  Andr\'e 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.
