inverse limit of product

• Nov 21st 2010, 12:53 AM
KaKa
inverse limit of product
I know that $\displaystyle \mathbb{Z}/(6)^i\mathbb{Z}\cong \mathbb{Z}/(2)^i\mathbb{Z}\times \mathbb{Z}/(3)^i\mathbb{Z}$ for any $\displaystyle i$.

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

Please give an explanation if it's right.
• Nov 21st 2010, 03:08 AM
tonio
Quote:

Originally Posted by KaKa
I know that $\displaystyle \mathbb{Z}/(6)^i\mathbb{Z}\cong \mathbb{Z}/(2)^i\mathbb{Z}\times \mathbb{Z}/(3)^i\mathbb{Z}$ for any $\displaystyle i$.

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

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