Periodic Reporting for period 1 - RRMAP (Riemann-Roch and motives for arithmetic problems)
Reporting period: 2021-01-01 to 2022-12-31
Mathematics was born in Greece 2500 years ago with the study of Geometry. Since then, Geometry has been at the heart of Western civilization and its Philosophy, as Plato prescribed at the Academia. Since then, our society has pursued to unveil the secrets of Geometry.
The development of Geometry, following Gauss, Riemann and Grothendieck, has been instrumental in our modern understanding of our world. From Einstein's General Relativity to the expected formalizations in a unifying theory of the macro and micro study of the world we live in, we rely on geometric tools to describe the phenomena we observe. The
The current research project is framed in the area of Arithmetic Geometry, the part of Mathematics which applies techniques and ideas borrowed from Algebraic Geometry to Number Theory problems, which should be considered as the direct heir to Greek reflections on Geometry. More concretely, during the nineties, arithmetic geometers of the previous century have successfully developed a satisfactory homotopy theory, called motivic homotopy theory. This new framework has been successful enough to provide the motivic analog of Grothendieck's six functors formalism, the basic formalism for satisfactory cohomology theory. These are the modern tools mathematicians use to define "shape" and Many Number Theory problems are related to cohomology theories such as higher K-theory or the Chow ring. A central theorem of the subject is the celebrated Grothendieck's Riemann-Roch theorem and Gillet's extension to higher K-theory. This theorem compares the direct image in cohomology of K-theory and the Chow
ring. Combined with regulators, the Riemann-Roch theorem provides formulas describing both arithmetic and geometric properties. In this research project, I propose to develop Riemann-Roch-type theorems and motivic techniques to attack arithmetic problems.
The development of Geometry, following Gauss, Riemann and Grothendieck, has been instrumental in our modern understanding of our world. From Einstein's General Relativity to the expected formalizations in a unifying theory of the macro and micro study of the world we live in, we rely on geometric tools to describe the phenomena we observe. The
The current research project is framed in the area of Arithmetic Geometry, the part of Mathematics which applies techniques and ideas borrowed from Algebraic Geometry to Number Theory problems, which should be considered as the direct heir to Greek reflections on Geometry. More concretely, during the nineties, arithmetic geometers of the previous century have successfully developed a satisfactory homotopy theory, called motivic homotopy theory. This new framework has been successful enough to provide the motivic analog of Grothendieck's six functors formalism, the basic formalism for satisfactory cohomology theory. These are the modern tools mathematicians use to define "shape" and Many Number Theory problems are related to cohomology theories such as higher K-theory or the Chow ring. A central theorem of the subject is the celebrated Grothendieck's Riemann-Roch theorem and Gillet's extension to higher K-theory. This theorem compares the direct image in cohomology of K-theory and the Chow
ring. Combined with regulators, the Riemann-Roch theorem provides formulas describing both arithmetic and geometric properties. In this research project, I propose to develop Riemann-Roch-type theorems and motivic techniques to attack arithmetic problems.
Intensive work around the Riemann-Roch theorem has been carried out during the period of the Marie-Curie IF grant. In particular, we have achieved a new proof of the Riemann-Roch theorem via Spanier-Whitehead duality was found. In addition, a rewriting proof of the Adams Riemann-Roch in the context of orientation theory was carried out, leading to a new statement for closed immersions. This result solves a question posed by Grothendieck at the landmark SGA 6, asking for a proof of the Riemann-Roch theorem without factorizing the morphism into a closed immersion and a projection. Also, inspired by the bivariant formalism of the Hochschild homology developed by Alonso-Jeremías-Lipman a new Riemann-Roch type theorem for the endomorphism of a cohomology has been proven
The impact of this project is purely scientific: the development of intersection theory tools with integral coefficients as well as the study of the relationship between K-theory and motivic cohomology. Several results have been proven, and new techniques in the study of cohomology haven been developped.