I am trying to prove that the primary decomposition of $(X^aY^b,(X+Y+Z)^c)$ in $k[X,Y,Z]$, for a,b,c positive integers, is $(X^a,(X+Y+Z)^c) \cap (Y^b,(X+Y+Z)^c)$. The equality of the ideal and the intersection is easy to see, but I am not succeeding in proving that the ideals $(X^a,(X+Y+Z)^c) $ and $ (Y^b,(X+Y+Z)^c)$ are primary, so I am not sure that I am on the right track. So the questions would be - is this a correct decomposition? How to see it?
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Hint. If $\phi:K[X,Y,Z]\to K[X,Y,Z]$ is given by $\phi(X)=X$, $\phi(Y)=Y$, and $\phi(Z)=Z-X-Y$, then $\phi$ is a $K$-automorphism, and the image by $\phi$ of your ideals are monomial.
user26857
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I just tried writing it down, but I can't see what you mean. If you could elaborate... – user198182 Jan 29 '15 at 23:50
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1$\phi((X^aY^b,(X+Y+Z)^c))=(X^aY^b,Z^c)$, and this is a monomial ideal whose primary decomposition is $(X^a,Z^c)\cap(Y^b,Z^c)$. Now go back with $\phi^{-1}$. – user26857 Jan 29 '15 at 23:57
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That is a great little trick! So much easier than trying to prove by hand. – user198182 Jan 30 '15 at 00:03
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I always wonder in these situations if I would have ever thought of that :) – user198182 Jan 30 '15 at 00:04