Lemma 10.71.8. Let $R$ be a ring. Let $M$, $N$ be $R$-modules. Any $x\in R$ such that either $xN = 0$, or $xM = 0$ annihilates each of the modules $\mathop{\mathrm{Ext}}\nolimits ^ i_ R(M, N)$.
Proof. Pick a free resolution $F_{\bullet }$ of $M$. Since $\mathop{\mathrm{Ext}}\nolimits ^ i_ R(M, N)$ is defined as the cohomology of the complex $\mathop{\mathrm{Hom}}\nolimits _ R(F_{\bullet }, N)$ the lemma is clear when $xN = 0$. If $xM = 0$, then we see that multiplication by $x$ on $F_{\bullet }$ lifts the zero map on $M$. Hence by Lemma 10.71.5 we see that it induces the same map on Ext groups as the zero map. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)
There are also: