Derived Functors of Inverse Limits Revisited
Derived Functors of Inverse Limits Revisited
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重温逆极限的派生函子
DOI：
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
J. Roos
中科院分区：
文献类型：
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作者：
J. Roos
我们证明、修正和推广了我们以前的一篇论文（利用和回顾了我们后来的一些论文）关于阿贝尔范畴中射影极限的派生函子的几个结果。特别地，我们证明了如果C是一个满足Grothendieck公理AB3和AB4*并且有一组生成子的阿贝耳范畴，那么在C中所谓的Mittag - Leffler序列上第一导得的射影极限函子就消失了。最近由Deligne和Neeman给出的例子证明了该范畴有一组生成子的条件是必要的。条件AB4*也是必要的，实际上我们为每个整数m小于1给出一个格罗滕迪克类别Cm和Cm中的Mittag‐Leffler序列的例子，其射影极限的衍生函子除m外在所有正度中消失。这导致了对格罗滕迪克类别中无限乘积的衍生函子的系统研究。本文还研究了这些函子应用的几个显式例子。
We prove, correct and extend several results of an earlier paper of ours (using and recalling several of our later papers) about the derived functors of projective limit in abelian categories. In particular we prove that if C is an abelian category satisfying the Grothendieck axioms AB3 and AB4* and having a set of generators then the first derived functor of projective limit vanishes on so‐called Mittag‐Leffler sequences in C. The recent examples given by Deligne and Neeman show that the condition that the category has a set of generators is necessary. The condition AB4* is also necessary, and indeed we give for each integer m ⩾ 1 an example of a Grothendieck category Cm and a Mittag‐Leffler sequence in Cm for which the derived functors of its projective limit vanish in all positive degrees except m. This leads to a systematic study of derived functors of infinite products in Grothendieck categories. Several explicit examples of the applications of these functors are also studied.