I am confused about a computation in exercise 24.4.L in Vakil's Foundations of Algebraic Geometry, where we are asked to explicitly compute a flat limit.
Let $X = \mathbb{A}^3 \times \mathbb{A}^1 \to Z = \mathbb{A}^1$ over a field $k$, where the coordinates on $\mathbb{A}^3$ are $x, y, z$, and the coordinates on $\mathbb{A}^1$ are $t$. Define $Y$ away from $t = 0$ as the union of the two lines $y = z = 0$ and $x = z – t = 0$. Find the flat limit at $t = 0$.
I think I am expected to get the union of two lines, plus some "nonreducedness" at the origin which points normal to the $xz$-plane due to the "motion" of the line $x = z – t = 0$ as we move $t \to 0$. However, my computation gives me the reduced union instead, and I'm not sure what I'm doing wrong.
Let $A = k[x, y, z, t]$ and $B = k[x, y, z, t, 1/t]/((y, z) \cap (x, z-t))$, so $B$ is the coordinate ring of $Y$. We first compute the scheme-theoretic closure of $Y$ in $X$, i.e. the kernel of the homomorphism $A \to B$. This is just the pullback of the ideal $(y, z) \cap (x, z-t) \subseteq A_t$ to $A$. Since $(y, z)$ and $(x, z-t)$ are both prime in $A_t$ and the primes of $A_t$ correspond bijectively to prime ideals in $A$ not containing $t$ via extension and contraction, the pullback of $(y, z) \cap (x, z – t)$ in $A$ is just $(y, z) \cap (x, z – t)$. The fiber is then computed by appending the equation $t = 0$, so the coordinate ring of $Y'\vert_0$ is $k[x, y, z, t]/((y, z) \cap (x, z – t), t) \cong k[x, y, z]/((y, z) \cap (x, z)) \cong k[x, y]/(xy)$, the reduced union of the two lines.
Surely the problem here arises from an erroneous computation of the pullback of $(y, z) \cap (x, z- t)$ to $A$, but I don't feel like my reasoning was flawed. What exactly is going wrong here?
Best Answer
$(y,z)\cap(x,z-t)=(xy,xz,y(t-z),z(z-t))=(xy,xz,yz-yt,z^2-tz)$, so $((y,z)\cap(x,z-t),t)=(xy,xz,yz,z^2,t)$, which is different from your conclusion of $(xy,z,t)$. The issue is that the ideal $(I\cap J)+K$ is not necessarily the same as $(I+K)\cap (J+K)$, which is a step that you used in your calculation with $I=(y,z)$, $J=(x,z-t)$, and $K=(t)$.