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SupermanifoldA supermanifold is a concept in noncommutative geometry. Recall that in noncommutative geometry, we don't look at point set spaces but instead, the algebra of functions over them. If M is a (smooth) manifold and H is an (smooth) algebra bundle over M with a Grassmann algebra as the fiber, then the space of (smooth) sections of M forms a supercommutative algebra under pointwise multiplication. We say that this algebra defines the supermanifold (which isn't a point set space). If M is a real manifold and we define an involution * over the fiber turning it into a * algebra, then the resulting algebra would define a real supermanifold. OX is the sheaf of smooth sections of the algebra bundle. This is a structure sheaf of noncommutative rings. See locally ringed space. Side comment This is different from the convention used by some people where using a fixed Grassmann algebra generated by a countable number of generators Λ, they define a supermanifold as a point set space using charts with the "even coordinates" taking values in the linear combination of elements of Λ with even grading and the "odd coordinates" taking values which are linear combinations of elements of Λ with odd grading. But then, you'd have to question what the physical meaning of all these Grassmann-valued variables are! Most physicists would claim they have none and they are purely formal, but then, this makes the definition in the main part of the article more preferable.
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