nLab
formal geometry

Contents

Idea

Several notions of formal geometry

Formal geometry is a highly overloaded term in mathematics, used in number of conceptually similar ways, usually meaning that we work in setup in which some crucial details of geometry or analysis are not present or satisfied, e.g.

  • we work with functions on “manifolds” but the functions do not necessarily converge, the geometry is rather based on topological algebras of formal power series; this is the formal geometry of Grothendieck school and the main notion is that of a formal scheme (or more general ind-schemes). There are also noncommutative versions like Kapranov's noncommutative geometry.

  • Gelfand’s formal geometry: study infinite dimensional manifolds of jet bundles and related objects coming from usual differential geometry, geometry of formal differential operators, study of related objects from homological algebra, including Gelfand’s formal manifold (homological vector field)

  • we talk about neighborhoods,or localizations, morphisms of spaces, but not about spectra and points (a part of noncommutative geometry is done in such style) – this is sometimes called “pseudogeometry”

  • in algebraic geometry: formal spectrum of an adic noetherian ring

Gelfand’s formal geometry

See MathOverflow: formal-geometry, how-do-i-describe-the-gl-n-torsor-attached-to-a-smooth-morphism-of-relative-dimen,

  • Israel Gelfand, David Kazhdan, Certain questions of differential geometry and the computation of the cohomologies of the Lie algebras of vector fields, Soviet Math. Doklady 12, 1367-1370 (1971)

For characteristic p>0p\gt 0 case see

  • Fedosov quantization in positive characteristics arXiv

In a similar formal context Gelfand and collaborators introduced (𝔄,𝒟)(\mathfrak{A},\mathcal{D})-systems

  • I. M. Gelʹfand, Yu. L. Daletskiĭ, Lie superalgebras and Hamiltonian operators, Rep. No. 16 Sem. Supermanifolds, Dept. Math. Univ. Stockholm, 1987, 26 p.
  • I. M. Gelʹfand, Yu. L. Daletskiĭ, B. L. Tsygan, On a variant of noncommutative differential geometry, Dokl. Akad. Nauk SSSR 308 (1989), no. 6, 1293–1297; translation in Soviet Math. Dokl. 40 (1990), no. 2, 422–426 MR91j:58015

The (𝔄,𝒟)(\mathfrak{A},\mathcal{D})-systems were partly motivated by the calculus of variations, formalizing further the setting of works of Gelfand and Dorfman.

  • И. М. Гельфанд, И. Я. Дорфман, Гамильтоновы операторы и бесконечномерные алгебры Ли, Функц. анализ и его прил., 15:3 (1981), 23–40, pdf; Гамильтоновы операторы и классическое уравнение Янга–Бакстера, Функц. анализ и его прил. 16:4 (1982), 1–9 pdf; Скобка Схоутена и гамильтоновы операторы, Функц. анализ и его прил., 14:3 (1980), 71–74 pdf
  • Ю. Л. Далецкий, Гамильтоновы операторы в градуированном формальном вариационном исчислении, pdf, transl. Yu. L. Daletskii, Hamiltonian operators in graded formal calculus of variations, Funct. Anal. Appl. 20:2, 1986, 136-138

Formal noncommutative symplectic geometry of Kontsevich

  • Maxim Kontsevich, Formal (non)-commutative symplectic geometry, The Gelfand Mathematical. Seminars, 1990-1992, Ed. L.Corwin, I.Gelfand, J.Lepowsky, Birkhauser 1993, 173-187, pdf

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
Revised on July 19, 2014 00:39:13 by Urs Schreiber (82.113.106.79)