# Mori's “projective manifolds with ample tangent bundles”, Proposition 3

See http://www.jstor.org/stable/1971241?seq=6#page_scan_tab_contents, pages 597 and 598.

At the bottom of page 597, we get the following diagram:

$\begin{array}{ccccccccc} A/I & \xrightarrow{nat.} & A/(I+M^n) \\ & {\alpha}\searrow & \uparrow nat. \\ & & A/(MI+M^n+(r_1,...,r_a)) \end{array}$

where the $r_i$ are in $I+M^n$. $A$ is a formal power series ring over $k$ and $M$ its maximal ideal. $I \subset M^2, I\cap M^n = M(I\cap M^{n-1}) \subset MI$ (for some n, following from the Artin-Rees lemma). I am not sure how much context is required, so for now I will refer to the link for more. Do feel free to ask.

The crucial claim, on which the entire argument seems to rely, is that "since $I \subset M^2$, $\alpha$ is surjective by the above diagram." The following part of the proof is then a mere formality.

Why is $\alpha$ surjective? This seems not at all as obvious as the statement of the claim makes it sound. I believe thus that I am completely missing something critical.

For what it is worth, I am not yet familiar enough with obstruction theory. Maybe this is where the crux lies (see the linked paper, on the same page)? If so, I would appreciate some pointers on where to educate myself about this.