Further remarks on systems of interlocking exact sequences
In a system of interlocking sequences, the assumption that three out of the four sequences are exact does not guarantee the exactness of the fourth. In 1967, Hilton proved that, with the additional condition that it is differential at the crossing points, the fourth sequence is also exact. In this p...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Wiley
2005-01-01
|
| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/IJMMS.2005.155 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | In a system of interlocking sequences, the assumption that three
out of the four sequences are exact does not guarantee the
exactness of the fourth. In 1967, Hilton proved that,
with the additional condition that it is differential at the
crossing points, the fourth sequence is also exact. In this paper,
we trace such a diagram and analyze the relation between the
kernels and the images, in the case that the fourth sequence is
not necessarily exact. Regarding the exactness of the fourth
sequence, we remark that the exactness of the other three
sequences does guarantee the exactness of the fourth at
noncrossing points. As to a crossing point p, we need
the extra criterion that the fourth sequence is differential. One notices that the condition, for the
fourth sequence, that kernel ⊇ image at
p turns out to be equivalent to the “opposite” condition kernel
⊆ image. Next, for the kernel and the image at p of the fourth sequence,
even though they may not coincide, they are not far
different—they always have the same cardinality as sets, and
become isomorphic after taking quotients by a subgroup which is
common to both. We demonstrate these phenomena with an example. |
|---|---|
| ISSN: | 0161-1712 1687-0425 |