Abstract: Complex algebraic geometry was born in the 19^th century over. Over the next century it was extended to any base field, such as finite fields, with in a polished treatment appearing in Weil’s Foundations in 1946. With the advent of scheme theory in the late 1950s, and the work of Grothendieck that followed, a fully syntactic approach to algebraic geometry became available. This allowed the base to be extended from any field to any ring, which is to say any algebra over Z. This led to an explosion in the arithmetic theory of polynomial equations which continues to this day.

There are some good signs, however, that a full syntactic theory of algebraic geometry can be developed over any semiring, or equivalently any algebra over the natural numbers, which is the most basic number system of them all. The point is to remove the crutch of subtraction from the foundations of algebraic geometry. At some places this is easy, and at others it isn’t.

For instance, without subtraction it’s not clear how to define syntactically the complement of a hypersurface defined by an equation f=g. This is important in Galois theory or covering space theory when we want to remove the branch locus. Another example is with smoothness, differentials, and deformation theory, which are all built on the notion of two elements having a nilpotent difference.

In this talk, I’ll explain how “complements of presheaves” give solutions (or probable solutions!) to these questions. In particular, I’ll give an example (the first!) of a nontrivial finite etale extension of “genuine” semirings.