Standard Vector Bundles

TL;DR

This paper develops a new categorical framework for vector bundles over schemes, emphasizing strict properties of tensor products and direct sums, which simplifies their algebraic manipulation.

Contribution

It introduces categories of standard vector bundles with strict tensor product and direct sum operations, enhancing the algebraic structure and functoriality compared to traditional approaches.

Findings

Categories of standard vector bundles are equivalent to classical categories.

Tensor product is strictly associative and commutative with line bundles.

Operations are strictly functorial on base change.

Abstract

We construct the categories of standard vector bundles over schemes and define direct sum and tensor product. These categories are equivalent to the usual categories of vector bundles with additional properties. The tensor product is strictly associative, strictly commutative with line bundles, and strictly functorial on base change.