$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\SW{SW}$Let $R$ be a commutative Noetherian ring. For $R$-modules $M,N$, let $\mathcal I_R(M,N)$ be the collection of all $f\in \text{Hom}_R(M,N)$ that factors through some projective $R$-module. Put $\underline{\Hom}_R(M,N):=\Hom_R(M,N)/\mathcal I_R(M,N)$ and for every $m,n\in \mathbb Z$, put $\Hom_{\SW(R)}(M[n],N[m]) :=\varinjlim_{i\geq m,n} \underline{\Hom}_R (\Omega_R^{i-n} M , \Omega^{i-m}N)$.
It is clear to see that $\underline{\Hom}_R(M,M)=0$ if and only if $M$ is projective.
My question is: What can we say about $M$ if $\Hom_{\SW(R)}(M[n],M[n])=0$ for some $n\in \mathbb Z$? I am willing to assume $M$ is finitely generated (but I think it won't make a difference for the answer).
Best Answer
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 $\smash{\widehat{\operatorname{Ext}}}^0_R(M,M)$.