# A motivic homotopy theory without $$\mathbb {A}^{1}$$-invariance

@article{Binda2019AMH,
title={A motivic homotopy theory without \$\$\mathbb \{A\}^\{1\}\$\$-invariance},
author={Federico Binda},
journal={Mathematische Zeitschrift},
year={2019}
}
• F. Binda
• Published 5 September 2019
• Mathematics
• Mathematische Zeitschrift
In this paper, we continue the program initiated by Kahn–Saito–Yamazaki by constructing and studying an unstable motivic homotopy category with modulus $$\overline{\mathbf{M}}\mathcal {H}(k)$$, extending the Morel–Voevodsky construction from smooth schemes over a field k to certain diagrams of schemes. We present this category as a candidate environment for studying representability problems for non $$\mathbb {A}^1$$-invariant generalized cohomology theories.
2 Citations

### Motives with modulus, III: The categories of motives

• Mathematics, Materials Science
Annals of K-Theory
• 2022
We construct and study a triangulated category of motives with modulus $\mathbf{MDM}_{\mathrm{gm}}^{\mathrm{eff}}$ over a field $k$ that extends Voevodsky's category

## References

SHOWING 1-10 OF 35 REFERENCES

### Triangulated categories of logarithmic motives over a field

• Mathematics
• 2020
In this work we develop a theory of motives for logarithmic schemes over fields in the sense of Fontaine, Illusie, and Kato. Our construction is based on the notion of finite log correspondences, the

### RELATIVE CYCLES WITH MODULI AND REGULATOR MAPS

• Mathematics
Journal of the Institute of Mathematics of Jussieu
• 2017
Let $\overline{X}$ be a separated scheme of finite type over a field $k$ and $D$ a non-reduced effective Cartier divisor on it. We attach to the pair $(\overline{X},D)$ a cycle complex with modulus,

### On 0-cycles with modulus

Given a smooth surface $X$ over a field and an effective Cartier divisor $D$, we provide an exact sequence connecting $CH_0(X,D)$ and the relative $K$-group $K_0(X,D)$. We use this exact sequence to

### A cycle class map from Chow groups with modulus to relative $K$-theory

Let $\bar{X}$ be a smooth quasi-projective $d$-dimensional variety over a field $k$ and let $D$ be an effective Cartier divisor on it. In this note, we construct cycle class maps from (a variant of)

### A module structure and a vanishing theorem for cycles with modulus

• Mathematics
• 2014
We show that the higher Chow groups with modulus of Binda-Kerz-Saito for a smooth quasi-projective scheme $X$ is a module over the Chow ring of $X$. From this, we deduce certain pull-backs, the

### Algebraic cycles and crystalline cohomology

• Mathematics
• 2015
We show that additive higher Chow groups on smooth varieties over a field of characteristic $p \not = 2$ induce a Zariski sheaf of pro-differential graded algebras, whose Milnor range is isomorphic

### Zero cycles with modulus and zero cycles on singular varieties

• Mathematics
Compositio Mathematica
• 2017
Given a smooth variety $X$ and an effective Cartier divisor $D\subset X$ , we show that the cohomological Chow group of 0-cycles on the double of $X$ along $D$ has a canonical decomposition in terms

### Symmetric monoidal structure on non-commutative motives

• Mathematics
Journal of K-Theory
• 2011
In this article we further the study of non-commutative motives, initiated in [12, 43]. Our main result is the construction of a symmetric monoidal structure on the localizing motivator Motlocdg of