Exercise 4, § 1

Let $A$ be a commutative algebra over $\mathbf{C}$, $I$ an ideal of A$,h$ the canonical injection of $I$ into $A$, and $S$ the set of $\chi \in \mathsf{X}'(A)$ which vanish on $I$. a) $\mathsf{X}'(h)$ is surjective.

b) $\mathsf{X}'(h)|(\mathsf{X}'(A)$ - S) is a homeomorphism of $\mathsf{X}'(A)$ - S onto $\mathsf{X}(I)$. (Let $\chi_0\in \mathsf{X}'(A)$ - S. Let $x_1, . . . , x_n\in A,\varepsilon >0$, and let V be the neighbourhood of $\chi_0$ in $\mathsf{X}'(A)$ defined by $|(\chi -\chi_0)(x_i)|\leqslant \varepsilon ,(i= 1, . . . , n)$. Let $u_0\in I$ such that $\chi_0(u_0) = 1$. We have $u_i=u_0x_i\in I$. Then if $|(\chi -\chi_0)(u_i)|\leqslant \delta (i= 0,1, . . . , n)$ with $\delta$ sufficiently small, we have $\chi \in V$.)