2 objects -> fields

I have an idea for the case of a submersion - maybe it is nonsense but maybe not.

In the algebraic case of nonsingular varieties X,Y over an algebraically closed field one can check smoothness (which implies flatness) of a morphism X -> Y in terms of the induced maps of the Zariski tangent spaces at closed points being surjective. So in particular, identifying the Zariski tangent spaces with the fibres of the tangent bundles the "algebraic version of submersion checked on closed points" implies flatness.

If we take the viewpoint of differentiable spaces and work with injectivity of the cotangent sheaf tensored with residue objects fields rather than surjectivity of the tangent bundle maybe it is possible to transfer the proof to the case you are interested in? All one really needs is to translate the statement that we have an injective morphism on the fibres of the cotangent bundle into one about regular sequences in the rings of germs and then hopefully one could use a version of the local criterion of flatness to conclude.

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I have an idea for the case of a submersion - maybe it is nonsense but maybe not.

In the algebraic case of nonsingular varieties X,Y over an algebraically closed field one can check smoothness (which implies flatness) of a morphism X -> Y in terms of the induced maps of the Zariski tangent spaces at closed points being surjective. So in particular, identifying the Zariski tangent spaces with the fibres of the tangent bundles the "algebraic version of submersion checked on closed points" implies flatness.

If we take the viewpoint of differentiable spaces and work with injectivity of the cotangent sheaf tensored with residue objects rather than surjectivity of the tangent bundle maybe it is possible to transfer the proof to the case you are interested in? All one really needs is to translate the statement that we have an injective morphism on the fibres of the cotangent bundle into one about regular sequences in the rings of germs and then hopefully one could use a version of the local criterion of flatness to conclude.