infinitesimal neighborhood

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

- 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

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

**Examples of sequences of local structures**

geometry | point | first order infinitesimal | $\subset$ | formal = arbitrary order infinitesimal | $\subset$ | local = stalkwise | $\subset$ | finite |
---|---|---|---|---|---|---|---|---|

$\leftarrow$ differentiation | integration $\to$ | |||||||

smooth functions | derivative | Taylor series | germ | smooth function | ||||

curve (path) | tangent vector | jet | germ of curve | curve | ||||

smooth space | infinitesimal neighbourhood | formal neighbourhood | germ of a space | open neighbourhood | ||||

function algebra | square-0 ring extension | nilpotent ring extension/formal completion | ring extension | |||||

arithmetic geometry | $\mathbb{F}_p$ finite field | $\mathbb{Z}_p$ p-adic integers | $\mathbb{Z}_{(p)}$ localization at (p) | $\mathbb{Z}$ integers | ||||

Lie theory | Lie algebra | formal group | local Lie group | Lie group | ||||

symplectic geometry | Poisson manifold | formal deformation quantization | local strict deformation quantization | strict deformation quantization |

- 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 December 30, 2014 14:17:06
by Urs Schreiber
(127.0.0.1)