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Let $R= k[x,y,z]/(xy,yz,zx)$. Let $I=(x)$. What are the irreducible components of $\mathrm{Spec}(R/I^n)$ where $n \geq 2$ and $k$ is a field?

For solving this problem I'm trying to use following exercise from Atiyah and Macdonald book:

Let $A$ be a commutative ring with unit, $X = \mathrm{Spec}(A) $ with the Zariski topology. Then the irreducible components of $X$ are $\lbrace V(p) : p\subset A \ \text{minimal prime ideal} \rbrace$ where $V(p) =\lbrace q \ \text{prime ideal } \mid p\subset q\rbrace$.

So, we need to find minimal prime ideals $p$ of $R/I^n$. Suppose $p$ is a minimal prime ideal of $R/I^n$, that means $p$ is a prime ideal of $k[x,y,z]$ that contains $(xy,yz,zx)$ and $(x)^n$, which is equivalent to say that $p$ contains $(x,y)$ or $(x,z)$. Also as $(x,y)$ and $(x,z)$ are prime ideals in $k[x,y,z]$ hence the irreducible components are $V(x,z)$ and $V(x,y)$. Is this solution correct?

user26857
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Arpit Kansal
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    Yes, your solution is absolutely correct: bravo! Sufficiently advanced readers will note with interest that the schemes $Spec (R/I^n)$ are all different for variable $n$ although they all have the same two irreducible components. There is nothing worrying about that: a scheme contains more information than just its underlying topological space ! – Georges Elencwajg Sep 14 '15 at 21:23

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