4

The statement of the Weierstrass Preparation Theorem is as follows:

Let $f = \sum_{i=0}^\infty a_iX^i \in K[[X]]$ for some field K where $a_h \neq 0$ and for every $n < h$, $a_n = 0$. Then the elements 1,$\bar x$, ... , $\bar x^{h-1}$ form a basis in $K[[X]]/(f)$ over $K$.

My question is how does one generalize and prove this statement for a ring of formal power series in n variables?

User20354
  • 982
  • 1
    You can replace $K$ by a complete local ring $A$ (for $n < h$, the $a_{n}$ should be contained in the maximal ideal of $A$), see for example these notes by Hochster. Thus you can apply this to $A = K[[X_{1},\dotsc,X_{r-1}]]$ and use that $(K[[X_{1},\dotsc,X_{r-1}]])[[X_{r}]] \simeq K[[X_{1},\dotsc,X_{r}]]$. – Minseon Shin Oct 22 '19 at 21:21

0 Answers0