Remark 3.9.3. The lemma above can also be proved using the reflection principle. However, one has to be careful. Namely, suppose the sentence $\phi _{scheme}(X)$ expresses the property “$X$ is a scheme”, then what does the formula $\phi _{scheme}^{V_\alpha }(X)$ mean? It is true that the reflection principle says we can find $\alpha $ such that for all $X \in V_\alpha $ we have $\phi _{scheme}(X) \leftrightarrow \phi _{scheme}^{V_\alpha }(X)$ but this is entirely useless. It is only by combining two such statements that something interesting happens. For example suppose $\phi _{red}(X, Y)$ expresses the property “$X$, $Y$ are schemes, and $Y$ is the reduction of $X$” (see Schemes, Definition 26.12.5). Suppose we apply the reflection principle to the pair of formulas $\phi _1(X, Y) = \phi _{red}(X, Y)$, $\phi _2(X) = \exists Y, \phi _1(X, Y)$. Then it is easy to see that any $\alpha $ produced by the reflection principle has the property that given $X \in \mathop{\mathrm{Ob}}\nolimits (\mathit{Sch}_\alpha )$ the reduction of $X$ is also an object of $\mathit{Sch}_\alpha $ (left as an exercise).

## Post a comment

Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)

There are also: