Flat Mittag-Leffler Modules, and Their Relative and Restricted Versions
The result's identifiers
Result code in 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>
Result on the web
<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>
Alternative languages
Result language
angličtina
Original language name
Flat Mittag-Leffler Modules, and Their Relative and Restricted Versions
Original language description
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 >= 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 >= 0.
Czech name
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Czech description
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Classification
Type
C - Chapter in a specialist book
CEP classification
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OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
<a href="/en/project/GA23-05148S" target="_blank" >GA23-05148S: Homological and structural theory in geometric contexts</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Others
Publication year
2024
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Book/collection name
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
Number of pages of the result
26
Pages from-to
223-248
Number of pages of the book
248
Publisher name
Springer Cham
Place of publication
Neuveden
UT code for WoS chapter
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