What you want is statement 4.2 in Kropholler's paper, "On groups of type $(FP)_\infty$" (1993). This states that your colimit is zero if and only if the module $M$ has finite projective dimension over $R$. No finite generation hypotheses are involved. His (and my) notation for this is $\mathop{\widehat{\rm Ext}}^0_R(M,M)$.
Dave Benson
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