# Rigidity theorem for presheaves with Witt-transfers

## TL;DR

This paper proves a rigidity theorem for homotopy invariant presheaves with Witt-transfers over fields of characteristic not 2, establishing isomorphisms between local and residue presheaf values, with applications to derived Witt-groups.

## Contribution

It introduces a rigidity theorem for presheaves with Witt-transfers, extending known results to this class and deriving corollaries for derived Witt-groups.

## Key findings

- Isomorphism between presheaf values at henselian neighborhoods and residue points.
- Rigidity theorem applies to homotopy invariant presheaves with Witt-transfers.
- Corollary results for derived Witt-groups $W^i(-)$.

## Abstract

The rigidity theorem for homotopy invariant presheaves with Witt-transfers on the category of smooth affine varieties over a field $k$ with characteristic not equal to 2 is proved. Namely for such a presheaf $\mathcal F$ the isomorphism $\mathcal F(U)\simeq \mathcal F(x)$ where $U$ is henseliation of a variety at smooth closed point with separable residue field (over $k$) is proved. The rigidity for presheaves $W^i(X\times -)$ where $X$ is smooth variety and $W^i(-)$ are derived Witt-groups ($i\in \mathbb Z/4\mathbb Z$) follows as corollary.