Exercise 29, § 5

Let $k$ be a field, $A$ a complete noetherian local $k$-algebra, $\kappa_A$ the residue field of $A$. a) Suppose that $\kappa_A$ is a separable extension of $k$. Show that there exists a homomorphism of $k$-algebras from $\kappa_A$ into $A$, section of the canonical homomorphism from $A$ to $\kappa_A$ (IX, § 3, no 2, prop. 1). b) Suppose that $\kappa_A$ is a purely inseparable extension of $k$. Show that there does not necessarily exist a section of $k$-algebras from $\kappa_A$ into $A$. (Take a field $k$ of characteristic 2, an element $a$ of $k$ which is not a square (for example $k = F_2(T), a = T$), the ideal $p$ of $k[X]$ generated by $X^2 + a$, and put $A = \widehat{k[X]_p}$) c) Suppose that $A$ is regular and $\kappa_A$ a separable extension of $k$. Show that $A$ is isomorphic as a $k$-algebra to $\kappa_A[[T_1, ..., T_n]]$ where $n = \dim(A)$. d) Show that this is not necessarily the case when $\kappa_A$ is not a separable extension of $k$.