Here is my question: If you were to make a presentation that require the use of tensor products (let's say for commutative rings) but your audience may not know about tensor product.
Since it is a short presentation, you would define it by its basis and properties of bilinearity: $$(a \cdot x + x') \otimes_A (b \cdot y + y') = ab \cdot (x \otimes y) + a \cdot (x \otimes y') + b \cdot (x' \otimes y) + (x' \otimes y')$$ And then shortly explain why it is useful for bilinear functions.
But let's be honest, at least half of the audience won't get it without a good explicit example, here is what I had in mind:
- The complexification of a $\mathbb{R}$-vector space $V$: $$V_\mathbb{C} := \mathbb{C} \otimes_\mathbb{R} V$$ which is very useful since you could also try to explain a bit of adjunctions properties of the tensor product by quickly explaining why $${V_\mathbb{C}}^\ast \simeq \hom_{\mathbb{R}} (V, \mathbb{C})$$
- The "gcd" tensor product: $$\mathbb{Z}/n\mathbb{Z} \otimes_\mathbb{Z} \mathbb{Z}/m\mathbb{Z} = \mathbb{Z}/(n \wedge m)\mathbb{Z}$$ Which simply seems right for me, the proof uses the simple mechanics of tensor product in their entirety.
EDIT: Two more interesting examples have arisen in the comments:
- The “kronecker” tensor product:
$$\mathbb{R}^n \otimes_\mathbb{R} \mathbb{R}^m \simeq \mathbb{R}^{nm}$$ Which seems relevant since it gives the dimension of the tensor product for vectors and matrices and gives credit to the kronecker product. - The “field ($\text{char} =0$) vs torsion” tensor product: $$ \mathbb{Q} \otimes_\mathbb{Z} \mathbb{Z}/n\mathbb{Z} \simeq \{0\}$$ Which puts in light the need for compatibility of the torsions of both modules (like the gcd tensor product did).
What do you think about these two examples? Do you have other example that are worth a little screen time in this presentation?
If this is a duplicate, I will kindly delete it but keep in mind that I don't want an explanation of what tensor products are, I'm at ease with them :)