关于无限积分扩域的一些同调问题的注记
Remarks on some Homological Problems regarding Infinite Integral Extensions
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中文总结 AI 辅助
本文应用收缩自同态的相关结果，在NBIM环正则性等同调问题上取得进展，解决了F-纯环的猜想，还在混合特征下对完美纯环证明了相关结论。
中文摘要 AI 辅助
设R是一个优秀局部环。若对某个i≥d:=dim(R)，有Tor_i^R(R⁺,k)=0，则称R为NBIM环。Bhatt、Iyengar和Ma提出：特征零的等特征NBIM环是否正则？若R是正特征环，Asgharzadeh和Mahdavi推测：对某个i>d，Ext_R^i(k,R^∞)=0意味着R是正则环。在正特征下，R⁺和R^∞是否是m-adic理想分离的，这是“平坦性局部判据”中的一个条件，这是Kunz定理的类似物，与代数几何中的同调猜想和奇点密切相关。我们应用Avramov、Hochster、Iyengar和Yao关于收缩自同态的结果，在上述前两个问题上取得进展。我们发现，这意味着有向NBIM环是正则环，且解决了F-纯环的猜想。这些改进是现有技术无法实现的，还为早期结果提供了新的简单证明。在混合特征下，我们对完美纯环证明了几个相关结果；当存在R→S在穿孔谱上有限平坦且S是正则环时，我们证明了上述第三个结论，这用到了S⁺的Cohen-Macaulay性质。
英文摘要
Let $R$ be an excellent local domain. $R$ is said to be $NBIM$ if $Tor_{i}^{R}(R^{+}, k) = 0$ for some $i\geq d:=\dim(R)$. Bhatt, Iyengar, and Ma ask if equi-characteristic zero $NBIM$ rings are regular. If $R$ is of positive characteristic, Asgharzadeh and Mahdavi conjecture that $Ext^{i}_{R}(k,R^{\infty}) = 0$ for some $i>d$ implies that $R$ is regular. It is an open question whether $R^{+}$ and $R^{\infty}$ are $\mathfrak{m}$-adically idealwise separated in positive characteristic, a condition from the `local criterion of flatness'. These are analogues of Kunz's theorem and intimately related to the homological conjectures and singularities in algebraic geometry. We apply a result of Avramov, Hochster, Iyengar, and Yao on contracting endomorphisms to make progress on the first two. We observe that it implies toric $NBIM$ rings are regular and solves the conjecture for $F$-pure rings. These improvements are inaccessible by previous techniques and give new and simple proofs of earlier results. In mixed characteristic, we show several linked results for perfectoid-pure rings. We show the third statement when there is $R\rightarrow S$ finite and flat on the punctured spectrum and $S$ is regular, this uses Cohen-Macaulayness of $S^{+}$.
