Free resolution

A free resolution is a projective resolution in which every resolving module is a free module.

Every finitely generated module with a grading over the polynomial ring $S=k[X_1,\dots ,X_n]$ has a finite graded free resolution of length at most $n$. The Koszul resolution of $k$ has length $n$, so ${\mathrm{Tor}}_i^S(M,k)=0$ for $i>n$. In a minimal graded free resolution, this Tor functor is $F_i/(X_1,\dots ,X_n)F_i$; the graded Nakayama lemma forces $F_i=0$ for $i>n$.

Over $S=k[X_1,\dots ,X_n]$ with its standard grading and $\mathfrak{m}=(X_1,\dots ,X_n)$, a graded free resolution is minimal when $d_i(F_i)\subseteq \mathfrak{m}{F}_{i-1}$ for every $i\ge 1$. It is constructed by choosing minimal homogeneous generators for each successive syzygy module, using the graded Nakayama lemma. After tensoring with $k=S/\mathfrak{m}$, all its differentials vanish, so $F_i/\mathfrak{m}{F}_i\cong {\mathrm{Tor}}_i^S(M,k)$; these dimensions and degrees are intrinsic to $M$.

Given a free resolution $\cdots \to F_1\to F_0\to M\to 0$, its first syzygy module is $\ker (F_0\to M)$, the module of relations among the chosen generators of $M$. Higher syzygy modules are successive kernels. They depend on the choice of resolution; a minimal graded free resolution gives canonical ranks and degrees of generators over a standard graded ring structure on a polynomial ring.

The bar resolution of an associative algebra $A$ over a field has degree-$n$ term $A^{\otimes (n+2)}$ as an $A^e$-module, with the outer factors carrying the left and right actions. Its boundary multiplies each consecutive pair with alternating signs, and its augmentation is multiplication $A\otimes A\to A$. It is exact: the $k$-linear contracting homotopy inserts $1$ at the left. Applying ${\mathrm{Hom}}_{A^e}(-,M)$ gives the Hochschild cochain complex of the bimodule $M$.