nLab
infinitesimal neighborhood

Infinitesimal neighbourhoods

Idea

An infinitesimal neighbourhood is a neighbourhood with infinitesimal diameter. These can be defined in several setups: nonstandard analysis, synthetic differential geometry, ringed spaces, ….

In nonstandard analysis

In nonstandard analysis, the monad of a standard point pp in a topological space (or even in a Choquet space) is the hyperset of all hyperpoint?s infinitely close to pp. It is the intersection of all of the standard neighbourhoods of pp and is itself a hyper-neighbourhood of pp, the infinitesimal neighbourhood of pp.

References

  • wikipedia Monad (non-standard analysis)
  • S. S. Kutateladze, Leibnizian, Robinsonian, and Boolean valued monads arxiv/1106.2755
  • Sergio Albeverio, Jens Erik Fenstad, Raphael Hoegh-Krohn, Nonstandard methods in stochastic analysis and mathematical physics, Academic Press 1986

For ringed spaces

Consider a morphism (f,f ):(Y,𝒪 Y)(X,𝒪 X)(f,f^\sharp):(Y,\mathcal{O}_Y)\to(X,\mathcal{O}_X) of ringed spaces for which the corresponding map F¯ :f *𝒪 X𝒪 Y\bar{F}^\sharp:f^*\mathcal{O}_X\to\mathcal{O}_Y of sheaves on YY is surjective. Let = f=Kerf¯ \mathcal{I} = \mathcal{I}_f = Ker\bar{f}^\sharp, then 𝒪 Y=f (𝒪 X)/ f\mathcal{O}_Y = f^\sharp(\mathcal{O}_X)/\mathcal{I}_f. The ring f *(𝒪 Y)f^*(\mathcal{O}_Y) has the \mathcal{I}-preadic filtration which has the associated graded ring Gr = n f n/ f n+1Gr_\bullet =\oplus_{n} \mathcal{I}_f^n/\mathcal{I}^{n+1}_f which in degree 11 gives the conormal sheaf Gr 1= f/ f 2Gr_1 = \mathcal{I}_f/\mathcal{I}^2_f of ff. The 𝒪 Y\mathcal{O}_Y-augmented ringed space (Y,f (𝒪 X)/ n+1)(Y,f^\sharp(\mathcal{O}_X)/\mathcal{I}^{n+1}) is called the nn-th infinitesimal neighborhood of YY along morphism ff. Its structure sheaf is called the nn-th normal invariant of ff.

Examples of sequences of local structures

geometrypointfirst order infinitesimal\subsetformal = arbitrary order infinitesimal\subsetlocal = stalkwise\subsetfinite
\leftarrow differentiationintegration \to
smooth functionsderivativeTaylor seriesgermsmooth function
curve (path)tangent vectorjetgerm of curvecurve
smooth spaceinfinitesimal neighbourhoodformal neighbourhoodgerm of a spaceopen neighbourhood
function algebrasquare-0 ring extensionnilpotent ring extension/formal completionring extension
arithmetic geometry𝔽 p\mathbb{F}_p finite field p\mathbb{Z}_p p-adic integers (p)\mathbb{Z}_{(p)} localization at (p)\mathbb{Z} integers
Lie theoryLie algebraformal grouplocal Lie groupLie group
symplectic geometryPoisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict deformation quantization

References

  • A. Grothendieck, Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Quatrième partie, Publications Mathématiques de l’IHÉS 32 (1967), p. 5-361, numdam

Revised on July 23, 2014 03:36:17 by Urs Schreiber (89.204.154.176)