On the structure of DeWitt supermanifolds

Abstract

We show that for a wide and most natural class of (possibly infinite-dimensional) Grassmannian algebras of coefficients, the structure sheaf of every smooth DeWitt supermanifold is acyclic (i.e. its cohomology vanishes in positive degree). This result was previously known for finite-dimensional ground algebras and is new even for the original DeWitt algebra of supernumbers /GL∞. From here we deduce that (equivalence classes of) smooth DeWitt supermanifolds over a fixed ground algebra and of graded smooth manifolds are in a natural bijection with each other. However, contrary to what was stated previously by some authors, this correspondence fails to be functorial; so it happens, for instance, for Rogers' ground algebra B∞. Finally, we observe that every DeWitt super Lie group is a deformation of a graded Lie group over the spectrum Spec /GL of the ground algebra.