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Martin Sleziak
Martin Sleziak
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Generators of a maximal ideal of k[X1$k[X_1,...\cdots,Xn]X_n]$

Hi. GivenGiven $k$ an algebraically closed field, I know that that a maximal ideal $\mathfrak{m}$ of $A = k[X_1,\cdots,X_n]$ is just a $\langle X_1-a_1,\cdots,X_n-a_n \rangle $ (Nullstellensatz). Knowing that, it seems intuitive that $\mathfrak{m}$ can not be generated by less than $n$ elements. Is that true  ? In that case, how can I show that ?

(Actually, I need that, just after using Nakayama's lemma, to show that $dim_{A/\mathfrak{m}=k} \mathfrak{m}/\mathfrak{m}^2$$\dim_{A/\mathfrak{m}=k} \mathfrak{m}/\mathfrak{m}^2$ is greater than $n$.)

(I've seen things that might be relevant like "Krull height theorem" but I think such things take place in a more general context and I have some difficulties, first to understand them, and second to adapt them...)

Thank you.

Generators of a maximal ideal of k[X1,...,Xn]

Hi. Given $k$ an algebraically closed field, I know that that a maximal ideal $\mathfrak{m}$ of $A = k[X_1,\cdots,X_n]$ is just a $\langle X_1-a_1,\cdots,X_n-a_n \rangle $ (Nullstellensatz). Knowing that, it seems intuitive that $\mathfrak{m}$ can not be generated by less than $n$ elements. Is that true  ? In that case, how can I show that ?

(Actually, I need that, just after using Nakayama's lemma, to show that $dim_{A/\mathfrak{m}=k} \mathfrak{m}/\mathfrak{m}^2$ is greater than $n$.)

(I've seen things that might be relevant like "Krull height theorem" but I think such things take place in a more general context and I have some difficulties, first to understand them, and second to adapt them...)

Thank you.

Generators of a maximal ideal of $k[X_1,\cdots,X_n]$

Given $k$ an algebraically closed field, I know that that a maximal ideal $\mathfrak{m}$ of $A = k[X_1,\cdots,X_n]$ is just a $\langle X_1-a_1,\cdots,X_n-a_n \rangle $ (Nullstellensatz). Knowing that, it seems intuitive that $\mathfrak{m}$ can not be generated by less than $n$ elements. Is that true? In that case, how can I show that ?

(Actually, I need that, just after using Nakayama's lemma, to show that $\dim_{A/\mathfrak{m}=k} \mathfrak{m}/\mathfrak{m}^2$ is greater than $n$.)

(I've seen things that might be relevant like "Krull height theorem" but I think such things take place in a more general context and I have some difficulties, first to understand them, and second to adapt them...)

Thank you.

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Mariano Suárez-Álvarez
Mariano Suárez-Álvarez
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Genrators Generators of a maximal ideal of k[X1,...,Xn]

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Laurent
Laurent
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Genrators of a maximal ideal of k[X1,...,Xn]

Hi. Given $k$ an algebraically closed field, I know that that a maximal ideal $\mathfrak{m}$ of $A = k[X_1,\cdots,X_n]$ is just a $\langle X_1-a_1,\cdots,X_n-a_n \rangle $ (Nullstellensatz). Knowing that, it seems intuitive that $\mathfrak{m}$ can not be generated by less than $n$ elements. Is that true ? In that case, how can I show that ?

(Actually, I need that, just after using Nakayama's lemma, to show that $dim_{A/\mathfrak{m}=k} \mathfrak{m}/\mathfrak{m}^2$ is greater than $n$.)

(I've seen things that might be relevant like "Krull height theorem" but I think such things take place in a more general context and I have some difficulties, first to understand them, and second to adapt them...)

Thank you.