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. 

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References

SHOWING 1-10 OF 35 REFERENCES

Triangulated categories of logarithmic motives over a field

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

  • F. BindaS. Saito
  • 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

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

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

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

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