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Assume $G$ is a group scheme over a field $k$ and consider the categories $Rep_G$ of finite dimensional representations of G and $REP_G$ of all representations of G. For two objects $A,B$ in $Rep_G$, is the following formula true?

$$Ext^n_{Rep_G}(A,B)=Ext^n_{REP_G}(A,B)$$

Or equivalently, does every (possibly infinite dimensional) extension

$$0\to B\to X_1\to\dots\to X_n\to A\to 0$$ equivalent to a finite dimensional one?

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  • $\begingroup$ Am I missing something? Doesn't the following work? The exact sequence is equivalent to $0 \to X_1/B \to X_2 \to \ldots\to X_n\to A\to 0$. Per induction all $X_i$ with $i>1$ and $X_1/B$ are finite dimensional. Thus $X_1$ is also finite dimensional. $\endgroup$
    – Johannes Hahn
    Commented Oct 8, 2014 at 13:31
  • $\begingroup$ @JohannesHahn When I use "equivalent", i mean the complex chain should have the two ends fixed...That is, $B$ and $A$ must remain... $\endgroup$
    – KylinChen
    Commented Oct 9, 2014 at 3:13
  • $\begingroup$ Alright maybe. But doesn't my argument show that the sequence is not only "equivalent" to a finite dimensional sequence but is finite dimensional itself. $\endgroup$
    – Johannes Hahn
    Commented Oct 9, 2014 at 14:36
  • $\begingroup$ @JohannesHahn There is no reason for $X_1/B$ to be finite dimensional, is it ? Thus you cannot use induction. $\endgroup$
    – Arkandias
    Commented Oct 9, 2014 at 19:48
  • $\begingroup$ @Arkandias: Oh, you're right. I knew it couldn't be that simple... $\endgroup$
    – Johannes Hahn
    Commented Oct 9, 2014 at 20:01

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