Flat Mittag-Leffler Modules, and Their Relative and Restricted Versions

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F24%3A10488126" target="_blank" >RIV/00216208:11320/24:10488126 - isvavai.cz</a>

Výsledek na webu

<a href="http://dx.doi.org/10.1007/978-3-031-53063-0" target="_blank" >http://dx.doi.org/10.1007/978-3-031-53063-0</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/978-3-031-53063-0" target="_blank" >10.1007/978-3-031-53063-0</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Flat Mittag-Leffler Modules, and Their Relative and Restricted Versions

Popis výsledku v původním jazyce

Assume that. R is a non-right perfect ring. Then there is a proper class ofclasses of (right .R-) modules closed under transfinite extensions lying between theclasses.P0 of projective modules, and.F0 of flat modules. These classes can be definedas variants of the class .F M of absolute flat Mittag-Leffler modules: either as theirrestricted versions (lying between.P0 and.F M), or their relative versions (between.F M and.F0). In this survey, we will deal with applications of these classes in relativehomological algebra and algebraic geometry. The classes .P0 and .F0 are known toprovide for approximations, and minimal approximations, respectively. We will showthat the classes of restricted flat Mittag-Leffler modules, and flat relative MittagLeffler modules, have rather different approximation properties: the former classesalways provide for approximations, but the latter do not, except for the boundary caseof .F0. The notion of an (infinite dimensional) vector bundle is known to be Zariskilocal for all schemes, the key point of the proof being that projectivity ascends anddescends along flat and faithfully flat ring homomorphisms, respectively. We willsee that the same holds for the properties of being a .κ-restricted flat Mittag-Lefflermodule for each cardinal.κ &gt;= ℵ0, and also a flat.Q-Mittag-Leffler module whenever.Q is a definable class of finite type. Thus, as in the model case of vector bundles,Zariski locality holds for flat quasi-coherent sheaves induced by each of these classesof modules. Moreover, we will see that the notion of a locally.n-tilting quasi-coherentsheaf is Zariski local for all.n &gt;= 0.

Název v anglickém jazyce

Flat Mittag-Leffler Modules, and Their Relative and Restricted Versions

Popis výsledku anglicky

Assume that. R is a non-right perfect ring. Then there is a proper class ofclasses of (right .R-) modules closed under transfinite extensions lying between theclasses.P0 of projective modules, and.F0 of flat modules. These classes can be definedas variants of the class .F M of absolute flat Mittag-Leffler modules: either as theirrestricted versions (lying between.P0 and.F M), or their relative versions (between.F M and.F0). In this survey, we will deal with applications of these classes in relativehomological algebra and algebraic geometry. The classes .P0 and .F0 are known toprovide for approximations, and minimal approximations, respectively. We will showthat the classes of restricted flat Mittag-Leffler modules, and flat relative MittagLeffler modules, have rather different approximation properties: the former classesalways provide for approximations, but the latter do not, except for the boundary caseof .F0. The notion of an (infinite dimensional) vector bundle is known to be Zariskilocal for all schemes, the key point of the proof being that projectivity ascends anddescends along flat and faithfully flat ring homomorphisms, respectively. We willsee that the same holds for the properties of being a .κ-restricted flat Mittag-Lefflermodule for each cardinal.κ &gt;= ℵ0, and also a flat.Q-Mittag-Leffler module whenever.Q is a definable class of finite type. Thus, as in the model case of vector bundles,Zariski locality holds for flat quasi-coherent sheaves induced by each of these classesof modules. Moreover, we will see that the notion of a locally.n-tilting quasi-coherentsheaf is Zariski local for all.n &gt;= 0.

Druh

C - Kapitola v odborné knize

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

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

Rok uplatnění

2024

Kód důvěrnosti údajů

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

Název knihy nebo sborníku

Functor Categories, Model Theory, Algebraic Analysis and Constructive Methods: FCMTCCT2 2022, Almería, Spain, July 11–15, Invited and Selected Contributions

ISBN

978-3-031-53063-0

Počet stran výsledku

26

Strana od-do

223-248

Počet stran knihy

248

Název nakladatele

Springer Cham

Místo vydání

Neuveden

Kód UT WoS kapitoly

—