几乎自由模与Mittag-Leffler条件
Almost free modules and Mittag--Leffler conditions
摘要：
Drinfeld最近建议在$X$上无限维向量丛的定义中用平坦的Mittag-Leffler模代替投射模。由此产生两个问题：（1）一般环上所有平坦Mittag-Leffler模的类$\mathcal D$的结构是什么？(2)平坦Mittag-Leffler模能否在$X$上的拟凝聚层的所有链复形范畴上建立Quillen模型范畴结构？我们通过证明一个模$M$是平坦的Mittag-Leffler，当且仅当$M$是Eklof和Mekler意义下的$\aleph_1$ -投射模来回答（1）。我们用它来表征环，使$\mathcal D$是封闭的产品，并涉及类的所有Mittag-Leffler，严格的Mittag-Leffler，和可分离的模块。然后证明了类$\mathcal D$对任何非右完满环都是不可分解的.因此，与所有投射模和平坦模的类不同，类$\mathcal D$不接受Hovey最近开发的同伦理论工具。这对（2）给出了否定的答案。
英文摘要：
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 $X$. Two questions arise: (1) What is the structure of the class $\mathcal D$ 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 $\aleph_1$--projective in the sense of Eklof and Mekler. We use this to characterize the rings such that $\mathcal D$ is closed under products, and relate the classes of all Mittag--Leffler, strict Mittag--Leffler, and separable modules. Then we prove that the class $\mathcal D$ is not deconstructible for any non--right perfect ring. So unlike the classes of all projective and flat modules, the class $\mathcal D$ does not admit the homotopy theory tools developed recently by Hovey . This gives a negative answer to (2).
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