Philipp Rothmaler
We study (relative) $\mathcal K$-Mittag-Leffler modules as was done in the author's habilitation thesis, rephrase old, unpublished results in terms of definable subcategories, and present newer ones, culminating in a characterization of countably generated $\cal K$-Mittag-Leffler modules.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We study (relative) K-Mittag-Leffler modules, with emphasis on the class K of absolutely pure modules. A final goal is to describe the K-Mittag-Leffler abelian groups as those that are, modulo their torsion part, aleph_1-free, Cor.6.12. Several more general results of independent interest are derived on the way. In particular, every flat K-Mittag-Leffler module (for K as before) is Mittag-Leffler, Thm.3.9. A question about the definable subcategories generated by the divisible modules and the torsion-free modules, resp., has been left open, Quest.4.6.
We study versions of strict Mittag-Leffler modules relativized to a class $\cK$ (of modules), that is, \emph{strict} versions (in the technical sense of Raynaud and Gruson) of $\cK$-Mittag-Leffler modules, as investigated in the preceding paper, {\em Mittag-Leffler modules and definable subcategories}, in this very series (as well as the arXiv).
We develop a structure theory for two classes of infinite dimensional modules over tame hereditary algebras: the Baer modules, and the Mittag-Leffler ones.
We show that over any ring, the double Ext-orthogonal class to all flat Mittag-Leffler modules contains all countable direct limits of flat Mittag-Leffler modules. If the ring is countable, then the double orthogonal class consists precisely of all flat modules and we deduce, using a recent result of Šaroch and Trlifaj, that the class of flat Mittag-Leffler modules is not precovering in Mod-R unless R is right perfect.
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 M…
We study Mittag-Leffler conditions on modules providing relative versions of classical results by Raynaud and Gruson. We then apply our investigations to several contexts. First of all, we give a new argument for solving the Baer splitting problem. Moreover, we show that modules arising in cotorsion pairs satisfy certain Mittag-Leffler conditions. In particular, this implies that tilting modules satisfy a useful finiteness condition over their endomorphism ring. In the final section, we focus on a special tilting cotorsion pair related to the pure-semisimplicity conjecture.
Let $R$ be a ring and denote by $\mathcal{FM}$ the class of all flat and Mittag-Leffler left $R$-modules. In \cite{BazzoniStovicek2} it is proved that, if $R$ is countable, the orthogonal class of $\mathcal{FM}$ consists of all cotorsion modules. In this note we extend this result to the class of all rings $R$ satisfying that each flat left $R$-module is filtered by totally ordered limits of projective modules. This class of rings contains all countable, left perfect and discrete valuation domains. Moreover, assuming that there do not exist inaccessible cardinals, we obtain that, over these ri…
The Mittag-Leffler condition descents via pure monomorphisms
本文澄清了 Raynaud 和 Gruson 关于 Mittag-Leffler 性质经纯单态射下降的证明，并证明纯投射性也经此下降。
Flat relative Mittag-Leffler modules and approximations
该研究刻画了介于平坦绝对Mittag-Leffler模与平坦模之间的平坦相对Mittag-Leffler模的局部结构，验证了Enochs猜想对相关模类成立，并将结果应用于f-投射模场景。
Mittag-Leffler modules occur naturally in algebra, algebraic geometry, and model theory, [18], [12], [17]. If $R$ is a non-right perfect ring, then it is known that in contrast with the classes of all projective and flat modules, the class of all flat Mittag-Leffler modules is not deconstructible [14], and it does not provide for approximations when $R$ has cardinality $\leq \aleph_0$, [6]. We remove the cardinality restriction on $R$ in the latter result. We also prove an extension of the Countable Telescope Conjecture [21]: a cotorsion pair $(\mathcal A,\mathcal B)$ is of countable type when…
Let $R$ be any ring. We prove that all direct products of flat right $R$-modules have finite flat dimension if and only if each finitely generated left ideal of $R$ has finite projective dimension relative to the class of all $\mathcal F$-Mittag-Leffler left $R$-modules, where $\mathcal F$ is the class of all flat right $R$-modules. In order to prove this theorem, we obtain a general result concerning global relative dimension. Namely, if $\mathcal X$ is any class of left $R$-modules closed under filtrations that contains all projective modules, then $R$ has finite left global projective dimen…
In this note I take the opportunity to correct the last statement of Part I of same title and continue the study of uniform purity of epimorphisms in order to derive the main result, which states that--provided $R_R\in \langle\cal K\rangle$, equivalently, $\langle \cal L\rangle$ (the definable subcategory generated by $\cal L$) contains all absolutely pure left modules--every countably generated $\cal K$-Mittag-Leffler module in $\langle \cal L\rangle$ is a direct summand of a $\langle \cal L\rangle$-preenvelope of a union of an $\cal L$-pure $ω$-chain of finitely presented modules. In conclus…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。