  The usual dictionary between geometry and commutative algebra is not
appropriate for Arithmetic geometry because addition is a singular operation at
the "Real prime". We replace Rings, with addition and multiplication, by Props
(=strict symmetric monoidal category generated by one object), or by Bioperad
(=two closed symmetric operads acting on each other): to a ring we associate
the prop of all matrices over it, with matrix multiplication and block direct
sums as the basic operations, or the bioperad consisting of all raw and column
vectors over it. We define the "commutative" props and bioperads, and using
them we develop a generalized algebraic geometry, following Grothendieck
footsteps closely. This new geometry is appropriate for Arithmetic (and
potentially also for Physics).
