# Do homogeneous ideals have homogeneous primary decomposition?

In proving closed algebraic set in $\mathbb P_n(k)$ has decomposition into irreducible algebraic sets, we can argue by decomposition in noetherian space.

Can we prove directly by decomposition of homogeneous ideal into homogeneous prime ideals?

I don't know if this holds. I tried to use the correspondence between ideals in $k[X_1,...,X_n]$ and homogeneous ideals in $k[X_0,...,X_n]$ by $I \rightarrow {}^hI$ and ${}^aJ \leftarrow J$. Suppose we have $J$ a homogeneous ideal in $k[X_0,...,X_n]$, then ${}^aJ = \mathfrak{q}_1\cap\cdots\cap\mathfrak{q}_k$ and we have ${}^h({}^aJ) = {}^h\mathfrak{q}_1\cap\cdots\cap{}^h\mathfrak{q}_k$ a primary decomposition. But ${}^h({}^aJ)=J$ may not hold. Take $J=(X_0,...,X_n)$ for example: ${}^h({}^aJ)=k[X_0,..,X_n]$

I saw tags here: "minimal primes of a homogeneous ideal are homogeneous" but the problem was not fully answered and the cited link is missing.

Can anyone answer that tag or help me with this question?

• May I ask another question? Is there any way to tell when do ${}^h{}^aJ$ and $J$ coincide? Thanks! Oct 14 '13 at 13:50