Lemma 93.8.8. In Example 93.8.1 let $P$ be a $k$-algebra. Let $J \subset P$ be an ideal. Denote $(P^ h, J^ h)$ the henselization of the pair $(P, J)$. There is a natural functor
of deformation categories.
Proof. Given a deformation of $P$ we can take the henselization of it to get a deformation of the henselization; this is clear and we encourage the reader to skip the proof. More precisely, let $(A, Q) \to (k, P)$ be a morphism in $\mathcal{F}$, i.e., an object of $\mathcal{D}\! \mathit{ef}_ P$. Denote $J_ Q \subset Q$ the inverse image of $J$ in $Q$. Let $(Q^ h, J_ Q^ h)$ be the henselization of the pair $(Q, J_ Q)$. Recall that $Q \to Q^ h$ is flat (More on Algebra, Lemma 15.12.2) and hence $Q^ h$ is flat over $A$. By More on Algebra, Lemma 15.12.7 we see that the map $Q^ h \to P^ h$ induces an isomorphism $Q^ h \otimes _ A k = Q^ h \otimes _ Q P = P^ h$. Hence $(A, Q^ h) \to (k, P^ h)$ is the desired object of $\mathcal{D}\! \mathit{ef}_{P^ h}$. $\square$
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)