Motives beyond A1-homotopy invariance

Información del proyecto

MbHI

Identificador del acuerdo de subvención: 896517

DOI

10.3030/896517

Proyecto cerrado

Fecha de la firma de la CE 9 Abril 2020

Fecha de inicio 1 Octubre 2020

Fecha de finalización 30 Septiembre 2022

Financiado con arreglo a

• EXCELLENT SCIENCE - Marie Skłodowska-Curie Actions

Coste total

€ 207 312,00

Aportación de la UE

€ 207 312,00

Coordinado por

KOBENHAVNS UNIVERSITET
Denmark

Periodic Reporting for period 1 - MbHI (Motives beyond A1-homotopy invariance)

Período documentado: 2020-10-01 hasta 2022-09-30

Resumen del contexto y de los objetivos generales del proyecto

Over the last couple of decades, the A1-local motivic homotopy theory initiated by Voevodsky has been creating quite a stir among the mathematics community. Combining components of algebra and topology, it studies algebraic varieties from a homotopy theoretic viewpoint. However, there is a fundamental issue in Voevodsky’s theory, namely, it depends on A1-homotopy theory and thus neglects infinitesimal quantity, which is essential for algebraic and arithmetic geometry. The objective of this project is to overcome this gap, that is, to develop a motivic homotopy theory beyond A1-homotopy invariance.

Trabajo realizado desde el comienzo del proyecto hasta el final del período abarcado por el informe y los principales resultados hasta la fecha

The first and most essential difficulty in carrying out the project’s objectives was to construct a “correct” category of motives. One of our main achievements is the construction of a nice category MSp with promising evidence that it is correct, where MSp stands for “motivic spectra”. All relevant cohomology theories in algebraic geometry, including étale cohomology, crystalline cohomology, syntomic cohomology, and algebraic K-theory, are representable in MSp, and thus it provides a unified way to study those cohomology theories, realizing the philosophy of motives. Then we have obtained several useful equivalences in MSp, in particular, an equivalence of the n-th grassmannian and the stack of rank n vector bundles is proved. Furthermore, it was applied to algebraic K-theory, establishing a beautiful new characterization of algebraic K-theory, namely, it is universal among Zariski sheaves of spectra which admit an action of the Picard stack and satisfy projective bundle formula.

Avances que van más allá del estado de la técnica e impacto potencial esperado (incluida la repercusión socioeconómica y las implicaciones sociales más amplias del proyecto hasta la fecha)

The results described above are certainly progress beyond the state of the art and open a new gateway to motivic homotopy theory.

Universality of K-theory