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The Zariski spectrum as a formal geometry

2008
*
Theoretical Computer Science
*

We next define

doi:10.1016/j.tcs.2008.06.030
fatcat:nddz2pkt2ngwdmmocqritef3bm
*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. ...##
###
On valuation spectra

1998
*
Banach Center Publications
*

For

doi:10.4064/-44-1-147-148
fatcat:nrkcllfrvffvbnvmye76ff2h3e
*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. ...##
###
Derived noncommutative Zariski immersion and an equivalent reformulation of Friedlander-Milnor conjecture
[article]

2016
*
arXiv
*
pre-print

We il- lustrate

arXiv:1603.01010v1
fatcat:mjvvbmjaazdjbhups6bmwxdqgy
*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 ...##
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"Brave New" Algebraic Geometry and global derived moduli spaces of ring spectra
[article]

2004
*
arXiv
*
pre-print

in Algebraic

arXiv:math/0309145v2
fatcat:th7ixp3nwfck5l6m7ubsvbgj5y
*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. ...##
###
Formal Zariski topology: Positivity and points

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. ...

##
###
The geometric semantics of algebraic quantum mechanics

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

doi:10.1098/rsta.2014.0245
pmid:26124252
fatcat:ohukkcsmgne5zodvkqgaurvmbi
*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 ...##
###
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

2015
*
Confluentes Mathematici (CM)
*

We regain moreover

doi:10.5802/cml.18
fatcat:mcdr3bhoynfa7pqo5kiq7bvfce
*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. ...##
###
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]

2014
*
arXiv
*
pre-print

This obstruction also applies to other spectra such

arXiv:1407.2745v1
fatcat:rnvcparml5erhb3hjcqwdgmtlq
*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*. ...##
###
Half a Century of Rigid Analytic Spaces

2009
*
Pure and Applied Mathematics Quarterly
*

Starting out from Tate's Harvard notes,

doi:10.4310/pamq.2009.v5.n4.a9
fatcat:k4s5pj33szdyzcbyq7r53xhxoa
*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 ...##
###
On the Sheafyness Property of Spectra of Banach Rings
[article]

2020
*
arXiv
*
pre-print

This permits to use

arXiv:2009.13926v2
fatcat:wyxbinnpjrdmnbusnlwyppstmi
*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. ...##
###
On the real spectrum of a ring and its application to semialgebraic geometry

1986
*
Bulletin of the American Mathematical Society
*

Their definition takes account of

doi:10.1090/s0273-0979-1986-15431-5
fatcat:g2ifk6jjlffkhjnngh6if5uapu
*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 ...##
###
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. ...

##
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Stratifying integral representations via equivariant homotopy theory
[article]

2022
*
arXiv
*
pre-print

We prove that

arXiv:2203.14946v1
fatcat:zdu7x7srxbhidm3omtkifg7j7m
*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. ...
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