= Cotangent space of a formal power series ring
{title2=$I/I^2\cong(\mathfrak m/\mathfrak m^2)\oplus k$}
For a <local ring> $(A,\mathfrak m)$ with <residue field> $k$, put $B=A[[x]]$ and $I=\mathfrak mB+xB$. There is a natural $k$-<vector space> <isomorphism> $I/I^2\cong(\mathfrak m/\mathfrak m^2)\oplus k$. The inverse sends $(\bar a,\bar b)$ to the class of $a+bx$. It is well-defined because changing $a$ by $\mathfrak m^2$ or $b$ by $\mathfrak m$ changes the result by $I^2$. It is <surjective> since every $f\in I$ is congruent modulo $x^2B\subseteq I^2$ to its constant and linear terms. As $I^2=\mathfrak m^2B+x\mathfrak mB+x^2B$, those coefficients of an element of $I^2$ lie in $\mathfrak m^2$ and $\mathfrak m$, respectively. This proves injectivity. Thus adjoining one formal variable increases finite <embedding dimension> by one.
Back to article page
