Exercise 111.5.1. Let $S$ be a multiplicative subset of the ring $A$.
For an $A$-module $M$ show that $S^{-1}M = S^{-1}A \otimes _ A M$.
Show that $S^{-1}A$ is flat over $A$.
111.5 Flat ring maps
Exercise 111.5.2. 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:
$A = k[x, y]$ and $N = (x, y) \subset A$. (Here and below $k$ is a field.)
$A = k[x, y]$ and $N = A/(x, y)$.
Exercise 111.5.3. Give an example of a ring $A$ and a finite $A$-module $M$ which is a flat but not a projective $A$-module.
Remark 111.5.4. 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 111.5.5. Find a flat but not free module over ${\mathbf Z}_{(2)}$.
Exercise 111.5.6. Flat deformations.
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 $B/\epsilon B$.
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 $.)
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 111.5.7. Let $(A, {\mathfrak m}, k)$ be a local ring and let $k'/k$ 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 111.5.8. The same result holds for arbitrary field extensions $K/k$.
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