AI 中文总结
该研究通过建立完美oid代数被根理想除所得商的投射与内射维数估计，得到了有限内射维数模等在完美oid代数方面的同调特征，是正特征相关特征的混合特征类似物。
AI 中文摘要
我们首先建立了关于完美oid代数被根理想除所得商的投射维数和内射维数的估计。作为应用，我们得到了有限内射维数模、（大）科恩 - 麦考利模、戈伦斯坦局部环以及混合特征下正则局部环在完美oid代数方面的同调特征。这些是正特征下通过弗罗贝尼乌斯态射相应特征的混合特征类似物。
英文摘要
We first establish estimates for the projective and injective dimensions for quotients of perfectoid algebras by radical ideals. As applications, we obtain homological characterizations of modules of finite injective dimension, (big) Cohen--Macaulay modules, Gorenstein local rings, and regular local rings in mixed characteristic in terms of perfectoid algebras. These are mixed characteristic analogues of the corresponding characterizations via Frobenius morphisms in positive characteristic.
CommentsSome special cases of the present results appeared in "Perfect Closure Detects Injective Dimension". The current project, however, has a broader mixed-characteristic scope. Due to substantial changes in content and structure, we present this as a separate paper
