Timeline for Classification of faithfully flat morphisms between formal power series
Current License: CC BY-SA 3.0
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| Mar 21, 2017 at 10:03 | comment | added | Piotr Achinger | 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. | |
| Mar 21, 2017 at 10:01 | comment | added | Jason Starr | 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. | |
| Mar 21, 2017 at 9:23 | history | asked | asv | CC BY-SA 3.0 |