模范畴的可数Telescope猜想
The countable Telescope Conjecture for module categories
摘要：
所谓模范畴的望远镜猜想，我们指的是如下的主张：“设R是任意环，（A，B）是Mod-R中的遗传余扭对，其中A和B在直极限下闭。则（A，B）是有限型的.“我们证明了这个猜想的一个修改，用'可数'代替'有限'。证明了任意环R上模的遗传余挠对（A，B）是由一组强可数表示模生成的，只要B在良序链的并下是闭的.我们还用可数表现模之间的态射刻画了B中的模和A中的可数表现模，并证明了（A，B）是由一个纯内射模生成的，只要A在直极限下是闭的.然后，我们将注意力转移到模范畴中的余挠对和紧生成的三角范畴中的局部化对之间的强类比。
英文摘要：
By the Telescope Conjecture for Module Categories, we mean the following claim: "Let R be any ring and (A, B) be a hereditary cotorsion pair in Mod-R with A and B closed under direct limits. Then (A, B) is of finite type." We prove a modification of this conjecture with the word 'finite' replaced by 'countable'. We show that a hereditary cotorsion pair (A, B) of modules over an arbitrary ring R is generated by a set of strongly countably presented modules provided that B is closed under unions of well-ordered chains. We also characterize the modules in B and the countably presented modules in A in terms of morphisms between finitely presented modules, and show that (A, B) is cogenerated by a single pure-injective module provided that A is closed under direct limits. Then we move our attention to strong analogies between cotorsion pairs in module categories and localizing pairs in compactly generated triangulated categories.
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