Some computations in the heart of the homotopy t-structure on logarithmic motives

TL;DR

This paper introduces a method for computing the zeroth homotopy group of effective log motives of smooth proper varieties over perfect fields, demonstrating its -invariance and implications for the stripping functor's full faithfulness.

Contribution

It provides a novel computational approach for -homotopy groups in logarithmic motives and establishes the full faithfulness of the stripping functor from log motives to Nisnevich sheaves.

Findings

The -invariance of the -homotopy group of effective log motives.

Explicit computation of -homotopy groups for -projective space.

Full faithfulness of the stripping functor from log motives to Nisnevich sheaves.

Abstract

In this note we will illustrate a method for computing the $\pi_0$ of the effective log motive of a smooth and proper variety over a perfect field $k$ and show that it is $\mathbf{A}^1$-invariant. We will apply this to compute the first homotopy groups of $\mathbf{P}^1$ to show that the stripping functor from log motivic sheaves to (usual) Nisnevich sheaves with transfers is fully faithful.