Reconstruction of the stacky approach to de Rham cohomology

Mondal, Shubhodip

doi:10.48550/arxiv.2202.07089

Cited by 1 publication

(1 citation statement)

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“…Secondly, there is a general method (due to Mondal [Mon22]) by which one can try to recover stacky approaches to cohomology theories from purely cohomological information. This method can be applied in the setting outlined in this article to show that Hodge filtered de Rham cohomology of smooth affine schemes can be used to recover the stacks X dR , while this method fails in Simpson's approach (and will fail for any approach which recovers unfiltered de Rham cohomology).…”

Section: Formentioning

confidence: 99%

Prismatization over $\mathbf{Z}$

Gurney¹

2023

Preprint

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The aim of this article is to given an extension of the prismatization functor for p-adic formal schemes (whose construction was first sketched by Drinfeld in [Dri20] and then given by Bhatt-Lurie in [BL22b]) to all schemes over Spec(Z). We then prove some basic properties of this extension (algebraicity, flatness for syntomic morphisms, perfectness of cohomology) and show that for smooth schemes over Q this construction recovers (a version of) the filtered de Rham stack. ContentsIntroduction 1. Preliminaries 1.1. Conventions 2. de Rham cohomology over Q 2.1. Reminders on filtrations and the Cartier stack 2.2. The de Rham stack 2.3. Cohomology of X dR 3. Witt vectors 3.1. Reminders 3.2. The map V(1) : V → W 3.3. The ring scheme W 4. The stack Σ Z 4.1. Hodge-Tate elements and ∆ Z 4.2. The stack Σ Z 5. Prismatization over Z 5.1. The ring stacks W E 5.2. Definition and algebraicity of X ∆ Z 5.3. The Hodge-Tate stack X HT 5.4. Presentations of X ∆ Z 5.5. Comparisons and the cohomology of X ∆ Z References

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Section: Formentioning

confidence: 99%