Tensor product of modules

Definition

The tensor product of modules is a construction that combines two modules over a ring to form a new module, which captures bilinear relationships between them. This operation is fundamental in abstract algebra, particularly in the study of linear algebra, as it allows for the creation of new modules from existing ones while preserving certain structural properties. The tensor product is denoted as $$M \otimes_R N$$ for modules $M$ and $N$ over a ring $R$ and has important implications in various areas such as homological algebra and algebraic geometry.

5 Must Know Facts For Your Next Test

1. The tensor product $$M \otimes_R N$$ is itself a module over the same ring $R$.
2. The elements of the tensor product are formal sums of elements of the form $$m \otimes n$$, where $$m \in M$$ and $$n \in N$$.
3. The tensor product satisfies the universal property, meaning any bilinear map from two modules factors through it uniquely.
4. The tensor product is associative: $$(M \otimes_R N) \otimes_R P \cong M \otimes_R (N \otimes_R P)$$ for any module $P$.
5. If either module is projective, the tensor product retains many properties, simplifying calculations and maintaining desirable features.

Review Questions

• How does the universal property relate to the tensor product of modules, and why is it important?
  • The universal property states that for any bilinear map from two modules to another module, there exists a unique module homomorphism from their tensor product to that module. This property is crucial because it allows us to construct new modules from existing ones while ensuring that all bilinear relationships are preserved. Essentially, it guarantees that the tensor product captures all necessary interactions between the two original modules in a coherent way.
• In what ways does the associativity of the tensor product benefit its application in abstract algebra?
  • The associativity of the tensor product means that when combining multiple modules, the order in which you take their products doesn't affect the result. This property simplifies computations in complex algebraic structures and makes it easier to work with sequences of modules. As a result, mathematicians can manipulate these combinations with more flexibility, leading to clearer and more efficient proofs and constructions in various mathematical contexts.
• Critically evaluate how the tensor product interacts with projective modules and its implications in homological algebra.
  • When dealing with projective modules, the tensor product retains many desirable properties such as flatness, which means that the resulting module behaves well under localization. This interaction has significant implications in homological algebra, where projective modules serve as building blocks for resolving other modules. The ability to maintain properties through tensoring allows mathematicians to construct resolutions and perform computations effectively, showcasing the central role of tensor products in understanding module categories and their relationships.

Related terms

Bilinear map: A function that is linear in each of its arguments separately, serving as a foundation for defining the tensor product.

Module: A generalization of vector spaces where scalars belong to a ring instead of a field, allowing for more complex algebraic structures.

Universal property: A defining characteristic of the tensor product that ensures any bilinear map from two modules can be uniquely factored through the tensor product.