Projective resolution

A projective resolution of an $R$-module $M$ is an exact sequencein which every $P_i$ is a projective module.

A finitely generated module over a Noetherian ring admits a possibly infinite free resolution whose every term is finitely generated. Surject a finite free module onto it; the kernel is finitely generated because a finite free module is Noetherian. Repeat this construction on the successive kernels. If the original module has finite projective dimension $n$, its $n$th syzygy is projective by dimension shifting of the Ext functor, giving a length-$n$ resolution with finite projective terms.

For a short exact sequence of modules with projective resolutions of its outer terms, there is a projective resolution of the middle term forming a degreewise split short exact sequence of resolutions. Lift the degree-zero projective cover of the quotient into the middle module and add the projective cover of the submodule; this surjects onto the middle module. Its kernel fits into the corresponding short exact sequence of kernels. Repeating this construction proves the claim. Additive contravariant functors preserve the degreewise split exact sequences with their direction reversed.

Set $A=kQ$, $P_i=A{e}_i$, $X_i=e_{i}X$. There is an exact sequencewhere $d(p\otimes x)=p\rho \otimes x-p\otimes f_{\rho }(x)$ in the source and target summands, and the augmentation is $p\otimes x\mapsto px$. Each $P_i$ is a summand of $A$, so the two terms are projective modules. This also works for infinite-dimensional modules.

Put $C^0={⨁}_i\mathrm{Hom}(X_i,Y_i)$ and $C^1={⨁}_{\rho :i\to j}\mathrm{Hom}(X_i,Y_j)$. The differential has components $g_{\rho }{h}_i-h_j{f}_{\rho }$. Its kernel is ${\mathrm{Hom}}_Q(X,Y)$ and its cokernel is ${\mathrm{Ext}}_Q^1(X,Y)$. In a vertexwise splitting of an extension, the off-diagonal arrow blocks change by these coboundaries when the splitting changes.

Restriction at one arrow induces a surjection ${\mathrm{Ext}}_Q^1(X,Y)\twoheadrightarrow {\mathrm{Hom}}_k(\ker f_{\rho },\mathrm{coker}{g}_{\rho })$. Coboundaries vanish on the kernel after quotienting by the image, while every map between these spaces can be lifted and extended. Nonzero kernel and cokernel therefore force a nonsplit extension. For $X=Y$, rigidity forces every arrow to have maximal rank.

A short exact sequence presenting a module as a quotient of a projective module. The kernel need not be projective. Continuing a projective presentation gives a projective resolution.

For a regular sequence $f_1,\dots ,f_r$ in a commutative ring $R$, its Koszul complex is the finite free projective resolution of $R/(f_1,\dots ,f_r)$ with terms ${\bigwedge }^j{R}^r$ and boundary map contracting by $(f_1,\dots ,f_r)$. Exactness follows by induction: append $f_r$ using the mapping cone for multiplication by $f_r$, which is injective on the preceding quotient. For a polynomial ring viewed as a bimodule, use the regular sequence $X_{i,\ell }-X_{i,r}$ in the enveloping algebra.

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 $M$.