Tensor Product

The tensor product of two modules is the universal recipient of bilinear maps, converting bilinear constructions into linear ones and enabling change of scalars between rings.

Definition

The tensor product of two modules over a commutative ring is a module together with a bilinear map into it that is universal: every bilinear map out of the pair of modules factors uniquely through it as a linear map.

Scope

This topic covers the construction and universal property of the tensor product of modules, its behavior on generators and relations, base change and extension of scalars, the tensor product of vector spaces and of algebras, and the right-exactness of the tensor functor.

Core questions

• How can bilinear maps be turned into linear maps?
• What universal property defines the tensor product?
• How does the tensor product implement change of scalars between rings?
• How does the tensor product interact with direct sums and exact sequences?

Key theories

Universal property of the tensor product

The tensor product is the unique module through which every bilinear map from a pair of modules factors as a linear map, which characterizes it up to isomorphism and governs all of its properties.

Extension of scalars

Tensoring a module with a larger ring along a ring homomorphism extends its scalars, turning a module over one ring into a module over another, the basic mechanism of base change in algebra and geometry.

Right-exactness of the tensor functor

Tensoring preserves cokernels and surjections but not in general injections, so it is right-exact; the failure of left-exactness is measured by the derived functors Tor, founding homological algebra.

Clinical relevance

Tensor products are ubiquitous: they construct multilinear algebra and the exterior and symmetric algebras, model composite quantum systems as tensor products of state spaces, implement base change in algebraic geometry, and underlie the tensors of differential geometry and machine learning.

History

Tensors arose in the work of Ricci and Levi-Civita on differential geometry and in Grassmann's exterior algebra, while the module-theoretic tensor product and its universal property were abstracted in the mid-twentieth century as homological algebra developed, becoming a standard tool through the work of Cartan, Eilenberg, and Mac Lane.

Key figures

• Hermann Grassmann
• Élie Cartan
• Emmy Noether
• Saunders Mac Lane

Seminal works

• dummit2004
• atiyah1969
• lang2002

Frequently asked questions

What problem does the tensor product solve?

It provides a single module through which all bilinear maps factor linearly, so that bilinear questions become linear ones. This universal property, not any explicit formula, is what makes the construction useful and well behaved.

Why is the tensor product only right-exact?

Tensoring preserves surjections and cokernels but can destroy injectivity, because relations among elements may collapse. The precise failure is captured by the Tor functors, which is why tensor products are studied alongside homological algebra.