Timeline for Classification of long exact sequences
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 13, 2013 at 12:17 | comment | added | Rasmus | As it turns out, this is not so hard and $\mathbb Z/8$-modules are enough: mathoverflow.net/questions/121606 | |
| Aug 17, 2012 at 7:44 | comment | added | Fernando Muro | @Will Sawin: looking at Ringel-Schmidmeier's paper one should be able to work out an example of a non-cyclic two-step indecomposable complex of $\mathbb{Z}/128$-module given by an injection. Perhapes even using $\mathbb{Z}/64$-modules, but probably not $\mathbb{Z}/32$-modules. This makes me guess that it would take hours, maybe days for me to come up with such example. | |
| Aug 16, 2012 at 20:04 | vote | accept | Rasmus | ||
| Aug 16, 2012 at 20:04 | comment | added | Rasmus | @tweetie-bird: In arxiv.org/abs/math/0409417 Schmidmeier and Ringel show that the category of finitely generated $\mathbb Z/p^n$-submodule inclusions is controlled $\mathbb Z/p$-wild. This seems to mean roughly that the classification of its objects is at least as complication as the classification of finitely generated modules over the free $\mathbb Z/p$-algebra on two generators. | |
| Aug 16, 2012 at 19:58 | comment | added | Fernando Muro | @tweetie-bird, wildness is a precise measure of intractability, take an introductory book on representation theory of algebras and you'll find explicit definitions and illustrative examples. | |
| Aug 16, 2012 at 19:30 | comment | added | Will Sawin | The classification for the case when each group is cyclic is very simple. Just take a sequence of numbers $(a_1,...,a_n)$ and consider the exact sequence: $0 \to \mathbb Z_{a_1} \to \mathbb Z_{a_1a_2} \to \mathbb Z_{a_2a_3} \to \dots \to \mathbb Z_{a_{n-1}a_n} \to \mathbb Z_{a_n} \to 0$ or alternately you can replace $0 \to \mathbb Z_{a_1} \to$ with $0 \to \mathbb Z \to^{a_1} \mathbb Z \to $. This answer would then imply that there must be non-cyclic indecomposables. Is it possible to give a reasonably easy description of some non-cyclic indecomposable exact sequence? | |
| Aug 16, 2012 at 19:26 | comment | added | tweetie-bird | What is the definition of "wild" in this context? Thanks in advance for your help in understanding the answer! | |
| Aug 16, 2012 at 17:18 | history | answered | Fernando Muro | CC BY-SA 3.0 |