On the non-existence of Tensor Products of Algebraic Cycles

TL;DR

This paper investigates the possibility of extending the tensor product map from line bundles to higher codimension algebraic cycles, proving such an extension cannot exist beyond codimension 1.

Contribution

It constructs an extension of the tensor product map for codimension 1 cycles and proves the non-existence of such an extension in higher codimensions.

Findings

Extension exists for codimension 1 cycles.

No extension exists for codimension greater than 1.

Provides theoretical proof of non-existence in higher codimensions.

Abstract

Let $\otimes$ be the map which classifies the tensor product of two line bundles, an extension of this map to the space of all codimension 1 algebraic cycles is constructed. It is proved that this extension cannot exist in codimension greater than 1.