MbHI · Motives beyond A1-homotopy invariance
Horizon 2020 — Marie Skłodowska-Curie Actions
- Duration
- 2020-10-01 → 2022-09-30
- EU contribution
- €207,312
- Participants
- 1
- Scheme
- MSCA-IF
Lines connect the coordinator with its partners.
Results in brief
Motives beyond A1-homotopy invariance
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.
Data: CORDIS, © European Union
Project objective
The proposed project is aimed at establishing new foundations of motivic homotopy theory, which enhances Voevodsky's motivic homotopy theory. Voevodsky's motivic homotopy theory is based on A1-homotopy theory, and thus it cannot capture non A1-homotopy invariant phenomena in algebraic geometry such as algebraic K-theory (for singular varieties), topological cyclic homology, logarithmic cohomology, deformation theory, (wild) ramification theory, and so on. Our new foundation is based on projective bundle formula instead of A1-homotopy invariance, so that it has a potential to capture aforementioned non A1-homotopy invariant phenomena. To overcome fundamental difficulties to use projective bundle formula as an input of homotopy theory, we use ``derived correspondence'', which is a derived version of framed correspondence. Another key input is the notion of derived blow-ups, which was used by Kerz, Strunk and Tamme to solve Weibel's conjecture. This project consists of the construction of a new motivic homotopy category and its applications. Applications would include a construction of motivic cohomology (for possibly singular varieties) together with a motivic spectral sequence to algebraic K-theory (Beilinson's conjecture), motivic interpretation of topological cyclic homology, and motivic interpretation of logarithmic cohomology.
Original text from CORDIS.
Participants
- KOBENHAVNS UNIVERSITET · KOBENHAVNCoordinatorDenmark
Links
Data: CORDIS, © European Union