A Foundation for Synthetic Algebraic Geometry

TL;DR

This paper develops a foundational framework for algebraic geometry within the Zariski topos using homotopy type theory, introducing higher types for cohomology and proposing axioms justified by a cubical model.

Contribution

It introduces a homotopical approach to algebraic geometry within the Zariski topos, utilizing higher types and axioms for cohomology computation.

Findings

Use of higher types to define and reason about cohomology.

Introduction of the 'Zariski local choice' axiom for cohomology calculation.

Justification of axioms via a cubical model of homotopy type theory.

Abstract

This is a foundation for algebraic geometry, developed internal to the Zariski topos, building on the work of Kock and 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, i.e. generating covers are given by localization maps $A\to A_{f_1}$ for finitely many elements $f_1,\dots,f_n$ that generate the ideal $(1)=A\subseteq A$. We use homotopy type theory together with three axioms as the internal language of a (higher) Zariski topos. One of our main contributions is the use of higher types -- in the homotopical sense -- to define and reason about cohomology. Actually computing cohomology groups, seems to need a principle along the lines of our ``Zariski local choice'' axiom, which we justify as well as the other axioms using a cubical model of homotopy type theory.