Pure semisimplicity conjecture and Artin problem for dimension sequences

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F21%3A10436286" target="_blank" >RIV/00216208:11320/21:10436286 - isvavai.cz</a>

Výsledek na webu

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=etEL.o3Yif" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=etEL.o3Yif</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.jpaa.2021.106745" target="_blank" >10.1016/j.jpaa.2021.106745</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Pure semisimplicity conjecture and Artin problem for dimension sequences

Popis výsledku v původním jazyce

Inspired by a recent paper due to Jose Luis Garcia, we revisit the attempt of Daniel Simson to construct a counterexample to the pure semisimplicity conjecture. Using compactness, we show that the existence of such counterexample would readily follow from the very existence of certain (countable set of) hereditary artinian rings of finite representation type. The existence of such rings is then proved to be equivalent to the existence of special types of embeddings, which we call tight, of division rings into simple artinian rings. Using the tools by Aidan Schofield from 1980s, we can show that such an embedding F -&gt; M-n (G) exists provided that n &lt; 5. As a byproduct, we obtain a division ring extension G subset of F such that the bimodule F-G(F) has the right dimension sequence (1,2,2,2,1,4). Finally, we formulate Conjecture A, which asserts that a particular type of adjunction of an element to a division ring can be made, and demonstrate that its validity would be sufficient to prove the existence of tight embeddings in general, and hence to disprove the pure semisimplicity conjecture. (C) 2021 Elsevier B.V. All rights reserved.

Název v anglickém jazyce

Pure semisimplicity conjecture and Artin problem for dimension sequences

Popis výsledku anglicky

Inspired by a recent paper due to Jose Luis Garcia, we revisit the attempt of Daniel Simson to construct a counterexample to the pure semisimplicity conjecture. Using compactness, we show that the existence of such counterexample would readily follow from the very existence of certain (countable set of) hereditary artinian rings of finite representation type. The existence of such rings is then proved to be equivalent to the existence of special types of embeddings, which we call tight, of division rings into simple artinian rings. Using the tools by Aidan Schofield from 1980s, we can show that such an embedding F -&gt; M-n (G) exists provided that n &lt; 5. As a byproduct, we obtain a division ring extension G subset of F such that the bimodule F-G(F) has the right dimension sequence (1,2,2,2,1,4). Finally, we formulate Conjecture A, which asserts that a particular type of adjunction of an element to a division ring can be made, and demonstrate that its validity would be sufficient to prove the existence of tight embeddings in general, and hence to disprove the pure semisimplicity conjecture. (C) 2021 Elsevier B.V. All rights reserved.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GA17-23112S" target="_blank" >GA17-23112S: Strukturní teorie reprezentací algeber (lokalizace a vychylující teorie)</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2021

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Journal of Pure and Applied Algebra

ISSN

0022-4049

e-ISSN

—

Svazek periodika

225

Číslo periodika v rámci svazku

11

Stát vydavatele periodika

NL - Nizozemsko

Počet stran výsledku

11

Strana od-do

106745

Kód UT WoS článku

000664028800009

EID výsledku v databázi Scopus

2-s2.0-85102556784