The saturation of a monomial ideal 
Let $ d >0 $ be an integer, and let $ I \subset K[x_1,...,x_n] $ be the monomial ideal
  $$I = ( x_1^{a_1}x_2^{a_2}...x_n^{a_n} : \ \sum_{i=1}^n a_i =d;  \ a_i < d\ \forall i).$$
(a) Compute the saturation $ \widetilde{I} $.
(b) The smallest integer $ k $ such that $ I : m^k = I : m^{k+1} $ is called the saturation number of $ I $. What is the saturation number of $ I $?

The saturation $ \widetilde{I} $ of $ I $ is the ideal $ I : m^{\infty} = \bigcup_{k=1}^{\infty} I : m^k$, where $ m = (x_1,...,x_n) $.
 A: Notice first that the saturation distributes over finite intersections, and for a primary monomial ideal $Q$, $\tilde{Q}=Q$ if $\sqrt{Q}\neq m$ and $\tilde{Q}=R$ if $\sqrt{Q}=m$.  Thus given a primary decomposition $I=\bigcap_{i=1}^{r}Q_{i}$, $\tilde{I}=\bigcap\{Q_{i}\mid \sqrt{Q_{i}}\neq m\}$. On the other hand, $I$ is of Veronese type indexed by $d$ and $(d-1,d-1,\cdots, d-1)\in\mathbb{N}^{n}$, and we have a primary decomposition $I=m^{d}\cap (\bigcap_{i=1}^{n}(x_{1},\cdots, x_{i-1}, x_{i+1}, \cdots, x_{n}))$ by Observation 9, page 8 of the following link. So $\tilde{I}=\bigcap_{i=1}^{n}(x_{1},\cdots, x_{i-1}, x_{i+1}, \cdots, x_{n})$. Moreover, since $m^{a}:m=m^{a-1}$ for each $a\in\mathbb{N}$ (You can show first that $m^{a}:x_{i}=m^{a-1}$ for each $1\le i\le n$, and use Proposition 1.2.2 of Monomial Ideals by Herzog and Hibi), we have that $I:m^{a}=m^{d-a} \cap \tilde{I}$ for each $a\in\mathbb{N}$. Finally, since $m^{r}\supset \tilde{I}$ if and only if $r\le2$, the saturation number of $I$ is $d-2$.
