nLab formal geometry

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 $p\gt 0$ case see

• Fedosov quantization in positive characteristics arXiv

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

See also

• Bernstein, Rozenfeld, Homogeneous spaces of infinite-dimensional lie algebras and characteristic classes of foliations, Uspehi Mat. Nauk, English pdf

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

Related concepts

• formal dg-algebra

• formal neighbourhood, formal disk

• Hasse local-global principle

• formal Picard group, formal Brauer group

• Kapranov's noncommutative geometry

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