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As a Marie S. Curie postdoctoral fellow I worked on establishing new foundations of motivic homotopy theory, which enhances Voevodsky's motivic homotopy theory. 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 has been 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 was to overcome this gap, that is, to develop a motivic homotopy theory beyond A1-homotopy invariance.
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 was 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.
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 was 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 used ``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 consisted 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. These results are progress beyond the state of the art and open a new gateway to motivic homotopy theory.
I was supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 896517.