On the generator of power of ideal

2012-10-09T01:07:29Z

Let $I$ be a graded ideal in a polynomial ring $R$, which is generated minimally by $x_1,...,x_k$. Then the power of $I$, i.e $I^t$ is generated by monomials of the form $x_{1}^{a_1}...x_{n}^{a_{n}}$ where $a_1+...+a_n=t$. Denote this set by $S$.

Can we say anything (others than above)about the minimal generating set of $I^{t}$? Is it $S$ ?

Given a minimal generating set for a graded ideal in a graded commutative ring, from these how much do we know about the minimal generating set for the power of it?

Edit : Here is an example for precising my question :

In the polynomial ring $k[x,y,z]$ let $I=(x^2, xy^3, y^2z^3)$, then $I^2=(x^4, x^2y^6, y^4z^6, x^3y^3, xy^5z^3, x^2y^2z^3)$

Is $\lbrace x^4, x^2y^6, y^4z^6, x^3y^3, xy^5z^3, x^2y^2z^3\rbrace$ a minimal generating set for $I^2$ ?

Update There are some typing mistake that I have not noticed. I have change my question. This time, the generating set of $I$ is minimal. So what can we say about the generating set for $I^2$ above ? Is it minimal? Thank you everyone for helping me answer my question!

Answer by Hailong Dao for On the generator of power of ideal

2012-10-12T05:31:03Z

I have to admire your persistence, perhaps you really want an answer (-:

In general, the answer to your first question (second paragraph) is NO, it is not $S$, even for monomial ideals in a polynomial rings. Take the ideal $I$ generated by $x_1 = a^4b, x_2=b^4a, x_3=a^3b^3$. Then $x_3^2$ is not a minimal generator for $I^2$ since it is divisible by $x_1x_2$.

One particular case when the answer is YES is when the $x_1,\cdots, x_n$ form a regular sequence.

For specific examples, it may be worth learning some program such as Macaulay 2. In your specific example $S$ is the minimal generating set for $I^2$, provided that you fix the fifth entry as Gerhard pointed out.

For a monomials ideal one can also visualize the minimal generators as the points on the convex hull of the set of degrees of the ideal.

As my first example shows, one can not make very good statement about a specific power of $I$. However, asymptotically we can say quite a bit:

The minimal number of generators of $I^n$ for $n>>0$ is a polynomial in $n$. The degree of this polynomial is called the analytic spread of $I$ (geometrically it is one more than the dimension of the exceptional fibre of the blow-up).

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On the generator of power of ideal

Answer by Hailong Dao for On the generator of power of ideal