Dear Eduardo and everybody: In one of your papers you used the term Nullstellensatz for a special case (in some sense an "algebraically closed"case). I propose to use that term in this more general case. The parameters for various traditional cases can be perhaps expessed by an essential connected morphism of toposes E->S. That is, a full inclusion of "relatively discrete" into "relatively continuous" which has both left adjoint ("connected components") and right adjoint ("points"). In that context there is a natural map from points to components; if it is epic, we can say that the Nullstellensatz holds for E->S. If S is just the category of abstract sets, one could think of E as algebraically closed if the Nullstellensatz holds. But as seems implicit in Galois theory, for algebraic geometry over a non-algebraically closed K, the appropriate base topos S consists not of abstract sets, but rather of sheaves on C = the opposite of the category of finite extensions of K, with every map covering. If E is the topos of sheaves on (finitely generated K-algebras )^op with respect to a topology that restricts to the above on C, I believe we have a classical example of both your formulation and mine. Bill PS There are other stronger results that also could be called Nullstellensatz, involving another topos F between E and S, such as the one generated by algebras that are finite dimensional as K-vector spaces, or one suggested by Birkhoff's SDI theorem. What is the appropriate statement for these results ? Quoting Eduardo Dubuc <edubuc@dm.uba.ar>:
hello:
Given a set CC of objects in a topos EE, consider the following property:
" X no= empty iff exists C \in CC, hom(C, X) no= empty "
example; CC = a set of generators
Has (this property) already a name ?
If not, can you suggest one ?
Any answer will be welcome.
(Notice that if CC is a set of points (instead of objects) we say that there are enough points)
Thanks Eduardo J. Dubuc