Motives beyond A1-homotopy invariance

ended

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.