The Stacks project

Lemma 27.11.6. With hypotheses and notation as in Lemma 27.11.1 above. Assume there exists a ring map $R \to A_0$ and a ring map $R \to R'$ such that $B = R' \otimes _ R A$. Then

  1. $U(\psi ) = Y$,

  2. the diagram

    \[ \xymatrix{ Y = \text{Proj}(B) \ar[r]_{r_\psi } \ar[d] & \text{Proj}(A) = X \ar[d] \\ \mathop{\mathrm{Spec}}(R') \ar[r] & \mathop{\mathrm{Spec}}(R) } \]

    is a fibre product square, and

  3. the maps $\theta : r_\psi ^*\mathcal{O}_ X(n) \to \mathcal{O}_ Y(n)$ are isomorphisms.

Proof. This follows immediately by looking at what happens over the standard opens $D_{+}(f)$ for $f \in A_{+}$. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 27.11: Functoriality of Proj

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Your email address will not be published.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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01N2. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 27.11.6 as pdf