Abstract representation theory of Dynkin quivers of type A
Identifikátory výsledku
Kód výsledku v IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F16%3A10330777" target="_blank" >RIV/00216208:11320/16:10330777 - isvavai.cz</a>
Výsledek na webu
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DOI - Digital Object Identifier
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Alternativní jazyky
Jazyk výsledku
angličtina
Název v původním jazyce
Abstract representation theory of Dynkin quivers of type A
Popis výsledku v původním jazyce
We study the representation theory of Dynkin quivers of type A in abstract stable homotopy theories, including those associated to fields, rings, schemes, differential-graded algebras, and ring spectra. Reflection functors, (partial) Coxeter functors, and Serre functors are defined in this generality and these equivalences are shown to be induced by universal tilting modules, certain explicitly constructed spectral bimodules. In fact, these universal tilting modules are spectral refinements of classical tilting complexes. As a consequence we obtain split epimorphisms from the spectral Picard groupoid to derived Picard groupoids over arbitrary fields. These results are consequences of a more general calculus of spectral bimodules and admissible morphisms of stable derivators. As further applications of this calculus we obtain examples of universal tilting modules which are new even in the context of representations over a field. This includes Yoneda bimodules on mesh categories which encode all the other universal tilting modules and which lead to a spectral Serre duality result. Finally, using abstract representation theory of linearly oriented A n -quivers, we construct canonical higher triangulations in stable derivators and hence, a posteriori, in stable model categories and stable oo -categories.
Název v anglickém jazyce
Abstract representation theory of Dynkin quivers of type A
Popis výsledku anglicky
We study the representation theory of Dynkin quivers of type A in abstract stable homotopy theories, including those associated to fields, rings, schemes, differential-graded algebras, and ring spectra. Reflection functors, (partial) Coxeter functors, and Serre functors are defined in this generality and these equivalences are shown to be induced by universal tilting modules, certain explicitly constructed spectral bimodules. In fact, these universal tilting modules are spectral refinements of classical tilting complexes. As a consequence we obtain split epimorphisms from the spectral Picard groupoid to derived Picard groupoids over arbitrary fields. These results are consequences of a more general calculus of spectral bimodules and admissible morphisms of stable derivators. As further applications of this calculus we obtain examples of universal tilting modules which are new even in the context of representations over a field. This includes Yoneda bimodules on mesh categories which encode all the other universal tilting modules and which lead to a spectral Serre duality result. Finally, using abstract representation theory of linearly oriented A n -quivers, we construct canonical higher triangulations in stable derivators and hence, a posteriori, in stable model categories and stable oo -categories.
Klasifikace
Druh
J<sub>x</sub> - Nezařazeno - Článek v odborném periodiku (Jimp, Jsc a Jost)
CEP obor
BA - Obecná matematika
OECD FORD obor
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Návaznosti výsledku
Projekt
<a href="/cs/project/GA14-15479S" target="_blank" >GA14-15479S: Teorie reprezentací (strukturní rozklady a jejich meze)</a><br>
Návaznosti
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Ostatní
Rok uplatnění
2016
Kód důvěrnosti údajů
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Údaje specifické pro druh výsledku
Název periodika
Advances in Mathematics
ISSN
0001-8708
e-ISSN
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Svazek periodika
2016
Číslo periodika v rámci svazku
293
Stát vydavatele periodika
US - Spojené státy americké
Počet stran výsledku
86
Strana od-do
856-941
Kód UT WoS článku
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EID výsledku v databázi Scopus
2-s2.0-84960145326