In mathematics, formal moduli are an aspect of the theory of

moduli space In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such ...

s (of

algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex number ...

vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to ev ...

s, for example), closely linked to

deformation theory In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution ''P'' of a problem to slightly different solutions ''P''ε, where ε is a small number, or a vector of small quantities. The infinitesi ...

and

formal geometry In mathematics, specifically in algebraic geometry, a formal scheme is a type of space which includes data about its surroundings. Unlike an ordinary scheme, a formal scheme includes infinitesimal data that, in effect, points in a direction off ...

. Roughly speaking, deformation theory can provide the Taylor polynomial level of information about deformations, while formal moduli theory can assemble consistent Taylor polynomials to make a

formal power series In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial s ...

theory. The step to moduli spaces, properly speaking, is an ''algebraization'' question, and has been largely put on a firm basis by Artin's approximation theorem. A formal universal deformation is by definition a formal scheme over a complete local ring, with special fiber the scheme over a field being studied, and with a

universal property In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently ...

amongst such set-ups. The local ring in question is then the carrier of the formal moduli.

References

* Moduli theory Algebraic geometry Geometric algebra {{Mathanalysis-stub