Let $A$ be a ring, and let $M$ be a right $A$-module. Then $M$ is flat if and only if for each left $A$-module $N$ we have that $Tor^1_A(M,N) = 0$. Becasuse $Tor$ commutes with filtered direct limits, and since every left module is such a limit of finitely presented $A$-modules, $M$ is flat if and only if $Tor^1_A(M,N) = 0$ for any finitely presented right $A$-module $N$.

Recall that $N$ is called an $FP_3$-module if there is an exact sequence

$F_2 \to F_1 \to F_0 \to N \to 0$ with $F_0,F_1,F_2$ finitely generated free $A$-modules. My question is: if we know that $Tor^1_A(M,N) = 0$ for all $FP_3$-modules $N$, can we also conclude that $M$ is flat?

Let $A$ be a ring, and let $M$ be a right $A$-module. Then $M$ is flat if and only if for each left $A$-module $N$ we have that $Tor^1_A(M,N) = 0$. Becasuse $Tor$ commutes with filtered direct limits, and since every left module is such a limit of finitely presented $A$-modules, $M$ is flat if and only if $Tor^1_A(M,N) = 0$ for any finitely presented right $A$-module $N$.

Recall that $N$ is called an $FP_3$-module if there is an exact sequence

$F_2 \to F_1 \to F_0 \to N \to 0$ 

with $F_0,F_1,F_2$ finitely generated free $A$-modules. My question is: if we know that $Tor^1_A(M,N) = 0$ for all $FP_3$-modules $N$, can we also conclude that $M$ is flat?