Corollary 2

Let A and B be noetherian local rings and $\sigma$ a local homomorphism of B into A. Suppose that A is regular and B complete. In order that $\sigma$ be bijective, it is necessary and sufficient that it induce bijections of $\kappa_B$ onto $\kappa_A$ and of $m_B/m_B^2$ onto $m_A/m_A^2$.

The stated condition is obviously necessary.

Suppose conversely that $\sigma$ induces isomorphisms of $\mathrm{gr}_0(B)$ onto $\mathrm{gr}_0(A)$ and of $\mathrm{gr}_1(B)$ onto $\mathrm{gr}_1(A)$. Since the ring $\mathrm{gr}(B)$ is generated by $\mathrm{gr}_0(B) \cup \mathrm{gr}_1(B)$ and $\mathrm{gr}(A)$ is the symmetric algebra of the vector space $\mathrm{gr}_1(A)$ over the field $\mathrm{gr}_0(A)$, the homomorphism $\mathrm{gr}(\sigma)$ is bijective. Hence $\sigma$ is bijective (III, § 2, No. 8, Corollary 3 to Theorem 1).