Flat comodules and contramodules as directed colimits, and cotorsion periodicity

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F24%3A00600359" target="_blank" >RIV/67985840:_____/24:00600359 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.1007/s40062-024-00358-1" target="_blank" >https://doi.org/10.1007/s40062-024-00358-1</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s40062-024-00358-1" target="_blank" >10.1007/s40062-024-00358-1</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Flat comodules and contramodules as directed colimits, and cotorsion periodicity

Popis výsledku v původním jazyce

This paper is a follow-up to Positselski and Št’ovíček (Flat quasi-coherent sheaves as directed colimits, and quasi-coherent cotorsion periodicity. Electronic preprint arXiv:2212.09639 [math.AG]). We consider two algebraic settings of comodules over a coring and contramodules over a topological ring with a countable base of two-sided ideals. These correspond to two (noncommutative) algebraic geometry settings of certain kind of stacks and ind-affine ind-schemes. In the context of a coring C over a noncommutative ring A, we show that all A-flat C-comodules are ℵ1-directed colimits of A-countably presentable A-flat C-comodules. In the context of a complete, separated topological ring R with a countable base of neighborhoods of zero consisting of two-sided ideals, we prove that all flat R-contramodules are ℵ1-directed colimits of countably presentable flat R-contramodules. We also describe arbitrary complexes, short exact sequences, and pure acyclic complexes of A-flat C-comodules and flat R-contramodules as ℵ1-directed colimits of similar complexes of countably presentable objects. The arguments are based on a very general category-theoretic technique going back to an unpublished 1977 preprint of Ulmer and rediscovered in Positselski (Notes on limits of accessible categories. Electronic preprint arXiv:2310.16773 [math.CT]). Applications to cotorsion periodicity and coderived categories of flat objects in the respective settings are discussed. In particular, in any acyclic complex of cotorsion R-contramodules, all the contramodules of cocycles are cotorsion.

Název v anglickém jazyce

Flat comodules and contramodules as directed colimits, and cotorsion periodicity

Popis výsledku anglicky

This paper is a follow-up to Positselski and Št’ovíček (Flat quasi-coherent sheaves as directed colimits, and quasi-coherent cotorsion periodicity. Electronic preprint arXiv:2212.09639 [math.AG]). We consider two algebraic settings of comodules over a coring and contramodules over a topological ring with a countable base of two-sided ideals. These correspond to two (noncommutative) algebraic geometry settings of certain kind of stacks and ind-affine ind-schemes. In the context of a coring C over a noncommutative ring A, we show that all A-flat C-comodules are ℵ1-directed colimits of A-countably presentable A-flat C-comodules. In the context of a complete, separated topological ring R with a countable base of neighborhoods of zero consisting of two-sided ideals, we prove that all flat R-contramodules are ℵ1-directed colimits of countably presentable flat R-contramodules. We also describe arbitrary complexes, short exact sequences, and pure acyclic complexes of A-flat C-comodules and flat R-contramodules as ℵ1-directed colimits of similar complexes of countably presentable objects. The arguments are based on a very general category-theoretic technique going back to an unpublished 1977 preprint of Ulmer and rediscovered in Positselski (Notes on limits of accessible categories. Electronic preprint arXiv:2310.16773 [math.CT]). Applications to cotorsion periodicity and coderived categories of flat objects in the respective settings are discussed. In particular, in any acyclic complex of cotorsion R-contramodules, all the contramodules of cocycles are cotorsion.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GA23-05148S" target="_blank" >GA23-05148S: Homologická a strukturní teorie v geometrických kontextech</a><br>

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2024

Kód důvěrnosti údajů

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

Název periodika

Journal of Homotopy and Related Structures

ISSN

2193-8407

e-ISSN

1512-2891

Svazek periodika

19

Číslo periodika v rámci svazku

4

Stát vydavatele periodika

GE - Gruzie

Počet stran výsledku

44

Strana od-do

635-678

Kód UT WoS článku

001329072200001

EID výsledku v databázi Scopus

2-s2.0-85206388319