Ample line bundles, global generation and $K_0$ on quasi-projective derived schemes

TL;DR

This paper extends classical results on ample line bundles and $K_0$ groups to quasi-projective derived schemes, showing vector bundles can be globally generated after twisting and providing a presentation of $K^0(X)$.

Contribution

It generalizes key properties of vector bundles and $K$-theory from classical to derived algebraic geometry for quasi-projective schemes.

Findings

Vector bundles on derived schemes with ample line bundles can be twisted to be globally generated.

$K^0(X)$ can be presented as the Grothendieck group of vector bundles modulo exact sequences.

Results extend classical algebraic geometry properties to the derived setting.

Abstract

The purpose of this note is to extend some classical results on quasi-projective schemes to the setting of derived algebraic geometry. Namely, we want to show that any vector bundle on a derived scheme admitting an ample line bundle can be twisted to be globally generated. Moreover, we provide a presentation of $K^0(X)$ as the Grothendieck group of vector bundles modulo exact sequences on any quasi-projective derived scheme $X$.