On intersection theory, Severi-Brauer varieties, and the intersection theory of Severi-Brauer varieties

• Author / Creator

  Mackall, Eoin

• This thesis investigates the Chow ring, and neighboring functors, of a Severi-Brauer variety. The approach taken here heavily depends on the computation of lower K-groups of a Severi-Brauer variety.We construct a functor (for an arbitrary scheme essentially of finite type over a field) that is a universal target for additive Chern classes and we compare this functor to the associated graded for the gamma filtration on the Grothendieck group of locally free sheaves via a Grothendieck-Riemann-Roch type theorem. When the Chow ring is generated by Chern classes our theorem reduces to the standard Grothendieck-Riemann-Roch.Following this we show that, for some Severi-Brauer varieties including the generic ones, the Chow ring is isomorphic with the associated graded of the gamma filtration on the Grothendieck ring. The theorem more generally involves Severi-Brauer varieties whose Chow rings are generated by Chern classes and whose associated algebra has index and exponent that differ very minimally (in the language of this section, for algebras of level 1). This prompts us to investigate the gamma filtration in its own right. We prove some results about the gamma filtration for a Severi-Brauer variety including results showing the gamma filtration depends only on primary division algebra factors of the central simple algebra of the Severi-Brauer variety.Lastly, we continue work on the picture for the diagonal K-cohomology groups which can be considered in degree one higher than the Chow ring. By assuming the vanishing of reduced Whitehead groups for certain algebras with equal index and exponent, we provide a complete description of the coniveau filtration on the first K-group in some cases.

• Subjects / Keywords

  • Chow ring
  • projective homogeneous varieties

• Graduation date

  Fall 2019

• Type of Item

  Thesis

• Degree

  Doctor of Philosophy

• DOI

  https://doi.org/10.7939/r3-0gps-fc31

• License

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