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Exercise 111.24.2. In each of the following cases determine whether (GU), (GD) holds, and explain why. (Use any Prop/Thm/Lemma you can find, but check the hypotheses in each case.)

  1. $k$ is a field, $A = k$, $B = k[x]$.

  2. $k$ is a field, $A = k[x]$, $B = k[x, y]$.

  3. $A = {\mathbf Z}$, $B = {\mathbf Z}[1/11]$.

  4. $k$ is an algebraically closed field, $A = k[x, y]$, $B = k[x, y, z]/(x^2-y, z^2-x)$.

  5. $A = {\mathbf Z}$, $B = {\mathbf Z}[i, 1/(2 + i)]$.

  6. $A = {\mathbf Z}$, $B = {\mathbf Z}[i, 1/(14 + 7i)]$.

  7. $k$ is an algebraically closed field, $A = k[x]$, $B = k[x, y, 1/(xy-1)]/(y^2-y)$.

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View Exercise 111.24.2 as pdf