  We develop algebraic geometry for general Segal's Gamma-rings and show that
this new theory unifies two approaches we had considered earlier on (for a
geometry under Spec Z). The starting observation is that the category obtained
by gluing together the category of commutative rings and that of pointed
commutative monoids, that we used in our previous work to define F1-schemes, is
naturally a full subcategory of the category of Segal's Gamma-rings. In this
paper we develop the affine case of this general algebraic geometry: one
distinctive feature is that the spectrum Spec(A) of a Gamma-ring is in general
a Grothendieck site rather than a point set endowed with a topology. Two
striking features of this new geometry are that it is the natural domain for
cyclic homology and for homological algebra, and that new operations, which do
not make sense in ordinary algebraic geometry, are here available. For
instance, in this new context, the quotient of a ring by a multiplicative
subgroup is still a Gamma-ring to which our general theory applies. Thus the
adele class space gives rise naturally to a Gamma-ring. Finally, we show that
our theory is not a special case of the T\"oen-Vaqui\' e general theory of
algebraic geometry under Spec Z.
