Hom and Ext functors

Resolve the first module $M$ by projective left modules in nonnegative chain degrees. Applying $\operatorname{Hom}_R(-,N)$ reverses the arrows, producing a cochain complex with degree-$i$ term $\operatorname{Hom}_R(P_i,N)$ and differential given by precomposition with the resolution differential. Its cohomology defines Ext. Degree zero recovers ordinary Hom because a map on $P_0$ descends to $M$ exactly when it vanishes on the image of $P_1$.

Alternatively, resolve $N$ by injectives in nonnegative cochain degrees and apply $\operatorname{Hom}_R(M,-)$. The resulting cohomology groups are naturally isomorphic to those obtained from the projective calculation. Module categories have both types of resolutions. In a more general abelian category one must specify enough projectives or enough injectives before using the respective construction. The two computations are compatible methods, not a permission to replace a projective resolution with an arbitrary exact complex.

A map $M'\to M$ induces Ext maps from the groups with first argument $M$ to those with first argument $M'$. A map $N\to N'$ induces maps in the same direction in the second argument. The comparison theorem makes these maps independent of chosen lifts. Ignoring this reversal can produce a plausible diagram with every first-argument arrow pointing incorrectly, so label the variable being changed before writing any derived map.

For a short exact sequence in the second argument, the first connecting map starts at a Hom group into the quotient and lands in a first Ext group into the submodule. It measures failure to lift that map. With the first argument varying, the Hom arrows reverse, and the corresponding connecting map measures failure to extend. If $M$ is projective, all its positive Ext groups vanish against every $N$. Dually, an injective second argument makes all positive Ext vanish against every first argument.

The group $\operatorname{Ext}_R^1(M,N)$ also classifies short exact sequences with submodule $N$ and quotient $M$, up to isomorphisms inducing the identities on those endpoints. Its zero class is represented by the split sequence. The group operation is the Baer sum, not the direct sum of middle modules alone. Two abstractly isomorphic middle modules can still represent different extension data when their specified inclusion and quotient maps are retained.

For an integer quotient in the first argument, applying Hom to its length-one free resolution yields multiplication by that integer on the second argument. Hom is its kernel and first Ext its cokernel; higher Ext vanishes. This resembles the earlier Tor calculation, but the meanings and variance differ. In particular, a torsion first argument has no nonzero maps into the integers, yet can have nonzero first Ext into the integers. Confusing Hom with Ext would erase exactly the obstruction the resolution was designed to retain.