Almost free modules and Mittag-Leffler conditions
Drinfeld recently suggested to replace projective modules by the flat Mittag-Leffler ones in the definition of an infinite dimensional vector bundle on a scheme X (Drinfeld, 2006 [8]). Two questions arise: (1) What is the structure of the class D of all flat Mittag-Leffler modules over a general rin...
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Published in | Advances in mathematics (New York. 1965) Vol. 229; no. 6; pp. 3436 - 3467 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
Elsevier Inc
01.04.2012
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Subjects | |
Online Access | Get full text |
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Summary: | Drinfeld recently suggested to replace projective modules by the flat Mittag-Leffler ones in the definition of an infinite dimensional vector bundle on a scheme X (Drinfeld, 2006 [8]). Two questions arise: (1) What is the structure of the class D of all flat Mittag-Leffler modules over a general ring? (2) Can flat Mittag-Leffler modules be used to build a Quillen model category structure on the category of all chain complexes of quasi-coherent sheaves on X?
We answer (1) by showing that a module M is flat Mittag-Leffler, if and only if M is ℵ1-projective in the sense of Eklof and Mekler (2002) [10]. We use this to characterize the rings such that D is closed under products, and relate the classes of all Mittag-Leffler, strict Mittag-Leffler, and separable modules. Then we prove that the class D is not deconstructible for any non-right perfect ring. So unlike the classes of all projective and flat modules, the class D does not admit the homotopy theory tools developed recently by Hovey (2002) [26]. This gives a negative answer to (2). |
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ISSN: | 0001-8708 1090-2082 |
DOI: | 10.1016/j.aim.2012.02.013 |