09 décembre 2020
Matthew Morrow est lauréat du programme ERC Consolidator Grant 2020 pour son projet ’Motivic Cohomology of Schemes’.
The project belongs to the field of arithmetic algebraic geometry and is centred around algebraic K-theory, motivic cohomology, and topological cyclic homology. The overall goal is to develop a general theory of motivic cohomology for arbitrary schemes, extending the existing theory of Bloch, Levine, Suslin, Voevodsky, and others in the special case of smooth algebraic varieties. This will describe non-connective algebraic K-theory via an Atiyah--Hirzebruch spectral sequence. The project relies on very recent breakthroughs in algebraic K-theory and topological cyclic homology In the case of singular algebraic varieties, our goal will be to develop a theory of motivic cohomology which both satisfies singular analogous of the Beilinson--Lichtenbaum conjectures and is also compatible with the trace maps to negative cyclic and topological cyclic homology. Its properties will refine those of K-theory in the presence of singularities; for example, we will study a motivic refinement of Weibel's vanishing conjecture and a theory of ``infinitesimal motivic cohomology'' satisfying cdh descent. In the case of regular arithmetic schemes we will propose a new approach to the theory of p-adic motivic cohomology, based on topological cyclic homology and syntomic cohomology, which works in much greater generality than previous approaches. Perfectoid techniques will play an important role and we will establish the p-adic Beilinson--Lichtenbaum and Bloch--Kato conjectures.
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