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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
$I^2=(x^4, x^2y^6, y^4z^6, x^3y^3, x^3y^5z^3xy^5z^3, x^2y^2z^3)$
Is $\lbrace x^4, x^2y^6, y^4z^6, x^3y^3, x^3y^5z^3xy^5z^3, x^2y^2z^3\rbrace$ 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!
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6
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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 $I=(x^2, x^3yxy^3, y^2z^3)$, then
$I^2=(x^4, x^6y^2x^2y^6, y^4z^6, x^5yx^3y^3, x^3y^3z^3x^3y^5z^3, x^2y^2z^3)$
Is $\lbrace x^4, x^6y^2x^2y^6, y^4z^6, x^5yx^3y^3, x^3y^3z^3x^3y^5z^3, x^2y^2z^3\rbrace$ a minimal generating set for $I^2$ ?
Update As Mariano Suarez Alvarez pointed out, we do There are some typing mistake that I have not need $x^6y^2$ since we noticed. I have $x^4$. But what about $x^5y$, since $x^4\times xy=x^5y$change my question. This time, and for $x^3y^3z^3$, the generating set of $x^2y^2z^3$. I$ is minimal. So , what can I talk we say about the minimal generating set for $I^2$ now above ? Is it minimal?
Thank you everyone for helping me answer my question!
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5
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Let $I$ be a graded ideal in a graded commutative 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 $I=(x^2, x^3y, y^2z^3)$, then
$I^2=(x^4, x^6y^2, y^4z^6, x^5y, x^3y^3z^3, x^2y^2z^3)$
Is $\lbrace x^4, x^6y^2, y^4z^6, x^5y, x^3y^3z^3, x^2y^2z^3\rbrace$ a minimal generating set for $I^2$ ?
Update As Mariano Suarez Alvarez pointed out, we do not need $x^6y^2$ since we have $x^4$. But what about $x^5y$, since $x^4\times xy=x^5y$, and for $x^3y^3z^3$, $x^2y^2z^3$. So, what can I talk about the minimal generating set for $I^2$ now ?
Thank you everyone for helping me answer my question!
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4
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Let $I$ be a graded ideal in a graded commutative 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 $I=(x^2, x^3y, y^2z^3)$, then
$I^2=(x^4, x^6y^2, y^4z^6, x^5y, x^3y^3z^3, x^2y^2z^3)$
Is $\lbrace x^4, x^6y^2, y^4z^6, x^5y, x^3y^3z^3, x^2y^2z^3\rbrace$ a minimal generating set for $I^2$ ?
Update As Mariano Suarez Alvarez pointed out, we do not need $x^6y^2$ since we have $x^4$. But what about $x^5y$, since $x^4\times xy=x^5y$, and for $x^3y^3z^3$, $x^2y^2z^3$. So, what can I talk about the minimal generating set for $I^2$ now ?
Thank you everyone for helping me answer my question!
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3
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Let $I$ be a graded ideal in a graded commutative 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 $I=(x^2, x^3y, y^2z^3)$, then
$I^2=(x^4, x^6y^2, y^4z^6, x^5y, x^3y^3z^3, x^2y^2z^3)$
Is $\lbrace x^4, x^6y^2, y^4z^6, x^5y, x^3y^3z^3, x^2y^2z^3\rbrace$ a minimal generating set for $I^2$ ?
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2
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Let $I$ be a graded ideal in a graded commutative ring $R$, which is generated minimally by $x_1,...,x_k$. Then the power of $I$ : 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 $S$ ?
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1
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On the generator of power of ideal
Let $I$ be a graded ideal in a graded commutative ring $R$, which is generated minimally by $x_1,...,x_k$. Then the power of $I$ : $I^t$ 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 $S$ ?
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