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Journal of the London Mathematical Society 1989 s2-39(2):347-368; doi:10.1112/jlms/s2-39.2.347
© 1989 by London Mathematical Society
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© Oxford University Press

De Witt Supermanifolds and Infinitedimensional Ground Rings

Peter Bryant

Department of Pure Mathematics and Mathematical Statistics 16 Mill Lane, Cambridge CB2 1SB

Batchelor compared graded manifolds with Rogers supermanifolds (for BL = {Lambda}RL, L < {infty}) using algebraic techniques. We investigate the extension to supermanifolds over an infinite-dimensional ground ring B (not necessarily of ‘Banach-Grassman’ type).

An ‘abstract’ supermanifold over B is a Z2-ringed space (S, Formula) with an appropriate local modelling property (so Formula is not necessarily a sheaf of B-valued functions). We show that the rule

(X, A) -> Alg(A(X), B)

(algebraic representation) gives an equivalence between graded manifolds and abstract de Witt supermanifolds if and only if B is reflexive in the sense that B = Bo*.

We recover Batchelor's results for B = BL with the fine topology since (Formula) is abstract de Witt if and only if S is H{infty}-de Witt (that is, Formula = Formula), provided that L exceeds the odd dimension of S. We can extend to L = {infty} provided that B{infty} is taken to be the (formal) Grassmann algebra B{infty} = {Lambda}R{infty} with its Frechet topology rather than Rogers's Banach-Grassmann subalgebra BG. The algebra B{infty} is reflexive but BG is not.

These results become relevant for comparing de Witt supermanifolds with Jadczyk-Pilch supermanifolds when we take a formal L -> {infty} limit.


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