2025-10-31

This following is a very brief summary of what happened in class on 2025-10-31.

After recapping our new algebraic structure, called an ... (ahem) ... algebra, we went on to construct a functor $T : R - Mod \to R - Alg$ left adjoint to the forgetful functor $U : R - Alg \to R - Mod$ . This functor is called the tensor algebra functor. To each $R$ -module, $M$ , it associates an $R$ -algebra, $T (M)$ , defined by

T (M) = R \oplus M \oplus (M \otimes_{R} M) \oplus (M \otimes_{R} M \otimes_{R} M) \oplus \dots

On simple tensors the product is given by "concatenation of tensors":

(m_{1} \otimes \dots \otimes m_{k}) \cdot (m_{1}^{'} \otimes \dots \otimes m_{l}^{'}) := m_{1} \otimes \dots \otimes m_{k} \otimes m_{1}^{'} \otimes \dots m_{l}^{'} .

The one "exception" to this definition is for the 0-tensors, which are the elements in $R$ in the first summand of $T (M)$ . For products involving elements from $R$ , the multiplication is just the $R$ -action on $k$ -tensors.

Although we did not verify that this construction is actually left adjoint to the forgetful functor (for time reasons), we did look at a few explicit examples.

Next time we will talk about the symmetric algebra and exterior algebra functors.

Concepts

References

Dummit & Foote, Abstract Algebra: Section 11.5