TRIPLES, ALGEBRAS AND COHOMOLOGY
TRIPLES, ALGEBRAS AND COHOMOLOGY
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三元组、代数和上同调
DOI：
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发表时间：
1967
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通讯作者：
J. Beck
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作者：
J. Beck
《范畴的理论与应用》的编辑们很高兴地使这篇论文能够普遍地发表。虽然论文的日期是1967年，但在1964年就有一个几乎完整的草稿。这篇论文是一个启示，我们这些谁感兴趣的同调代数的时间。虽然世界上第一个三元组（现在更常被称为“单子”）在这个论文的意义上是非加性的，并用于构建层的松弛决议（[Godement（1958）]），当时流行的信念是，理论的三元组有一个使用同调代数只有通过加性三元组的交换范畴，典型的是像Λ <$Λ <$Λ −，对范畴的Λ-Λ双模。事实上，[艾伦伯格和摩尔（1965年b）]竟然根据他们的相对同调代数的三元组是添加剂和保存的内核。因此，有相当大的惊讶时，乔恩贝克，在目前的工作中，不仅能够定义上同调由一个三元组的类别的对象的利益（而不是阿贝尔类别的系数模块），但甚至证明在广泛的一般性，第一上同调群分类奇异扩展的一个模块。贝克在这项工作中取得的最大成就是他对模块、奇异扩展和派生模块的描述。这些描述的简单性和说服力仍然是本论文更令人惊讶的特点之一。
It is with great pleasure that the editors of Theory and Applications of Categories make this dissertation generally available. Although the date on the thesis is 1967, there was a nearly complete draft circulated in 1964. This thesis was a revelation to those of us who were interested in homological algebra at the time. Although the world’s very first triple (now more often called “monad”) in the sense of this thesis was non-additive and used to construct flabby resolutions of sheaves ([Godement (1958)]), the then-prevailing belief was that the theory of triples had a use in homological algebra only via additive triples on abelian categories, typically something like Λ⊗Λ⊗Λ −, on the category of Λ-Λ bimodules. In fact, [Eilenberg & Moore (1965b)] went so far as to base their relative homological algebra on triples that were additive and preserved kernels. Thus there was considerable astonishment when Jon Beck, in the present work, was able not only to define cohomology by a triple on the category of objects of interest (rather than the abelian category of coefficient modules) but even prove in wide generality that the first cohomology group classifies singular extensions by a module. Not the least of Beck’s accomplishments in this work are his telling, and general, axiomatic descriptions of module, singular extension, and derivation into a module. The simplicity and persuasiveness of these descriptions remains one of the more astonishing features of this thesis.