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Math Help - inverse limit of product

  1. #1
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    inverse limit of product

    I know that \mathbb{Z}/(6)^i\mathbb{Z}\cong \mathbb{Z}/(2)^i\mathbb{Z}\times \mathbb{Z}/(3)^i\mathbb{Z} for any i.

    But \varprojlim_{i\in I} \mathbb{Z}/(6)^i\mathbb{Z}\cong \varprojlim_{i\in I} \mathbb{Z}/(2)^i\mathbb{Z}\times \varprojlim_{i\in I} \mathbb{Z}/(3)^i\mathbb{Z}??

    Please give an explanation if it's right.
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  2. #2
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    Quote Originally Posted by KaKa View Post
    I know that \mathbb{Z}/(6)^i\mathbb{Z}\cong \mathbb{Z}/(2)^i\mathbb{Z}\times \mathbb{Z}/(3)^i\mathbb{Z} for any i.

    But \varprojlim_{i\in I} \mathbb{Z}/(6)^i\mathbb{Z}\cong \varprojlim_{i\in I} \mathbb{Z}/(2)^i\mathbb{Z}\times \varprojlim_{i\in I} \mathbb{Z}/(3)^i\mathbb{Z}??

    Please give an explanation if it's right.

    To have an inverse limit you must have, besides a set of groups, a corresponding directed set of indexes and a

    corresdponding set of homomorphisms between those indexed groups with some properties.

    You must check these are fulfilled in this case.

    Tonio
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