Loading [MathJax]/extensions/tex2jax.js

The Stacks project

Lemma 37.76.7. Let $A \to B$ be a ring map which is finite and of finite presentation. There exists a finitely presented ring map $A \to A_{univ}$ and an idempotent $e_{univ} \in B \otimes _ A A_{univ}$ such that for any ring map $A \to A'$ and idempotent $e \in B \otimes _ A A'$ there is a ring map $A_{univ} \to A'$ mapping $e_{univ}$ to $e$.

Proof. Choose $b_1, \ldots , b_ n \in B$ generating $B$ as an $A$-module. For each $i$ choose a monic $P_ i \in A[x]$ such that $P_ i(b_ i) = 0$ in $B$, see Algebra, Lemma 10.36.3. Thus $B$ is a quotient of the finite free $A$-algebra $B' = A[x_1, \ldots , x_ n]/(P_1(x_1), \ldots , P_ n(x_ n))$. Let $J \subset B'$ be the kernel of the surjection $B' \to B$. Then $J =(f_1, \ldots , f_ m)$ is finitely generated as $B$ is a finitely generated $A$-algebra, see Algebra, Lemma 10.6.2. Choose an $A$-basis $b'_1, \ldots , b'_ N$ of $B'$. Consider the algebra

\[ A_{univ} = A[z_1, \ldots , z_ N, y_1, \ldots , y_ m]/I \]

where $I$ is the ideal generated by the coefficients in $A[z_1, \ldots , z_ n, y_1, \ldots , y_ m]$ of the basis elements $b'_1, \ldots , b'_ N$ of the expression

\[ (\sum z_ j b'_ j)^2 - \sum z_ j b'_ j + \sum y_ k f_ k \]

in $B'[z_1, \ldots , z_ N, y_1, \ldots , y_ m]$. By construction the element $\sum z_ j b'_ j$ maps to an idempotent $e_{univ}$ in the algebra $B \otimes _ A A_{univ}$. Moreover, if $e \in B \otimes _ A A'$ is an idempotent, then we can lift $e$ to an element of the form $\sum b'_ j \otimes a'_ j$ in $B' \otimes _ A A'$ and we can find $a''_ k \in A'$ such that

\[ (\sum b'_ j \otimes a'_ j)^2 - \sum b'_ j \otimes a'_ j + \sum f_ k \otimes a''_ k \]

is zero in $B' \otimes _ A A'$. Hence we get an $A$-algebra map $A_{univ} \to A$ sending $z_ j$ to $a'_ j$ and $y_ k$ to $a''_ k$ mapping $e_{univ}$ to $e$. This finishes the proof. $\square$


Comments (0)


Your email address will not be published.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.