldi, Giovanni Sambin, Peter Schuster
2015 Confluentes Mathematici (CM)  
We regain moreover the link with the usual notion in algebraic geometry, where the points of the prime spectrum are the prime ideals of the ring.  ...  Spatiality and reducibility of the formal Zariski topology As shown in the previous section, the formal points of the basic Zariski topology and of the formal Zariski topology coincide.  ... 
doi:10.5802/cml.18 fatcat:mcdr3bhoynfa7pqo5kiq7bvfce

Page 2339 of Mathematical Reviews Vol. , Issue 87e [page]

1987 Mathematical Reviews  
The bonus of this second proof is that it provides an elementary argument for the quasicompactness of X(R), as well as providing a concrete ring whose spectrum is homeomorphic to X(R).  ...  Assume that (A, M) is a formally equidimensional local Noetherian ring with infinite residue field A/M.  ... 

Extending obstructions to noncommutative functorial spectra [article]

Benno van den Berg, Chris Heunen
2014 arXiv   pre-print
This obstruction also applies to other spectra such as those named after Zariski, Stone, and Pierce.  ...  Any functor from the category of C*-algebras to the category of locales that assigns to each commutative C*-algebra its Gelfand spectrum must be trivial on algebras of nxn-matrices for n at least 3.  ...  Introduction The spectrum of a commutative ring is a leading tool of commutative algebra and algebraic geometry.  ... 
arXiv:1407.2745v1 fatcat:rnvcparml5erhb3hjcqwdgmtlq

Half a Century of Rigid Analytic Spaces

Siegfried Bosch
2009 Pure and Applied Mathematics Quarterly  
Starting out from Tate's Harvard notes, the classical approach by the school of Grauert and Remmert is covered, as well as the approach through formal schemes following Raynaud, including a glimpse on  ...  We explain the basic ideas and facts in rigid geometry from today's point of view.  ...  The formal R-scheme Spf A associated to A is the locally ringed space (X, O X ), where X is a (true) topological space, namely the prime spectrum Spec A/(I) endowed with the Zariski topology, and where  ... 
doi:10.4310/pamq.2009.v5.n4.a9 fatcat:k4s5pj33szdyzcbyq7r53xhxoa

On the Sheafyness Property of Spectra of Banach Rings [article]

Federico Bambozzi, Kobi Kremnizer
2020 arXiv   pre-print
This permits to use the tools of derived geometry to understand the geometry of Spa(R) also when H^0(O_Spa(R)) is not a sheaf.  ...  We prove that to R one can associate a homotopical Huber spectrum Spa^h(R) via the introduction of the notion of derived rational localizations.  ...  The authors are very grateful to Peter Scholze for pointing out a mistake on a first version of this paper.  ... 
arXiv:2009.13926v2 fatcat:wyxbinnpjrdmnbusnlwyppstmi

On the real spectrum of a ring and its application to semialgebraic geometry

Eberhard Becker
1986 Bulletin of the American Mathematical Society  
Their definition takes account of the entire structure of the real numbers as an ordered field. 1. From the solution of Hubert's 17th problem to the real spectrum.  ...  This paper is meant as an introduction and a guide to some recent developments in real algebraic geometry -more precisely, in semialgebraic geometry.  ...  Today it seems that the notion of the real spectrum of a ring may serve as a building block for a general semialgebraic geometry in the same way that the Zariski spectrum of a ring did for Grothendieck's  ... 
doi:10.1090/s0273-0979-1986-15431-5 fatcat:g2ifk6jjlffkhjnngh6if5uapu

Page 1394 of Mathematical Reviews Vol. , Issue 89C [page]

1989 Mathematical Reviews  
The smooth Zariski topos, Z, is defined as a Grothendieck topos with the category of the duals of C™-rings as site of definition.  ...  of manifolds into the smooth Zariski topos.  ... 

Stratifying integral representations via equivariant homotopy theory [article]

Tobias Barthel
2022 arXiv   pre-print
We prove that the derived category of R-linear representations of a finite group G is stratified for any regular commutative ring R.  ...  As an application, we obtain a classification of localizing tensor ideals of ordinary R-linear G-representations whose underlying R-module is projective.  ...  As an important approximation to the Balmer spectrum, Balmer [Bal10] constructed a comparison map between Spc(K) and the Zariski spectrum of the ring of endomorphisms of the unit 1 of K.  ... 
arXiv:2203.14946v1 fatcat:zdu7x7srxbhidm3omtkifg7j7m
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