Motivic Steenrod problem away from the characteristic

TL;DR

The paper proves that the motivic Steenrod property can fail for singular varieties away from the characteristic, by constructing explicit obstruction classes in motivic cohomology that do not lift to algebraic cobordism and cannot be realized as pushforwards along projective derived-lci maps. Leveraging motivic Milnor operations and a suspension-into-singularity technique, it shows CH^n of some singular schemes cannot be captured by the universal precobordism ring, thereby ruling out a simple cycle-theoretic model for the Chow ring in the singular setting. The work clarifies the relationship between motivic cohomology, algebraic cobordism, and Chow rings, and raises open questions about cycle-theoretic models and transfers, as well as the behavior at the characteristic.

Abstract

In topology, the Steenrod problem asks whether every singular homology class is the pushforward of the fundamental class of a closed oriented manifold. Here, we introduce an analogous question in algebraic geometry: is every element on the Chow line of the motivic cohomology of $X$ the pushforward of a fundamental class along a projective derived-lci morphism? If $X$ is a smooth variety over a field of characteristic $p \geq 0$, then a positive answer to this question follows up to $p$-torsion from resolution of singularities by alterations. However, if $X$ is singular, then this is no longer necessarily so: we give examples of motivic cohomology classes of a singular scheme $X$ that are not $p$-torsion and are not expressible as such pushforwards. A consequence of our result is that the Chow ring of a singular variety cannot be expressed as a quotient of its algebraic cobordism ring, as suggested by the first-named-author in his thesis.

Theorems & Definitions (92)

• Theorem : Standard
• Theorem 1.2: Main theorem
• Remark 1.3: Proper maps versus projective maps
• Theorem 2.1
• Theorem 2.2
• Remark 2.3
• Theorem 2.4: Tang
• Definition 2.5
• Lemma 2.6
• proof