formal differences?

Question

Hi, given a local ring $A$ with maximal ideal $m$ which are differences between $Spec(\hat{A})$ ($\hat{A}$ completion of $A$ along $m$) and $Spf(A)$?

Answer 1

The difference is total, the spaces and the sheaves are different. The underlying space of $Spf(\hat{A})$ is just a point (in general, if $m$ is not maximal, it is all the prime ideals containing $m$, i.e. the open ideals. The underlying space of $Spec(\hat{A})$ is made of all prime ideals. Therefore $Spf(\hat{A})$ is one-point space, thus the strcture sheaf is just the ring.

In general, if you consider a general ideal $I$ and denote the $I$-adic completion of a noetherian ring $A$ by $\hat{A}$, the underlying space of $Spf(\hat{A})$ is $Spec(A/I)$ while $Spec(\hat{A})$ is formed by all prime ideals in $\hat{A}$ (they are different from the ones in $Spec(A)$ but some features, like dimension, are preserved). Also, if $f \notin I$ the sections of the structural sheaf of $Spec(\hat{A})$ along the principal open subset determined by $f$ is the ring $S^{-1}\hat{A}$, a localization wrt $S = 1,f, f^2...$. While the sections of the structural sheaf of $Spf(\hat{A})$ along the principal open subset determined by $f$ is the completion of the ring $S^{-1}A$ for the $S^{-1}I$-adic topology.

There is however a completion map of ringed spaces $$ \kappa \colon Spf(\hat{A}) \to Spec(\hat{A}) $$ that permits some comparisons of the respective sheaf theories.

Answer 2

There is a nice introduction to formal schemes in Luc Illusie's section of the book ``Fundamental Algebraic Geometry (Grothendieck's FGA Explained)''. Recommend!