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

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.