Basic properties of complete intersection ideals












1












$begingroup$


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







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$endgroup$

















    1












    $begingroup$


    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







    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      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







      share|cite|improve this question









      $endgroup$




      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






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      asked Dec 2 '18 at 13:33









      christmas_lightchristmas_light

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          $begingroup$

          I am working under the assumption that $T$ is a polynomial ring i.e. $y_i$ are algebraically independent elements.




          1. 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 $ht(J) = mu(J) = n$.


          2. 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.


          3. 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.







          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            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
            $endgroup$
            – christmas_light
            Dec 2 '18 at 16:40






          • 1




            $begingroup$
            @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.
            $endgroup$
            – random123
            Dec 2 '18 at 16:49













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          active

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          2












          $begingroup$

          I am working under the assumption that $T$ is a polynomial ring i.e. $y_i$ are algebraically independent elements.




          1. 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 $ht(J) = mu(J) = n$.


          2. 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.


          3. 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.







          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            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
            $endgroup$
            – christmas_light
            Dec 2 '18 at 16:40






          • 1




            $begingroup$
            @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.
            $endgroup$
            – random123
            Dec 2 '18 at 16:49


















          2












          $begingroup$

          I am working under the assumption that $T$ is a polynomial ring i.e. $y_i$ are algebraically independent elements.




          1. 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 $ht(J) = mu(J) = n$.


          2. 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.


          3. 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.







          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            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
            $endgroup$
            – christmas_light
            Dec 2 '18 at 16:40






          • 1




            $begingroup$
            @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.
            $endgroup$
            – random123
            Dec 2 '18 at 16:49
















          2












          2








          2





          $begingroup$

          I am working under the assumption that $T$ is a polynomial ring i.e. $y_i$ are algebraically independent elements.




          1. 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 $ht(J) = mu(J) = n$.


          2. 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.


          3. 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.







          share|cite|improve this answer











          $endgroup$



          I am working under the assumption that $T$ is a polynomial ring i.e. $y_i$ are algebraically independent elements.




          1. 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 $ht(J) = mu(J) = n$.


          2. 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.


          3. 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.








          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Dec 2 '18 at 16:51

























          answered Dec 2 '18 at 15:55









          random123random123

          1,2601720




          1,2601720












          • $begingroup$
            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
            $endgroup$
            – christmas_light
            Dec 2 '18 at 16:40






          • 1




            $begingroup$
            @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.
            $endgroup$
            – random123
            Dec 2 '18 at 16:49




















          • $begingroup$
            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
            $endgroup$
            – christmas_light
            Dec 2 '18 at 16:40






          • 1




            $begingroup$
            @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.
            $endgroup$
            – random123
            Dec 2 '18 at 16:49


















          $begingroup$
          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
          $endgroup$
          – christmas_light
          Dec 2 '18 at 16:40




          $begingroup$
          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
          $endgroup$
          – christmas_light
          Dec 2 '18 at 16:40




          1




          1




          $begingroup$
          @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.
          $endgroup$
          – random123
          Dec 2 '18 at 16:49






          $begingroup$
          @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.
          $endgroup$
          – random123
          Dec 2 '18 at 16:49




















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