Title: Scheme theory over semirings and narrow class groups

Abstract: Usual scheme theory can be viewed as the syntactic theory of polynomial equations with integer coefficients. But none of its most fundamental ingredients, such as faithfully flat descent, require subtraction. So we can set up a scheme theory over semirings ("rings but not necessarily with subtraction", such as the non-negative integers or reals), thus bringing positivity in to the foundations of scheme theory. It is reasonable to view non-negativity as integrality at the infinite place, the Boolean semiring as the residue field, and the non-negative reals as the completion.

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.