The Tensor Product and Induced Modules

Nayab Khalid

The Tensor Product The Tensor Product A Construction (via the Universal Property) Properties

Examples References Nayab Khalid

PPS Talk University of St. Andrews

October 16, 2015 The Tensor Product and Induced Modules Overview of the Presentation

Nayab Khalid

The Tensor Product

A 1 The Tensor Product Construction

Properties

Examples 2 A Construction

References

3 Properties

4 Examples

5 References The Tensor Product and Induced Modules The Tensor Product

Nayab Khalid

The Tensor Product

A Construction

Properties The tensor product of modules is a construction that allows

Examples multilinear maps to be carried out in terms of linear maps.

References The tensor product can be constructed in many ways, such as using the basis of free modules. However, the standard, more comprehensive, definition of the tensor product stems from category theory and the universal property. The Tensor Product and Induced Modules The Tensor Product (via the Nayab Khalid Universal Property)

The Tensor Product

A Construction

Properties Examples Definition References Let R be a commutative ring. Let E1, E2 and F be modules over R. Consider bilinear maps of the form

f : E1 × E2 → F . The Tensor Product and Induced Modules The Tensor Product (via the Nayab Khalid Universal Property)

The Tensor Product Definition A Construction The tensor product E1 ⊗ E2 is the module with a bilinear map Properties Examples λ: E1 × E2 → E1 ⊗ E2 References such that there exists a unique homomorphism which makes the following diagram commute. The Tensor Product and Induced Modules A Construction of the Nayab Khalid Tensor Product

The Tensor Product Here is a more constructive definition of the tensor product. A Construction Definition Properties Let M be the free module generated by the set of all pairs Examples (x1, x2), xi ∈ Ei , i.e. generated by the set E1 × E2. References Let N be the submodule of M generated by all elements fulfilling the following properties: 0 0 • (x1 + x1, x2) − (x1, x2) − (x1, x2) 0 0 • (x1, x2 + x2) − (x1, x2) − (x1, x2) • (ax1, x2) − a(x1, x2)

• (x1, ax2) − a(x1, x2) 0 for all xi , xi ∈ Ei and a ∈ R. The Tensor Product and Induced Modules A Construction of the Nayab Khalid Tensor Product

The Tensor Product

A Construction

Properties Examples Definition References Now consider the canonical injection E1 × E2 → M and the canonical map M → M/N. The composition of these maps

φ: E1 × E2 → M/N

is bilinear and a tensor product. The Tensor Product and Induced Modules A Construction of the Nayab Khalid Tensor Product

The Tensor Product

A Construction Properties The module M/N is denoted by Examples 2 References O E1 ⊗ E2 or Ei . i=1 We have constructed a specific tensor product in the isomorphism class of tensor products, and we shall call it the tensor product of E1 and E2. The Tensor Product and Induced Modules Properties of the Tensor Product

Nayab Khalid

The Tensor Product For x ∈ E , we can write A i i Construction Properties φ(x1, x2) = x1 ⊗ x2. Examples References Then the following properties hold 0 0 • (x1 + x1, x2) = (x1, x2) + (x1, x2) 0 0 • (x1, x2 + x2) = (x1, x2) + (x1, x2) • (ax1, x2) = a(x1, x2)

• (x1, ax2) = a(x1, x2) 0 for xi , xi ∈ Ei and a ∈ R. The Tensor Product and Induced Modules Properties of the Tensor Product

Nayab Khalid

The Tensor Product

A Construction Properties We note that every element of E1 and E2 can be written as a Examples sum of terms x ⊗ y with x ∈ E1 and y ∈ E2, because such References terms generate E1 ⊗ E2, and

a(x ⊗ y) = ax ⊗ y = x ⊗ ay

for a ∈ R. The Tensor Product and Induced Modules Example 1: Nayab Khalid Tensor Product over a Field

The Tensor Product

A Construction

Properties Consider a field F . Consider a vector space V over F with Examples

References dimension m and basis {v1, ..., vm}, and a vector space W over F with dimension n and basis {w1, ..., wn}.

Then the tensor product V ⊗ W is the vector space spanned by the elements vi ⊗ wj (1 ≤ i ≤ m, 1 ≤ j ≤ n) and has dimension mn. The Tensor Product and Induced Modules Example 2: Nayab Khalid Collapsing Between Modules

The Tensor Product

A Construction Properties The tensor product can involve a great deal of collapsing Examples between modules. References

Consider the tensor product over Z of Z/nZ and Z/mZ, where m, n are integers > 1 and are relatively prime. Then the tensor product Z/nZ ⊗ Z/mZ = 0. The Tensor Product and Induced Modules References

Nayab Khalid

The Tensor Product Dummit, D. and Foote, R. A Construction Abstract Algebra Properties (2004) New Jersey: Wiley. Examples Lang, S. References Algebra (1965) Reading, Mass.: Addison-Wesley Pub. Co.

Armstrong, J. Tensor Products over Group Algebras (2010) [online] The Unapologetic Mathematician. Available at: https://unapologetic.wordpress.com/2010/11/09/tensor-products- over-group-algebras/ The Tensor Product and Induced Modules

Nayab Khalid

The Tensor Product

A Construction

Properties Examples The End References