Math 726

Fall 2002

Homework 3

 

Due Monday, November 4, 2002. Do not forget to mark your “favorite” problem.

 

1.      Let  and  be an ascending chain of subsets of a set  Compute

 

2.      Let  be any module. Show that  where  is the family of finitely-generated submodules of

 

3.      Show that if  is an additive, right exact functor that preserves sums (e.g. ), then  preserves direct limits.

 

4.      Let  the ordinary integers and let  Compute the adic completion of

 

5.      Consider the ring  with ideal  Compute the adic completion of