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Jun
5
comment Why is it Artinian?
Thank you for your help. I understand that for each product $x_1 x_2 \cdots x_n$ where $x_i \in m_i$ there exists $i$ such that $x_i \in p$. But how do we know that $p \supset m_i$ for some $i$.
Jun
5
revised Why is it Artinian?
added 2 characters in body
Jun
5
asked Why is it Artinian?
May
6
comment Proof of a corollary of the Noether normalisation lemma
@Cantlog : There is an addendum between the lemma and the corollary saying "under the condition of the lemma, if furthermore $k$ is algebraiccaly closed, and $A$ is an integral domain with field of fractions $K$ then ...". So, they may be the additional conditions.
May
5
asked Proof of a corollary of the Noether normalisation lemma
Apr
11
accepted If $X \times \{0\} \cup A \times I$ is closed in $X \times I$. Then, is $A$ closed in $X$?
Apr
11
comment If $X \times \{0\} \cup A \times I$ is closed in $X \times I$. Then, is $A$ closed in $X$?
Am I right in thinking that $B\times (0,1]$ is the complement of $X \times \{0\} \cup A \times I$ in $X \times I$. If it is true, then it follows that $B\times (0,1]$ is open.
Apr
11
comment Is the length of the composition series of a free module identical to the number of its bases?
It is the page 118 of 1969 version.
Apr
11
comment Is the length of the composition series of a free module identical to the number of its bases?
It works. Then, the description of the example in this famous book is not correct. Anyway, thank you for your help.
Apr
11
asked If $X \times \{0\} \cup A \times I$ is closed in $X \times I$. Then, is $A$ closed in $X$?
Apr
11
comment Is the length of the composition series of a free module identical to the number of its bases?
In the example, $A_n$ is the set of the homogeneous polynomials of degree $n$.
Apr
11
comment Is the length of the composition series of a free module identical to the number of its bases?
$A$ is a Noetherian graded ring $A=\oplus_{n=0}^\infty A_n$. $M$ is a finitely-generated graded $A$-module $M=\oplus_{n=0}^\infty M_n$.
Apr
11
revised Is the length of the composition series of a free module identical to the number of its bases?
added 4 characters in body
Apr
11
revised Is the length of the composition series of a free module identical to the number of its bases?
added 4 characters in body
Apr
11
asked Is the length of the composition series of a free module identical to the number of its bases?
Mar
25
comment If a chain of distinct irreducible closed subsets of a quasi-affine variety $Y$ is maximal, a chain of their closures is maximal in $\overline{Y}$?
Thank you again. I understand completely this time.
Mar
24
accepted If a chain of distinct irreducible closed subsets of a quasi-affine variety $Y$ is maximal, a chain of their closures is maximal in $\overline{Y}$?
Mar
24
comment If a chain of distinct irreducible closed subsets of a quasi-affine variety $Y$ is maximal, a chain of their closures is maximal in $\overline{Y}$?
Thank you for answering my question. I didn't notice that $W'$ is an open subset of W. One thing is not clear to me. You wrote $Z_i$ is not only irreducible in $Y$ but also in $X$. Does the irreducibility of a subset depend on the space containing it ?
Mar
22
asked If a chain of distinct irreducible closed subsets of a quasi-affine variety $Y$ is maximal, a chain of their closures is maximal in $\overline{Y}$?
Mar
7
awarded  Yearling