Talk:Middle-dimensional surgery kernel (Ex)
From Manifold Atlas
Revision as of 11:20, 3 April 2012 by Fabian Hebestreit (Talk | contribs)
First a little lemma: \\
Let be a chain complex with projective, and a direct summand. Then also is a direct summand. \\
Proof:
Note that is a direct summand iff the sequence
splits. By exactness at however, we have which is projective being a direct summand of a projective module.
ad(1): Iterating the lemma we find that is a direct summand of if the same statement holds for some lower . However eventually both terms are zero, since the complex is finite. Being a direct summand in a finitely generated module is then itself finitely generated, and hence also . The second assertion follows immediately from the universal coefficient theorem.