Title: Simutaneously vanishing higher derived limits without large cardinals Show Chinese title

Title: 同时消失的高阶导出极限而不使用大基数

Abstract: A question dating to Sibe Marde\v{s}i\'{c} and Andrei Prasolov's 1988 work Strong homology is not additive, and motivating a considerable amount of set theoretic work in the ensuing years, is that of whether it is consistent with the ZFC axioms for the higher derived limits $\mathrm{lim}^n$ $(n>0)$ of a certain inverse system $\mathbf{A}$ indexed by ${^\omega}\omega$ to simultaneously vanish. An equivalent formulation of this question is that of whether it is consistent for all $n$-coherent families of functions indexed by ${^\omega}\omega$ to be trivial. In this paper, we prove that, in any forcing extension given by adjoining $\beth_\omega$-many Cohen reals, $\mathrm{lim}^n \mathbf{A}$ vanishes for all $n > 0$. Our proof involves a detailed combinatorial analysis of the forcing extension and repeated applications of higher dimensional $\Delta$-system lemmas. This work removes all large cardinal hypotheses from the main result of arXiv:1907.11744 and substantially reduces the least value of the continuum known to be compatible with the simultaneous vanishing of $\mathrm{lim}^n \mathbf{A}$ for all $n > 0$. Show Chinese abstract

Abstract: 一个可以追溯到西贝·马尔德什奇和安德烈·普拉索洛夫1988年工作的问题是：强同调不是可加的，这促使在随后的几年中进行了大量集合论的工作，即是否与ZFC公理一致，对于由${^\omega}\omega$索引的某个逆系统$\mathbf{A}$的高阶导出极限$\mathrm{lim}^n$ $(n>0)$ 同时消失。这个问题的等价表述是：是否一致地所有由${^\omega}\omega$索引的$n$-一致函数族都是平凡的。 In this paper, we prove that, in any forcing extension given by adjoining $\beth_\omega$-many Cohen reals, $\mathrm{lim}^n \mathbf{A}$ vanishes for all $n > 0$. Our proof involves a detailed combinatorial analysis of the forcing extension and repeated applications of higher dimensional $\Delta$-system lemmas. This work removes all large cardinal hypotheses from the main result of arXiv:1907.11744 and substantially reduces the least value of the continuum known to be compatible with the simultaneous vanishing of $\mathrm{lim}^n \mathbf{A}$ for all $n > 0$.