Basic properties of complete intersection ideals
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The title is quite self-explanatory. I'm studying, for an exam, the paper "The solution to Waring's problem for monomials" (this is the link https://arxiv.org/abs/1110.0745 ).
At the bottom of page 2, Lemma 2.2, we consider $J=(y_1,y_2^{a_2},ldots, y_n^{a_n}) subset T=k[y_1,ldots,y_n]$ ideal with $2leq a_2leqldotsleq a_n$.
The authors note that $J$ is homogeneous (and I agree) and complete intersection, for which they assert $$A=T/J$$ is an artinian Gorenstein ring, with $dim A_tau neq 0$ for $tau=a_2+ldots+a_n-(n-1)$.
My question are:
1) Why we can simply note $J$ is complete intersection?
2) Where can I find a proof of the statement "$J$ homogeneous and complete intersection $implies$ A artinian and Gorenstein"?
3) I know what a Gorenstein ring is, but I can't see why $dim A_tau neq 0$ for this $textit{specific}$ $tau$.
I know there's a lot of stuff in this question, but I hope someone can help me understaning, or suggesting me some references. Thanks in advance.
algebraic-geometry commutative-algebra ideals
asked Dec 2 at 13:33
christmas_light
797
add a comment |
up vote
1
down vote
favorite
The title is quite self-explanatory. I'm studying, for an exam, the paper "The solution to Waring's problem for monomials" (this is the link https://arxiv.org/abs/1110.0745 ).
At the bottom of page 2, Lemma 2.2, we consider $J=(y_1,y_2^{a_2},ldots, y_n^{a_n}) subset T=k[y_1,ldots,y_n]$ ideal with $2leq a_2leqldotsleq a_n$.
The authors note that $J$ is homogeneous (and I agree) and complete intersection, for which they assert $$A=T/J$$ is an artinian Gorenstein ring, with $dim A_tau neq 0$ for $tau=a_2+ldots+a_n-(n-1)$.
My question are:
1) Why we can simply note $J$ is complete intersection?
2) Where can I find a proof of the statement "$J$ homogeneous and complete intersection $implies$ A artinian and Gorenstein"?
3) I know what a Gorenstein ring is, but I can't see why $dim A_tau neq 0$ for this $textit{specific}$ $tau$.
I know there's a lot of stuff in this question, but I hope someone can help me understaning, or suggesting me some references. Thanks in advance.
algebraic-geometry commutative-algebra ideals
asked Dec 2 at 13:33
christmas_light
797
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
The title is quite self-explanatory. I'm studying, for an exam, the paper "The solution to Waring's problem for monomials" (this is the link https://arxiv.org/abs/1110.0745 ).
At the bottom of page 2, Lemma 2.2, we consider $J=(y_1,y_2^{a_2},ldots, y_n^{a_n}) subset T=k[y_1,ldots,y_n]$ ideal with $2leq a_2leqldotsleq a_n$.
The authors note that $J$ is homogeneous (and I agree) and complete intersection, for which they assert $$A=T/J$$ is an artinian Gorenstein ring, with $dim A_tau neq 0$ for $tau=a_2+ldots+a_n-(n-1)$.
My question are:
1) Why we can simply note $J$ is complete intersection?
2) Where can I find a proof of the statement "$J$ homogeneous and complete intersection $implies$ A artinian and Gorenstein"?
3) I know what a Gorenstein ring is, but I can't see why $dim A_tau neq 0$ for this $textit{specific}$ $tau$.
I know there's a lot of stuff in this question, but I hope someone can help me understaning, or suggesting me some references. Thanks in advance.
algebraic-geometry commutative-algebra ideals
asked Dec 2 at 13:33
christmas_light
797
The title is quite self-explanatory. I'm studying, for an exam, the paper "The solution to Waring's problem for monomials" (this is the link https://arxiv.org/abs/1110.0745 ).
At the bottom of page 2, Lemma 2.2, we consider $J=(y_1,y_2^{a_2},ldots, y_n^{a_n}) subset T=k[y_1,ldots,y_n]$ ideal with $2leq a_2leqldotsleq a_n$.
The authors note that $J$ is homogeneous (and I agree) and complete intersection, for which they assert $$A=T/J$$ is an artinian Gorenstein ring, with $dim A_tau neq 0$ for $tau=a_2+ldots+a_n-(n-1)$.
My question are:
1) Why we can simply note $J$ is complete intersection?
2) Where can I find a proof of the statement "$J$ homogeneous and complete intersection $implies$ A artinian and Gorenstein"?
3) I know what a Gorenstein ring is, but I can't see why $dim A_tau neq 0$ for this $textit{specific}$ $tau$.
I know there's a lot of stuff in this question, but I hope someone can help me understaning, or suggesting me some references. Thanks in advance.
algebraic-geometry commutative-algebra ideals
algebraic-geometry commutative-algebra ideals
asked Dec 2 at 13:33
christmas_light
797
asked Dec 2 at 13:33
christmas_light
797
asked Dec 2 at 13:33
christmas_light
797
asked Dec 2 at 13:33
christmas_light
797
asked Dec 2 at 13:33
christmas_light
797
797
add a comment |
add a comment |
1 Answer
1
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oldest
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up vote
2
down vote
accepted
I am working under the assumption that $T$ is a polynomial ring i.e. $y_i$ are algebraically independent elements.
The ideal $J$ is a complete intersection ideal iff $ht(J) = mu(J)$. Here $mu(J)$ stands for the minimal number of generators of the ideal $J$. It is clear that $ht(J) = n$ and $mu(J) leq n$, since $J$ can evidently be generated by $n$ elements. But we know that $ht(J) leq mu(J)$. If we combine the previous observations together we have $n = ht(J) leq mu(J) leq n$. Thus $ht(J) = mu(J) = n$.
Consider the ideal $mathfrak{m} = (y_1, dots , y_n)$. Then $T_{mathfrak{m}}$ is a regular local ring and hence Gorenstein. Since $y_1, y_2^{a_2}, dots y_n^{a_n}$ form a regular sequence, we have that $T_{mathfrak{m}}/J$ is gorenstein. Now note that since $y_i$ are nilpotent in $A$, every element not in image of $mathfrak{m}$ is already invertible in $A$, thus we get that $A = T_{mathfrak{m}}/J$. Thus $A$ is Gorenstien. Here I have used Lemma 45.21.3. and Lemma 45.21.6. from the following link https://stacks.math.columbia.edu/tag/0DW6.
I claim that for $a =y_2^{a_2-1}. y_3^{a_3-1} dots y_n^{a_n-1}$, we have that the image $overline{a}$ is a non-zero element in $A$. If not then $a in J$. That will give a polynomial equation in $y_i$ but they are transcendental.
answered Dec 2 at 15:55
random123
1,127720
thank you! just to be sure, you denote by $mu(J)$ the minim number of generators of $J$, right? because I'm not sure is a universall accepted notation
– christmas_light
Dec 2 at 16:40
1
@christmas_light Yes. Thats right. I thought it is the most commonly used notation for minimum number of generators. I will add this info in the edit.
– random123
Dec 2 at 16:49
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
I am working under the assumption that $T$ is a polynomial ring i.e. $y_i$ are algebraically independent elements.
The ideal $J$ is a complete intersection ideal iff $ht(J) = mu(J)$. Here $mu(J)$ stands for the minimal number of generators of the ideal $J$. It is clear that $ht(J) = n$ and $mu(J) leq n$, since $J$ can evidently be generated by $n$ elements. But we know that $ht(J) leq mu(J)$. If we combine the previous observations together we have $n = ht(J) leq mu(J) leq n$. Thus $ht(J) = mu(J) = n$.
Consider the ideal $mathfrak{m} = (y_1, dots , y_n)$. Then $T_{mathfrak{m}}$ is a regular local ring and hence Gorenstein. Since $y_1, y_2^{a_2}, dots y_n^{a_n}$ form a regular sequence, we have that $T_{mathfrak{m}}/J$ is gorenstein. Now note that since $y_i$ are nilpotent in $A$, every element not in image of $mathfrak{m}$ is already invertible in $A$, thus we get that $A = T_{mathfrak{m}}/J$. Thus $A$ is Gorenstien. Here I have used Lemma 45.21.3. and Lemma 45.21.6. from the following link https://stacks.math.columbia.edu/tag/0DW6.
I claim that for $a =y_2^{a_2-1}. y_3^{a_3-1} dots y_n^{a_n-1}$, we have that the image $overline{a}$ is a non-zero element in $A$. If not then $a in J$. That will give a polynomial equation in $y_i$ but they are transcendental.
answered Dec 2 at 15:55
random123
1,127720
thank you! just to be sure, you denote by $mu(J)$ the minim number of generators of $J$, right? because I'm not sure is a universall accepted notation
– christmas_light
Dec 2 at 16:40
1
@christmas_light Yes. Thats right. I thought it is the most commonly used notation for minimum number of generators. I will add this info in the edit.
– random123
Dec 2 at 16:49
add a comment |
up vote
2
down vote
accepted
I am working under the assumption that $T$ is a polynomial ring i.e. $y_i$ are algebraically independent elements.
The ideal $J$ is a complete intersection ideal iff $ht(J) = mu(J)$. Here $mu(J)$ stands for the minimal number of generators of the ideal $J$. It is clear that $ht(J) = n$ and $mu(J) leq n$, since $J$ can evidently be generated by $n$ elements. But we know that $ht(J) leq mu(J)$. If we combine the previous observations together we have $n = ht(J) leq mu(J) leq n$. Thus $ht(J) = mu(J) = n$.
Consider the ideal $mathfrak{m} = (y_1, dots , y_n)$. Then $T_{mathfrak{m}}$ is a regular local ring and hence Gorenstein. Since $y_1, y_2^{a_2}, dots y_n^{a_n}$ form a regular sequence, we have that $T_{mathfrak{m}}/J$ is gorenstein. Now note that since $y_i$ are nilpotent in $A$, every element not in image of $mathfrak{m}$ is already invertible in $A$, thus we get that $A = T_{mathfrak{m}}/J$. Thus $A$ is Gorenstien. Here I have used Lemma 45.21.3. and Lemma 45.21.6. from the following link https://stacks.math.columbia.edu/tag/0DW6.
I claim that for $a =y_2^{a_2-1}. y_3^{a_3-1} dots y_n^{a_n-1}$, we have that the image $overline{a}$ is a non-zero element in $A$. If not then $a in J$. That will give a polynomial equation in $y_i$ but they are transcendental.
answered Dec 2 at 15:55
random123
1,127720
thank you! just to be sure, you denote by $mu(J)$ the minim number of generators of $J$, right? because I'm not sure is a universall accepted notation
– christmas_light
Dec 2 at 16:40
1
@christmas_light Yes. Thats right. I thought it is the most commonly used notation for minimum number of generators. I will add this info in the edit.
– random123
Dec 2 at 16:49
add a comment |
up vote
2
down vote
accepted
up vote
2
down vote
accepted
I am working under the assumption that $T$ is a polynomial ring i.e. $y_i$ are algebraically independent elements.
The ideal $J$ is a complete intersection ideal iff $ht(J) = mu(J)$. Here $mu(J)$ stands for the minimal number of generators of the ideal $J$. It is clear that $ht(J) = n$ and $mu(J) leq n$, since $J$ can evidently be generated by $n$ elements. But we know that $ht(J) leq mu(J)$. If we combine the previous observations together we have $n = ht(J) leq mu(J) leq n$. Thus $ht(J) = mu(J) = n$.
Consider the ideal $mathfrak{m} = (y_1, dots , y_n)$. Then $T_{mathfrak{m}}$ is a regular local ring and hence Gorenstein. Since $y_1, y_2^{a_2}, dots y_n^{a_n}$ form a regular sequence, we have that $T_{mathfrak{m}}/J$ is gorenstein. Now note that since $y_i$ are nilpotent in $A$, every element not in image of $mathfrak{m}$ is already invertible in $A$, thus we get that $A = T_{mathfrak{m}}/J$. Thus $A$ is Gorenstien. Here I have used Lemma 45.21.3. and Lemma 45.21.6. from the following link https://stacks.math.columbia.edu/tag/0DW6.
I claim that for $a =y_2^{a_2-1}. y_3^{a_3-1} dots y_n^{a_n-1}$, we have that the image $overline{a}$ is a non-zero element in $A$. If not then $a in J$. That will give a polynomial equation in $y_i$ but they are transcendental.
answered Dec 2 at 15:55
random123
1,127720
I am working under the assumption that $T$ is a polynomial ring i.e. $y_i$ are algebraically independent elements.
The ideal $J$ is a complete intersection ideal iff $ht(J) = mu(J)$. Here $mu(J)$ stands for the minimal number of generators of the ideal $J$. It is clear that $ht(J) = n$ and $mu(J) leq n$, since $J$ can evidently be generated by $n$ elements. But we know that $ht(J) leq mu(J)$. If we combine the previous observations together we have $n = ht(J) leq mu(J) leq n$. Thus $ht(J) = mu(J) = n$.
Consider the ideal $mathfrak{m} = (y_1, dots , y_n)$. Then $T_{mathfrak{m}}$ is a regular local ring and hence Gorenstein. Since $y_1, y_2^{a_2}, dots y_n^{a_n}$ form a regular sequence, we have that $T_{mathfrak{m}}/J$ is gorenstein. Now note that since $y_i$ are nilpotent in $A$, every element not in image of $mathfrak{m}$ is already invertible in $A$, thus we get that $A = T_{mathfrak{m}}/J$. Thus $A$ is Gorenstien. Here I have used Lemma 45.21.3. and Lemma 45.21.6. from the following link https://stacks.math.columbia.edu/tag/0DW6.
I claim that for $a =y_2^{a_2-1}. y_3^{a_3-1} dots y_n^{a_n-1}$, we have that the image $overline{a}$ is a non-zero element in $A$. If not then $a in J$. That will give a polynomial equation in $y_i$ but they are transcendental.
answered Dec 2 at 15:55
random123
1,127720
edited Dec 2 at 16:51
answered Dec 2 at 15:55
random123
1,127720
answered Dec 2 at 15:55
random123
1,127720
answered Dec 2 at 15:55
random123
1,127720
1,127720
thank you! just to be sure, you denote by $mu(J)$ the minim number of generators of $J$, right? because I'm not sure is a universall accepted notation
– christmas_light
Dec 2 at 16:40
1
@christmas_light Yes. Thats right. I thought it is the most commonly used notation for minimum number of generators. I will add this info in the edit.
– random123
Dec 2 at 16:49
add a comment |
thank you! just to be sure, you denote by $mu(J)$ the minim number of generators of $J$, right? because I'm not sure is a universall accepted notation
– christmas_light
Dec 2 at 16:40
1
@christmas_light Yes. Thats right. I thought it is the most commonly used notation for minimum number of generators. I will add this info in the edit.
– random123
Dec 2 at 16:49
thank you! just to be sure, you denote by $mu(J)$ the minim number of generators of $J$, right? because I'm not sure is a universall accepted notation
– christmas_light
Dec 2 at 16:40
thank you! just to be sure, you denote by $mu(J)$ the minim number of generators of $J$, right? because I'm not sure is a universall accepted notation
– christmas_light
Dec 2 at 16:40
1
1
@christmas_light Yes. Thats right. I thought it is the most commonly used notation for minimum number of generators. I will add this info in the edit.
– random123
Dec 2 at 16:49
@christmas_light Yes. Thats right. I thought it is the most commonly used notation for minimum number of generators. I will add this info in the edit.
– random123
Dec 2 at 16:49
add a comment |
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