On the direct summand conjecture and its derived variant

被引:31
作者
Bhatt, Bhargav [1 ]
机构
[1] Univ Michigan, Dept Math, Ann Arbor, MI 48109 USA
基金
美国国家科学基金会;
关键词
EXTENSIONS; COHOMOLOGY; RINGS;
D O I
10.1007/s00222-017-0768-7
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
Andre recently gave a beautiful proof of Hochster's direct summand conjecture in commutative algebra using perfectoid spaces; his two main results are a generalization of the almost purity theorem (the perfectoid Abhyankar lemma) and a construction of certain faithfully flat extensions of perfectoid algebras where "discriminants" acquire all p-power roots. In this paper, we explain a quicker proof of Hochster's conjecture that circumvents the perfectoid Abhyankar lemma; instead, we prove and use a quantitative form of Scholze's Hebbarkeitssatz (the Riemann extension theorem) for perfectoid spaces. The same idea also leads to a proof of a derived variant of the direct summand conjecture put forth by de Jong.
引用
收藏
页码:297 / 317
页数:21
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