Lifting to truncated Brown-Peterson spectra and Hodge-de Rham degeneration in characteristic $p>0$

Sanath K. Devalapurkar

Lifting to truncated Brown-Peterson spectra and Hodge-de Rham degeneration in characteristic $p>0$

Sanath K. Devalapurkar

Abstract

The goal of this note is to prove that Hodge-de Rham degeneration holds for smooth and proper $\mathbf{F}_p$-schemes $X$ with $\dim(X)<p^n$ as soon as its category of quasicoherent sheaves admits a lift to the truncated Brown-Peterson spectrum $\mathrm{BP}\langle n-1\rangle$, and the Hochschild-Kostant-Rosenberg spectral sequence for $X$ degenerates at the $E_2$-page. This is obtained from a noncommutative version, whose proof is essentially the same as Mathew's argument in arXiv:1710.09045.

Lifting to truncated Brown-Peterson spectra and Hodge-de Rham degeneration in characteristic $p>0$

Abstract

The goal of this note is to prove that Hodge-de Rham degeneration holds for smooth and proper

-schemes

with

as soon as its category of quasicoherent sheaves admits a lift to the truncated Brown-Peterson spectrum

, and the Hochschild-Kostant-Rosenberg spectral sequence for

degenerates at the

-page. This is obtained from a noncommutative version, whose proof is essentially the same as Mathew's argument in arXiv:1710.09045.

Paper Structure (7 theorems, 16 equations)

This paper contains 7 theorems, 16 equations.

Key Result

Theorem 2

Let $n \leq \infty$, and let $X$ be a smooth and proper scheme overHere, $\mathbf{F}_p$ could be replaced by any perfect field of characteristic $p>0$; we only use $\mathbf{F}_p$ to avoid introducing conceptually unnecessary notation.$\mathbf{F}_p$ of dimension $<p^n$. Suppose that: Then the Hodge-de Rham spectral sequence collapses at the $E_1$-page, and the de-Rham-to-$\mathrm{HP}$ spectral se

Theorems & Definitions (17)

Theorem 2
Proposition 3
Remark 4
Remark 5
Remark 6
Remark 7
Proposition 8
Proposition 9
proof
Lemma 10
...and 7 more