On the direct summand conjecture and its derived variant

By B. Bhatt. 2016

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André 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cd70d7024fcea2ebe0a9a9adf4a0e6087689a718

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Citations in the corpus

Relative p-adic Hodge theory, II: Imperfect period rings (2016, methodology): “During the preparation of this paper, the work of André [5, 6] and Bhatt [18] became available.”