For an arbitrary rational polyhedron we consider its decompositions into
Minkowski summands and, dual to this, so-called free extensions of the
associated pair of semigroups. Being free for a pair of semigroups is
equivalent to flatness for the corresponding algebras. Our main result
is phrased in this dual setup: the category of free extensions always
contains an initial object which can be described explicitly.
This result is in contrast to the algebro-geometric version dealing with
deformation theory. The existence of initial free extensions could be
interpreted as the lack of obstructions.
This is joined work with Alexandru Constantinescu and Matej Filip.