# Behavior of Ext under base change

Let $$x$$ be a nonzerodivisor on a local Noetherian ring $$(R,m).$$ Let $$M,N$$ be finitely generated $$R/xR$$-modules.

How to show the existence of the following exact sequence

$$\cdots\longrightarrow Ext_{R/xR}^{n}(M,N)\longrightarrow Ext_{R}^{n}(M,N)\longrightarrow Ext_{R/xR}^{n-1}(M,N)\longrightarrow \cdots.$$

Any reference will be useful.

• I think the correct exact sequence is: $$\cdots\rightarrow \operatorname{Ext}^n_{R/xR}(M,N)\rightarrow \operatorname{Ext}^n_{R}(M,N) \rightarrow \operatorname{Ext}^{n-1}_{R/xR}(M,N) \rightarrow \operatorname{Ext}^{n+1}_{R/xR}(M,N)\rightarrow \operatorname{Ext}^{n+1}_{R}(M,N)\rightarrow \cdots$$ It is a particular case of the exact sequence (4) p. 44 of Bourbaki's Algèbre commutative X (unfortunately not yet translated into English, as far as I know). – abx Jan 2 at 16:46
• Thank you. I have edited the question. – Cusp Jan 2 at 17:20
• @abx Could you please explain the answer. – Cusp Jan 2 at 17:25
• Explain what? I gave you a reference, I thought this is what you were asking for. – abx Jan 2 at 17:46