On the effectivity of spectra representing motivic cohomology theories

TL;DR

This paper establishes a criterion for spectra to be effective in motivic homotopy theory and demonstrates the equivalence of two recent generalized motivic cohomology theories over an infinite perfect field.

Contribution

It provides a general effectiveness criterion for spectra in the motivic stable homotopy category and proves the equivalence of two recent motivic cohomology theories.

Findings

A criterion for effectiveness of spectra in motivic homotopy theory

Proof that two recent generalized motivic cohomology theories coincide

Enhanced understanding of the structure of motivic spectra

Abstract

Let k be an infinite perfect field. We provide a general criterion for a spectrum in the stable homotopy category over k to be effective, i.e. to be in the localizing subcategory generated by the suspension spectra of smooth schemes. As a consequence, we show that two recent versions of generalized motivic cohomology theories coincide.