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.

  • 8
    $\begingroup$ 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). $\endgroup$ – abx Jan 2 at 16:46
  • $\begingroup$ Thank you. I have edited the question. $\endgroup$ – Cusp Jan 2 at 17:20
  • $\begingroup$ @abx Could you please explain the answer. $\endgroup$ – Cusp Jan 2 at 17:25
  • 7
    $\begingroup$ Explain what? I gave you a reference, I thought this is what you were asking for. $\endgroup$ – abx Jan 2 at 17:46

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