Families of algebraic varieties parametrized by topological spaces

TL;DR

This paper explores families of algebraic varieties parametrized by topological spaces, extending classical algebraic geometry results and establishing an equivalence between certain sheaf categories.

Contribution

It generalizes classical results like Hilbert Nullstellensatz and primary decomposition to parametrized families and proves an equivalence between bivariant coherent sheaves and sheaves of finitely generated modules.

Findings

Generalization of Hilbert Nullstellensatz to parametrized families

Extension of primary decomposition results

Equivalence between bivariant coherent sheaves and finitely generated module sheaves

Abstract

We study families of algebraic varieties parametrized by topological spaces and generalize some classical results such as Hilbert Nullstellensatz and primary decomposition of commutative rings. We show that there is an equivalence between the category of bivariant coherent sheaves and the category of sheaves of finitely generated modules.