Classical Deformation Theory is used for determining the
completions of the local rings of an eventual moduli space. When a
moduli variety exists, a main result in the book is that the local
ring in a closed point can be explicitly computed as an
algebraization of the pro-representing hull (therefore, called the
local formal moduli) of the deformation functor for the
corresponding closed point.The book gives explicit computational
methods and includes the most necessary prerequisites. It focuses
on the meaning and the place of deformation theory, resulting in a
complete theory applicable to moduli theory. It answers the
question 'why moduli theory' and it give examples in mathematical
physics by looking at the universe as a moduli of molecules.
Thereby giving a meaning to most noncommutative theories.The book
contains the first explicit definition of a noncommutative scheme,
covered by not necessarily commutative rings. This definition does
not contradict any of the previous abstract definitions of
noncommutative algebraic geometry, but rather gives interesting
relations to other theories which is left for further
investigation.
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