2019/04/17 by Bhargav Bhatt, Matthew Morrow, Peter Scholze · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.1007/s10240-019-00106-9
openalex publication_date 2019/04/17 · crossref created 2019/04/17 · crossref issued 2019/06/01 · crossref published 2019/06/01 · crossref published-online 2019/06/01 · openalex created_date 2019/07/30 · crossref deposited 2026/02/17 · openalex updated_date 2026/07/28 · crossref indexed 2026/08/04
In mixed characteristic and in equal characteristic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> we define a filtration on topological Hochschild homology and its variants. This filtration is an analogue of the filtration of algebraic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> -theory by motivic cohomology. Its graded pieces are related in mixed characteristic to the complex <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>Ω</mml:mi> </mml:mrow> </mml:math> constructed in our previous work, and in equal characteristic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> to crystalline cohomology. Our construction of the filtration on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>THH</mml:mi> </mml:math> is via flat descent to semiperfectoid rings. As one application, we refine the construction of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mi>Ω</mml:mi> </mml:mrow> </mml:math> -complex by giving a cohomological construction of Breuil–Kisin modules for proper smooth formal schemes over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>𝒪</mml:mi> <mml:mi>K</mml:mi> </mml:msub> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> is a discretely valued extension of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>𝐐</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> with perfect residue field. As another application, we define syntomic sheaves <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>𝐙</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> for all <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> on a large class of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>𝐙</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> -algebras, and identify them in terms of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> -adic nearby cycles in mixed characteristic, and in terms of logarithmic de Rham-Witt sheaves in equal characteristic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>p</mml:mi> </mml:math> .