Ext, obstruction for what is Ext?, Tor is an obstruction for being flat

Question

Let $M,N$ be in $R-\mathbb{Mod}$.

Here on page 28 is written

Think of $\mathbb{Tor}$ as describing how far $N$ is from being flat.

Now, for what $\mathbb{Ext}$ measures how far is $N$ from?

Answer 1

The functor $\mathrm{Tor}^R(N, -)$ is the derived functor of $N \otimes_R -$, so $\mathrm{Tor}^R(N, -)$ is the zero functor if and only if $N \otimes_R -$ is an exact functor. We call $N$ for which $N \otimes_R -$ is exact flat modules.

The functor $\mathrm{Ext}_R(N, -)$ is the derived functor of $\mathrm{Hom}(N, -)$ and modules for which this functor is exact are the projective modules, so $\mathrm{Ext}_R(N, -)$ measures how far a module is from being projective.

The functor $\mathrm{Ext}_R(-, N)$ is the derived functor of $\mathrm{Hom}(-, N)$ and modules for which this functor is exact are the injective modules, so $\mathrm{Ext}_R(-, N)$ measures how far a module is from being injective.

Answer 2

Let $M,N$ be $R$-modules.

$Ext^{i}_{R}(M,N)=0$ for $i>0$ if $M$ is projective or $N$ is injective, this combined with the fact that these are the derived functors of $Hom$ functors makes me conclude that it is a measure of how far a module is from being projective or injective.