I'm wondering if finite unramified morphism between reduced schemes decomposes as closed immersions and etale morphisms. Suppose I have a morphism between reduced schemes which is finite, surjective and unramified, is it necessarily etale? I think this is certainly true if both source and target are curves, but I'm not sure about higher dimensional examples. Thanks
EDIT: to avoid trivial example let's assume the source and target are connected. What I'm wondering is precisely when one can deduce flatness from these conditions.