  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.
