On the closed subschemes of a scheme

TL;DR

This paper introduces algebraic operations on closed subschemes of a scheme, defining addition and multiplication that correspond to union and intersection, and shows these form functors to commutative monoids.

Contribution

It defines natural addition and multiplication on closed subschemes, linking them to scheme-theoretic union and intersection, and establishes functorial properties.

Findings

Multiplication coincides with scheme-theoretic intersection.

Addition coincides with scheme-theoretic union.

These structures form contravariant functors to commutative monoids.

Abstract

In this paper, we obtain some new results on closed subschemes. Specially, we define natural addition and multiplication on the closed subschemes of a scheme. It is shown that "the multiplication" precisely coincides with the well known notion of "the scheme-theoretic intersection". Dually, "the addition" coincides with "the scheme-theoretic union". It is also proved that these structures naturally provide contravariant functors from the category of schemes to the category of commutative monoids.