Di. 05.09.2023, Oberseminar Algebra

Thema: A Foundation for Synthetic Algebraic Geometry Referent: Dr. Felix Cherubini (Chalmers/University of Gothenburg) Raum: ENC-D 201, Zeit: 11:00 Uhr s.t.

Oberseminar Algebra

A Foundation for Synthetic Algebraic Geometry.

Dienstag, den 5. September 2023, 11:00 Uhr s.t., Raum ENC-D 201 ,

Referent: Dr. Felix Cherubini
  (Chalmers/University of Gothenburg)

Abstrakt:

We construct a novel candidate foundation for algebraic geometry, de- veloped in an internal language of the Zariski topos. Varying the rules of mathematical practice in this way, admits a reduction of the complexity of the algebraic geometry we have explored so far, at the price of a very unfamiliar perspective. We build on existing work of Anders Kock and Ingo Blechschmidt. The Zariski topos consists of sheaves on the site opposite to the category of finitely presented algebras over a fixed ring, with the Zariski topology. Classically, this category appears in the functor of points approach to algebraic geometry, which is of a similar flavor as our setup. We deviate from previous work in this direction by using homotopy type theory as the internal language of a (higher) Zariski topos. One of our main contributions is the use of higher types to define and reason about cohomology. Actually computing cohomology groups, seems to need a principle along the lines of our SZariski local choiceäxiom, which is justified as well, as the other axioms we use, by a cubical model of homotopy type theory. This work is the basis of a larger ongoing project, which has an accompanying formalization project. We will briefly report on these projects. .