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.)

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