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Let $\mathbb{C}[[z_1,\dots,z_n]]$ denote the algebra of formal power series. I am interested in faithfully flat morphisms $$Spec(\mathbb{C}[[z_1,\dots,z_m]])\to Spec(\mathbb{C}[[z_1,\dots,z_n]]),\, m\geq n$$ up to formal diffeomorphisms of the source and the target.

Is there anything known about classification of such objects?

Unfortunately I am unable to formulate my question more precisely.

Example. If $m=n=1$ then any such morphism is equivalent (in the above sense) to the morphism given by $z\mapsto z^k$ for some $k\in \mathbb{N}$.

A reference would be helpful.

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    $\begingroup$ Every local homomorphism of local rings induces a graded homomorphism of the associated graded rings. Thus, there are "at least as many" local homomorphisms from the power series ring $\mathbb{C}[[z_1,\dots,z_n]]$ to itself as there are finite morphisms from $\mathbb{P}_{\mathbb{C}}^{n-1}$ to itself. Already for $n=2$, there are moduli spaces of such maps, even after forming the quotient by pre- and post-composition with $\text{Aut}(\mathbb{P}_{\mathbb{C}}^1)$. The dimensions of the moduli spaces increase to infinity. $\endgroup$
    – Jason Starr
    Commented Mar 21, 2017 at 10:01
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    $\begingroup$ I think such a morphism is automatically flat if the closed fiber has dimension $m-n$ by `miracle flatness' (Matsumura, Thm. 23.1). Also, I guess you mean continuous morphisms ($\rm Spf$ instead of $\rm Spec$), in which case 'faithfully' should be automatic. This suggests that a classification should be beyond reach in higher dimensions. $\endgroup$
    – Piotr Achinger
    Commented Mar 21, 2017 at 10:03

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