再论半交换范畴中的正合偶
Exact couples in semiabelian categories revisited
摘要：
考虑Palamodov意义下的半交换范畴中的正合偶，即，在加法范畴中，每个态射都有一个核和一个上核，并且上像和像之间的诱导态射总是monic和epic的。假设偶中的态射是严格的，即，它们甚至在它们对应的余象和象之间诱导同构。我们表明，在这种情况下，埃克曼和希尔顿的经典建设产生两个派生的夫妇是由一个自然的双态。这两对对应于先验不同的上同调对象，左对应于左。右上同调，与初始正合偶有关。推导过程可以在额外的假设下迭代。
英文摘要：
Consider an exact couple in a semiabelian category in the sense of Palamodov, i.e., in an additive category in which every morphism has a kernel as well as a cokernel and the induced morphism between coimage and image is always monic and epic. Assume that the morphisms in the couple are strict, i.e., they induce even isomorphisms between their corresponding coimages and images. We show that the classical construction of Eckmann and Hilton in this case produces two derived couples which are connected by a natural bimorphism. The two couples correspond to the a priori distinct cohomology objects, the left resp. right cohomology, associated with the initial exact couple. The derivation process can be iterated under additional assumptions.
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