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.
Best Answer
Finite, surjective, and unramified does not imply etale. E.g. suppose that $Y$ is a proper closed subscheme of $X$, and we consider the map $X \coprod Y \to X$ defined as the disjoint union of the identity on $X$, and the given closed immersion $Y \to X$ on $Y$.
Then this map is finite, unramified, and surjective, but not etale. (See Sandor's answer for the missing condition, which is flatness!)
Added: A more interesting example is given by letting $X$ be a nodal cubic, letting $\tilde{X}$ be the normalization, and considering the natural map $\tilde{X} \to X.$ This map is not flat and certainly not etale, but it is unramified. (Formally, each branch through the node maps by a closed immersion into the nodal curve.)