2025-10-13

This following is a very brief summary of what happened in class on 2025-10-13.

We picked up where we left off last class, recalling the construction of the $S$-module $S{\otimes }_{R}M$ (for a fixed $R$-module $M$ and rings $R\subseteq S$). We then briefly outlined:

• The connection between $M$ and $S{\otimes }_{R}M$, namely the $R$-module morphism ${\eta }_M:M\to U(S{\otimes }_{R}M$) defined by $m\mapsto 1_S\otimes m$. (This is a component of the unit of the adjunction.)
• We didn't mention the explicits, but we do indeed have an actual functor $S{\otimes }_R-:R\mathbf{\text{-Mod}}\to S\mathbf{\text{-Mod}}$. For the arrow map, given $f:M_1\to M_2$ the corresponding $S$-module morphism $S{\otimes }_{R}f:S{\otimes }_R{M}_1\to S{\otimes }_R{M}_2$ is given on simple tensors by $s\otimes m_1\mapsto s\otimes f(m_1)$.
• We revisited some previous examples and noted some future results, such as the fact that for any finite abelian group $A$ we (unfortunately) have $\mathbf{Q}{\otimes }_{\mathbf{Z}}A\simeq 0$.

Rather than diving into the details, we noted that we will be better off in the long run first generalizing everything to bimodules. So we spent the rest of the hour defining bimodules and the morphisms between them, and then outlining how this new notion generalizes all of our previous types of modules.

Concepts

Tensor Products I - Extending scalars
Bimodules
Bimodule morphisms

References

• Dummit & Foote, Abstract Algebra: Section 10.4