Let ( A , m ) be a local Noetherian ring and let x 1 , … , x d be a system of parameters, i.e. m = ( x 1 , … , x d ). Then dim ⁡ A / ( x 1 , … , x i ) = d − i i = 1 , … , dI know just a few basic facts about dimension theory. I think I can prove the inequality ≤ via Krull's Hauptidealsatz in this way: the maximal ideal of A / ( x 1 , … , x i ) is m i := ( x i + 1 ¯ , … , x d ¯ ) .So it must be h t ( m i ) ≤ d − i.But how to prove the other inequality? I think I should do it by induction, but I cannot understand how to begin. So, if what I said so far is right, my question is: how can I prove that dim ⁡ A / ( x 1 ) ≥ d − 1 ?

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Let   (  A  ,  m  ) be a local Noetherian ring and let       x    1    ,  …  ,      x    d   be a system of parameters, i.e.   m  =  (      x    1    ,  …  ,      x    d    ). Then  dim  ⁡  A  /  (      x    1    ,  …  ,      x    i    )  =  d  −  i  i  =  1  ,  …  ,  dI know just a few basic facts about dimension theory. I think I can prove the inequality   ≤ via Krull&#039;s Hauptidealsatz in this way: the maximal ideal of   A  /  (      x    1    ,  …  ,      x    i    ) is        m    i    :=  (            x              i        +        1              ¯    ,  …  ,            x      d        ¯    )  .So it must be       h    t    (        m    i    )  ≤  d  −  i.But how to prove the other inequality? I think I should do it by induction, but I cannot understand how to begin. So, if what I said so far is right, my question is: how can I prove that  dim  ⁡  A  /  (      x    1    )  ≥  d  −  1  ?

EreneDreaceaw · Accepted Answer

Your proof for the 1st part is not true:      x    1    ,  …  ,      x    d   being system of parameters does not mean   m  =  (      x    1    ,  …  ,      x    d    )  ,, So   (            x              i        +        1              ¯    ,  …  ,            x      d        ¯    ) isn&#039;t necessarily the maximal ideal of   A  /  (      x    1    ,  …  ,      x    i    )  .

Let ( A , <mi mathvariant="fraktur">m <mo mathvariant="fraktur" stretchy="false">) be a

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