Bhargav Bhatt

Projectivity of the Witt vector affine Grassmannian

We prove that the Witt vector affine Grassmannian, which parametrizes W(k)-lattices in

Local cohomology modules of a smooth Z-algebra have finitely many associated primes

W(k)[\frac{1}{p}]^n

The dualizing complex of F-injective and Du Bois singularities

W(k)[1p]n for a perfect field k of characteristic p, is representable by an ind-(perfect scheme) over k. This improves on previous results of Zhu by constructing a natural ample line bundle. Along the way, we establish various foundational results on perfect schemes, notably h-descent results for vector bundles.

The Pro-étale topology for schemes

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.

Integral p-adic Hodge theory

Let R be a commutative Noetherian ring that is a smooth

p-adic derived de Rham cohomology

\mathbb {Z}

Algebraization and Tannaka duality

-algebra. For each ideal

Completions and derived de Rham cohomology

\mathfrak {a}