In mathematics, formal moduli are an aspect of the theory of moduli spaces (of algebraic varieties or vector bundles, for example), closely linked to deformation theory and formal geometry. 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 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 amongst such set-ups. The local ring in question is then the carrier of the formal moduli.
Famous quotes containing the word formal:
“The spiritual kinship between Lincoln and Whitman was founded upon their Americanism, their essential Westernism. Whitman had grown up without much formal education; Lincoln had scarcely any education. One had become the notable poet of the day; one the orator of the Gettsyburg Address. It was inevitable that Whitman as a poet should turn with a feeling of kinship to Lincoln, and even without any association or contact feel that Lincoln was his.”
—Edgar Lee Masters (1869–1950)