# Triangulated categories of logarithmic motives over a field

@article{Binda2020TriangulatedCO, title={Triangulated categories of logarithmic motives over a field}, author={Federico Binda and Doosung Park and Paul arne Ostvaer}, journal={arXiv: Algebraic Geometry}, year={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 dividing Nisnevich topology on log schemes, and the idea of parameterizing homotopies by $\overline{\square}$, i.e. the projective line with respect to its compactifying logarithmic structure at infinity. Hodge cohomology of log schemes is an example of an $\overline{\square}$-invariant theory that… Expand

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#### References

SHOWING 1-10 OF 110 REFERENCES

Triangulated Categories of Mixed Motives

- Mathematics
- Springer Monographs in Mathematics
- 2019

This book discusses the construction of triangulated categories of mixed motives over a noetherian scheme of finite dimension, extending Voevodsky's definition of motives over a field. In particular,… Expand

Étale motives

- Mathematics
- Compositio Mathematica
- 2015

We define a theory of étale motives over a noetherian scheme. This provides a system of categories of complexes of motivic sheaves with integral coefficients which is closed under the six operations… Expand

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,… Expand

Log Betti cohomology, log étale cohomology, and log de Rham cohomology of log schemes over C

- Mathematics
- 1999

where Ωχ/c is the de Rham complex of X ([Gr2]). In this paper, we prove generalizations of these results to schemes over C endowed with logarithmic structures in the sense of Fontaine-Illusie. Let X… Expand

Triangulated Categories of Motives over fs Log Schemes

- Mathematics
- 2016

Author(s): Park, Doosung | Advisor(s): Olsson, Martin | Abstract: This thesis is devoted to constructing the triangulated categories of motives over fs log schemes with rational coefficients and… Expand

Torsion and divisibility for reciprocity sheaves and 0-cycles with modulus

- Mathematics
- 2015

The notion of modulus is a striking feature of Rosenlicht-Serre's theory of generalized Jacobian varieties of curves. It was carried over to algebraic cycles on general varieties by Bloch-Esnault,… Expand

Motivic complexes and special values of zeta functions

- Mathematics
- 2013

Beginning with the conjecture of Artin and Tate in 1966, there has been a series of successively more general conjectures expressing the special values of the zeta function of an algebraic variety… Expand

Introduction to toric varieties

- Mathematics
- 2004

The course given during the School and Workshop “The Geometry and Topology of Singularities”, 8-26 January 2007, Cuernavaca, Mexico is based on a previous course given during the 23o Coloquio… Expand

Motivic complexes of Suslin and Voevodsky

- Mathematics
- 1997

In this report we sketch some of the insights and consequences of recent work by Andrei Suslin and Vladimir Voevodsky concerning algebraic K-theory and motivic cohomology. We can trace these… Expand

Modules over motivic cohomology

- Mathematics
- 2008

Abstract For fields of characteristic zero, we show that the homotopy category of modules over the motivic ring spectrum representing motivic cohomology is equivalent to Voevodsky's big category of… Expand