TABLE OF CONTENTS Supergeometry is an ingredient in many quantum models, e.g., SUSY mechanics and BRST formalism. Supergeometry is phrased in terms of ? 2-graded modules and sheaves over ? 2-graded commutative algebras. Their algebraic properties naturally generalize those of modules and sheaves over commutative algebras, but this is not a particular case of non-commutative geometry because of the peculiar definition of graded derivations. Here, we restrict our consideration to geometry of graded manifolds. They are not supermanifolds, though every graded manifold determines a DeWitt H ?-supermanifold, and vice versa [21] (see Theorem 6.9.5 below).
Unless otherwise stated, by a graded structure throughout this Chapter is meant a ? 2-graded structure, and the symbol [.] stands for the ? 2-graded parity. Let us recall some basic notions of the graded tensor calculus [21; 102].
An algebra 
is called graded if it is endowed with a grading automorphism ? such that ? 2 = Id. A graded algebra seen as a ?-module falls into the direct sum 
of two ?-modules 
and 
of even and odd elements such that
One calls 
and 
the even and odd parts of 
, respectively. In particular, if ? = Id, then 
. Since
we have
where 
, 
. It follows that 
is a subalgebra of 
and 
is an 
-module. If 
is a graded ring, then [ 1] = 0.
A...
