|
| | | | |
| | | |
| Stat |
Members: 3645
Articles: 2'503'724
Articles rated: 2609
24 April 2024 |
|
| | | | |
|
Article overview
| |
|
|
On the infinite tame-wild dichotomy, the Brauer-Thrall 3 conjectures, and related problemns | | M.C.Iovanov
; | | Date: |
1 Mar 2018 | | Abstract: | We prove the tame-wild dichotomy conjecture, due to D. Simson, for infinite
dimensional algebras and coalgebras. The key part of the approach is proving
new representation theoretic characterizations for local finiteness. Among
other, we show that the Ext quiver of the category ${
m f.d.-}A$ of finite
dimensional representations of an arbitrary algebra $A$ is locally finite (i.e.
$dim({
m Ext}^1(S,T))<infty$ for all simple finite dimensional $A$-modules
$S,T$) if and only if for every dimension vector $underline{d}$, the
representations of $A$ of dimension vector $underline{d}$ are all contained in
a finite subcategory (a category of modules over a finite dimensional quotient
algebra). This allows one reduce the tame/wild problem to the finite
dimensional case and Drozd’s classical result. We also show that these two
properties are local in the sense of localization: a category of comodules is
tame/not wild if and only if every "finite" localization is so, and give the
relations to Simson’s f.c.tame/f.c.wild dichotomy. We use the methods and
various embeddings we obtain, to give a proof for the Brauer-Thrall 3
conjecture, also raised by Simson, for the class of all wild algebras, thus
covering "almost" all algebras. We list several questions that seem to arise
naturally from this study. | | Source: | arXiv, 1803.0173 | | Services: | Forum | Review | PDF | Favorites | |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
browser Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)
|
| |
|
|
|
|
News, job offers and information for researchers and scientists:
| |
REGISTER