The Stacks project

Example 14.34.4. Let $R$ be a ring. As an example of the above we can take $i : \text{Mod}_ R \to \textit{Sets}$ to be the forgetful functor and $F : \textit{Sets} \to \text{Mod}_ R$ to be the functor that associates to a set $E$ the free $R$-module $R[E]$ on $E$. For an $R$-module $M$ the simplicial $R$-module $X(M)$ will have the following shape

\[ X(M) = \left( \ldots \xymatrix{ R[R[R[M]]] \ar@<2ex>[r] \ar@<0ex>[r] \ar@<-2ex>[r] & R[R[M]] \ar@<1ex>[r] \ar@<-1ex>[r] \ar@<1ex>[l] \ar@<-1ex>[l] & R[M] \ar@<0ex>[l] } \right) \]

which comes with an augmentation towards $M$. We will also show this augmentation is a homotopy equivalence of sets. By Lemmas 14.30.8, 14.31.9, and 14.31.8 this is equivalent to asking $M$ to be the only nonzero cohomology group of the chain complex associated to the simplicial module $X(M)$.

Comments (0)

There are also:

  • 2 comment(s) on Section 14.34: Standard resolutions

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Your email address will not be published.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 0G5T. Beware of the difference between the letter 'O' and the digit '0'.