We construct examples of bounded below, noncontractible, acyclic complexes of finitely generated projective modules over some rings S, as well as bounded above, noncontractible, acyclic complexes of injective modules. The rings S are certain rings of infinite matrices with entries in the rings of commutative polynomials or formal power series in infinitely many variables. In the world of comodules or contramodules over coalgebras over fields, similar examples exist over the cocommutative symmetric coalgebra of an infinite-dimensional vector space. A simpler, universal example of a bounded below, noncontractible, acyclic complex of free modules with one generator, communicated to the author by Canonaco, is included at the end of the paper.

我们构造了一些环S上有限生成射影模的下界、不可收缩、无环复形的示例，以及单射模的上界、不可收缩、无环复形。环S是无限矩阵的某些环，其条目位于无限多个变量的交换多项式或形式幂级数的环中。在域上的余代数的余模或反模的世界中，在无限维向量空间的共交换对称余代数上也存在类似的例子。论文末尾包含了一个更简单、通用的示例，该示例是由 Canonaco 向作者传达的带有一个生成器的自由模块的有界下界、不可收缩、非循环复合体。