The homological conjectures, which date back to Peskine, Szpiro and Hochster in the late 60’s, make fundamental predictions about syzygies and intersection problems in commutative algebra. They were settled long ago in the presence of a base field and led to tight closure theory, a powerful tool to investigate singularities in characteristic p.Recently, perfectoid techniques coming from p-adic Hodge theory have allowed us to get rid of any base field; this solves the direct summand conjecture and establishes the existence and weak functoriality of big Cohen-Macaulay algebras, which solve in turn the homological conjectures in general. This also opens the way to the study of singularities in mixed characteristic.We sketch a broad outline of this story, taking lastly a glimpse at ongoing work by L. Ma and K. Schwede, which shows how such a study could build a bridge between singularity theory in characteristic p and in characteristic 0.

中文翻译：

---

混合特征中的奇点。完美法

同源猜想可以追溯到60年代末的Peskine，Szpiro和Hochster，它们对交换代数中的合音和交点问题做出了基本的预测。他们早就解决了在基地场的存在，并导致关闭严密的理论，一个强大的工具来调查特征奇异p。近日，perfectoid技术来自哪里p-adic Hodge理论使我们摆脱了任何基础领域；这解决了直接求和猜想，并建立了大Cohen-Macaulay代数的存在性和弱函数性，从而反过来又解决了一般的猜想。这也为研究混合特征中的奇点开辟了道路。我们概述了这个故事的概要，最后瞥见了L.Ma和K.Schwede正在进行的工作，这表明了这种研究如何在两者之间架起桥梁特征p和特征0中的奇点理论。