Abstract: 

Usual scheme theory can be viewed as the syntactic theory of polynomial equations with coefficients in a ring. But none of its most fundamental ingredients, such as faithfully flat descent, require subtraction. So we can set up a scheme theory over semirings (i.e. ``rings but possibly without additive inverses’’, such as the non-negative integers or reals). This brings positivity phenomena in to the foundations of scheme theory. One may think of non-negativity as integrality at the infinite place, the Boolean semiring as its residue field, and the non-negative reals as the completion there.

In this talk, I'll discuss some recent developments in module theory over semirings. While the classical definitions of ``vector bundle'' are not all equivalent over semirings, the classical definitions of ``line bundle'' are all equivalent, which allows us to define Picard groups and Picard stacks. The narrow class group of a number field can be recovered as the reflexive class group of the semiring of its totally nonnegative integers. This gives a scheme-theoretic definition of the narrow class group, as was done for the ordinary class group a long time ago.

I’ll also give a number of basic open questions.

This is based on arXiv:2405.18645, which is joint work with Jaiung Jun, and on a forthcoming paper with Johan de Jong and Ivan Zelich.