On Canonical Homomorphisms of Tensor Sheaves

TL;DR

This paper introduces canonical homomorphisms between tensor sheaves of Schur type, generalizing linear algebra results to algebraic geometry, and constructs complexes that can become split exact sequences under certain conditions.

Contribution

It defines tensor modules of Schur type and constructs canonical homomorphisms between them, extending classical linear algebra results to sheaves in algebraic geometry.

Findings

Constructed canonical homomorphisms between tensor sheaves

Established conditions for isomorphisms when sheaves are locally free

Developed canonical complexes that can be split exact sequences

Abstract

In this paper we define tensor modules(sheaves) of Schur type,or of generalized Schur type associated with the give module(sheaf), using the so-called Schur functors. Then using global method we construct canonical homomorphisms between these modules(sheaves). We will get canonical isomorphisms if the original sheaf is locally free using idea of algebraic geometry, which is in fact a generalization of result in linear algebra. In the final section, we give canonical complexes using homomorphisms constructed before, and these complexes will become split exact sequence if further condition holds. And we could use local method to give concrete descriptions of these canonical homomorphisms.