The Stacks project

Lemma 90.11.8. Let $R$ be an $S$-algebra, and let $\mathcal{C}$ be a strictly full subcategory of $S\text{-Alg}/R$ containing $R[M]$ for all $M \in \mathop{\mathrm{Ob}}\nolimits (\text{Mod}^{fg}_ R)$. Let $F: \mathcal{C} \to \textit{Sets}$ be a functor. Suppose that $F(R)$ is a one element set and that for any $M, N \in \mathop{\mathrm{Ob}}\nolimits (\text{Mod}^{fg}_ R)$, the induced map

\[ F(R[M] \times _ R R[N]) \to F(R[M]) \times F(R[N]) \]

is a bijection. Then $F(R[M])$ has a natural $R$-module structure for any $M \in \mathop{\mathrm{Ob}}\nolimits (\text{Mod}^{fg}_ R)$.

Proof. Note that $R \cong R[0]$ and $R[M] \times _ R R[N] \cong R[M \times N]$ hence $R$ and $R[M] \times _ R R[N]$ are objects of $\mathcal{C}$ by our assumptions on $\mathcal{C}$. Thus the conditions on $F$ make sense. The functor $\text{Mod}_ R \to S\text{-Alg}/R$ of Lemma 90.11.7 restricts to a functor $\text{Mod}^{fg}_ R \to \mathcal{C}$ by the assumption on $\mathcal{C}$. Let $L$ be the composition $\text{Mod}^{fg}_ R \to \mathcal{C} \to \textit{Sets}$, i.e., $L(M) = F(R[M])$. Then $L$ preserves finite products by Lemma 90.11.7 and the assumption on $F$. Hence Lemma 90.11.4 shows that $L(M) = F(R[M])$ has a natural $R$-module structure for any $M \in \mathop{\mathrm{Ob}}\nolimits (\text{Mod}^{fg}_ R)$. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 06IA. Beware of the difference between the letter 'O' and the digit '0'.