The Stacks project

109.5 Flat ring maps

Exercise 109.5.1. Let $S$ be a multiplicative subset of the ring $A$.

  1. For an $A$-module $M$ show that $S^{-1}M = S^{-1}A \otimes _ A M$.

  2. Show that $S^{-1}A$ is flat over $A$.

Exercise 109.5.2. Find an injection $M_1 \to M_2$ of $A$-modules such that $M_1\otimes N \to M_2 \otimes N$ is not injective in the following cases:

  1. $A = k[x, y]$ and $N = (x, y) \subset A$. (Here and below $k$ is a field.)

  2. $A = k[x, y]$ and $N = A/(x, y)$.

Exercise 109.5.3. Give an example of a ring $A$ and a finite $A$-module $M$ which is a flat but not a projective $A$-module.

Remark 109.5.4. If $M$ is of finite presentation and flat over $A$, then $M$ is projective over $A$. Thus your example will have to involve a ring $A$ which is not Noetherian. I know of an example where $A$ is the ring of ${\mathcal C}^\infty $-functions on ${\mathbf R}$.

Exercise 109.5.6. Flat deformations.

  1. Suppose that $k$ is a field and $k[\epsilon ]$ is the ring of dual numbers $k[\epsilon ] = k[x]/(x^2)$ and $\epsilon = \bar x$. Show that for any $k$-algebra $A$ there is a flat $k[\epsilon ]$-algebra $B$ such that $A$ is isomorphic to $B/\epsilon B$.

  2. Suppose that $k = {\mathbf F}_ p = {\mathbf Z}/p{\mathbf Z}$ and

    \[ A = k[x_1, x_2, x_3, x_4, x_5, x_6]/ (x_1^ p, x_2^ p, x_3^ p, x_4^ p, x_5^ p, x_6^ p). \]

    Show that there exists a flat ${\mathbf Z}/p^2{\mathbf Z}$-algebra $B$ such that $B/pB$ is isomorphic to $A$. (So here $p$ plays the role of $\epsilon $.)

  3. Now let $p = 2$ and consider the same question for $k = {\mathbf F}_2 = {\mathbf Z}/2{\mathbf Z}$ and

    \[ A = k[x_1, x_2, x_3, x_4, x_5, x_6]/ (x_1^2, x_2^2, x_3^2, x_4^2, x_5^2, x_6^2, x_1x_2 + x_3x_4 + x_5x_6). \]

    However, in this case show that there does not exist a flat ${\mathbf Z}/4{\mathbf Z}$-algebra $B$ such that $B/2B$ is isomorphic to $A$. (Find the trick! The same example works in arbitrary characteristic $p > 0$, except that the computation is more difficult.)

Exercise 109.5.7. Let $(A, {\mathfrak m}, k)$ be a local ring and let $k \subset k'$ be a finite field extension. Show there exists a flat, local map of local rings $A \to B$ such that ${\mathfrak m}_ B = {\mathfrak m} B$ and $B/{\mathfrak m} B$ is isomorphic to $k'$ as $k$-algebra. (Hint: first do the case where $k \subset k'$ is generated by a single element.)

Remark 109.5.8. The same result holds for arbitrary field extensions $k \subset K$.

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 0279. Beware of the difference between the letter 'O' and the digit '0'.