The Stacks project

Lemma 27.22.12. Let $X$ be a scheme. Assume $X$ is quasi-compact and quasi-separated. Let $\mathcal{A}$ be a finite quasi-coherent $\mathcal{O}_ X$-algebra. Then $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits \mathcal{A}_ i$ is a directed colimit of finite and finitely presented quasi-coherent $\mathcal{O}_ X$-algebras such that all transition maps $\mathcal{A}_{i'} \to \mathcal{A}_ i$ are surjective.

Proof. By Lemma 27.22.8 there exists a finitely presented $\mathcal{O}_ X$-module $\mathcal{F}$ and a surjection $\mathcal{F} \to \mathcal{A}$. Using the algebra structure we obtain a surjection

\[ \text{Sym}^*_{\mathcal{O}_ X}(\mathcal{F}) \longrightarrow \mathcal{A} \]

Denote $\mathcal{J}$ the kernel. Write $\mathcal{J} = \mathop{\mathrm{colim}}\nolimits \mathcal{E}_ i$ as a filtered colimit of finite type $\mathcal{O}_ X$-submodules $\mathcal{E}_ i$ (Lemma 27.22.7). Set

\[ \mathcal{A}_ i = \text{Sym}^*_{\mathcal{O}_ X}(\mathcal{F})/(\mathcal{E}_ i) \]

where $(\mathcal{E}_ i)$ indicates the ideal sheaf generated by the image of $\mathcal{E}_ i \to \text{Sym}^*_{\mathcal{O}_ X}(\mathcal{F})$. Then each $\mathcal{A}_ i$ is a finitely presented $\mathcal{O}_ X$-algebra, the transition maps are surjections, and $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits \mathcal{A}_ i$. To finish the proof we still have to show that $\mathcal{A}_ i$ is a finite $\mathcal{O}_ X$-algebra for $i$ sufficiently large. To do this we choose an affine open covering $X = U_1 \cup \ldots \cup U_ m$. Take generators $f_{j, 1}, \ldots , f_{j, N_ j} \in \Gamma (U_ i, \mathcal{F})$. As $\mathcal{A}(U_ j)$ is a finite $\mathcal{O}_ X(U_ j)$-algebra we see that for each $k$ there exists a monic polynomial $P_{j, k} \in \mathcal{O}(U_ j)[T]$ such that $P_{j, k}(f_{j, k})$ is zero in $\mathcal{A}(U_ j)$. Since $\mathcal{A} = \mathop{\mathrm{colim}}\nolimits \mathcal{A}_ i$ by construction, we have $P_{j, k}(f_{j, k}) = 0$ in $\mathcal{A}_ i(U_ j)$ for all sufficiently large $i$. For such $i$ the algebras $\mathcal{A}_ i$ are finite. $\square$


Comments (0)


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Your email address will not be published.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 086N. Beware of the difference between the letter 'O' and the digit '0'.