Non-commutative nature of $\ell$-adic vanishing cycles

TL;DR

This paper explores the non-commutative aspects of $\, ext{l}$-adic vanishing cycles in algebraic geometry, linking singularity categories with $\, ext{l}$-adic cohomology and computing related K-theory invariants.

Contribution

It establishes a novel connection between the singularity category of the special fiber and $\, ext{l}$-adic vanishing cohomology, and computes associated algebraic K-theory with compact support.

Findings

Singularity category recovers $\, ext{l}$-adic vanishing cohomology.

Computed homotopy-invariant algebraic K-theory for certain embeddings.

Established link between non-commutative geometry and $\, ext{l}$-adic cohomology.

Abstract

Let $p:X \rightarrow S$ be a flat, proper and regular scheme over a strictly henselian discrete valuation ring. We prove that the singularity category of the special fiber with its natural two-periodic structure allows to recover the $\ell$-adic vanishing cohomology of $p$. Along the way, we compute homotopy-invariant non-connective algebraic K-theory with compact support of certain embeddings $X_t \hookrightarrow X_T$ in terms of the motivic realization of the dg category of relatively perfect complexes.