  We present a formalization of quasi-compact and quasi-separated schemes
(qcqs-schemes) in the Cubical Agda proof assistant. We follow Grothendieck's
functor of points approach, which defines schemes, the quintessential notion of
modern algebraic geometry, as certain well-behaved functors from commutative
rings to sets. This approach is often regarded as conceptually simpler than the
standard approach of defining schemes as locally ringed spaces, but to our
knowledge it has not yet been adopted in formalizations of algebraic geometry.
We build upon a previous formalization of the so-called Zariski lattice
associated to a commutative ring in order to define the notion of compact open
subfunctor. This allows for a concise definition of qcqs-schemes, streamlining
the usual presentation as e.g. given in the standard textbook of Demazure and
Gabriel. It also lets us obtain a fully constructive proof that compact open
subfunctors of affine schemes are qcqs-schemes.
