Resolution (algebra)

Given a module M over a ring R, a left resolution (or simply resolution) of M is an exact sequence (possibly infinite) of R-modules

${\displaystyle \cdots {\overset {d_{n+1}}{\longrightarrow }}E_{n}{\overset {d_{n}}{\longrightarrow }}\cdots {\overset {d_{3}}{\longrightarrow }}E_{2}{\overset {d_{2}}{\longrightarrow }}E_{1}{\overset {d_{1}}{\longrightarrow }}E_{0}{\overset {\varepsilon }{\longrightarrow }}M\longrightarrow 0.}$

The homomorphisms d_i are called boundary maps. The map ε is called an augmentation map. For succinctness, the resolution above can be written as

${\displaystyle E_{\bullet }{\overset {\varepsilon }{\longrightarrow }}M\longrightarrow 0.}$

The dual notion is that of a right resolution (or coresolution, or simply resolution). Specifically, given a module M over a ring R, a right resolution is a possibly infinite exact sequence of R-modules

${\displaystyle 0\longrightarrow M{\overset {\varepsilon }{\longrightarrow }}C^{0}{\overset {d^{0}}{\longrightarrow }}C^{1}{\overset {d^{1}}{\longrightarrow }}C^{2}{\overset {d^{2}}{\longrightarrow }}\cdots {\overset {d^{n-1}}{\longrightarrow }}C^{n}{\overset {d^{n}}{\longrightarrow }}\cdots ,}$

where each Cⁱ is an R-module (it is common to use superscripts on the objects in the resolution and the maps between them to indicate the dual nature of such a resolution). For succinctness, the resolution above can be written as

${\displaystyle 0\longrightarrow M{\overset {\varepsilon }{\longrightarrow }}C^{\bullet }.}$

A (co)resolution is said to be finite if only finitely many of the modules involved are non-zero. The length of a finite resolution is the maximum index n labeling a nonzero module in the finite resolution.

In many circumstances conditions are imposed on the modules E_i resolving the given module M. For example, a free resolution of a module M is a left resolution in which all the modules E_i are free R-modules. Likewise, projective and flat resolutions are left resolutions such that all the E_i are projective and flat R-modules, respectively. Injective resolutions are right resolutions whose Cⁱ are all injective modules.

Every R-module possesses a free left resolution.[3] A fortiori, every module also admits projective and flat resolutions. The proof idea is to define E₀ to be the free R-module generated by the elements of M, and then E₁ to be the free R-module generated by the elements of the kernel of the natural map E₀ → M etc. Dually, every R-module possesses an injective resolution. Projective resolutions (and, more generally, flat resolutions) can be used to compute Tor functors.

Projective resolution of a module M is unique up to a chain homotopy, i.e., given two projective resolutions P₀ → M and P₁ → M of M there exists a chain homotopy between them.

Resolutions are used to define homological dimensions. The minimal length of a finite projective resolution of a module M is called its projective dimension and denoted pd(M). For example, a module has projective dimension zero if and only if it is a projective module. If M does not admit a finite projective resolution then the projective dimension is infinite. For example, for a commutative local ring R, the projective dimension is finite if and only if R is regular and in this case it coincides with the Krull dimension of R. Analogously, the injective dimension id(M) and flat dimension fd(M) are defined for modules also.

The injective and projective dimensions are used on the category of right R modules to define a homological dimension for R called the right global dimension of R. Similarly, flat dimension is used to define weak global dimension. The behavior of these dimensions reflects characteristics of the ring. For example, a ring has right global dimension 0 if and only if it is a semisimple ring, and a ring has weak global dimension 0 if and only if it is a von Neumann regular ring.

Let M be a graded module over a graded algebra, which is generated over a field by its elements of positive degree. Then M has a free resolution in which the free modules E_i may be graded in such a way that the d_i and ε are graded linear maps. Among these graded free resolutions, the minimal free resolutions are those for which the number of basis elements of each E_i is minimal. The number of basis elements of each E_i and their degrees are the same for all the minimal free resolutions of a graded module.

If I is a homogeneous ideal in a polynomial ring over a field, the Castelnuovo-Mumford regularity of the projective algebraic set defined by I is the minimal integer r such that the degrees of the basis elements of the E_i in a minimal free resolution of I are all lower than r-i.

A classic example of a free resolution is given by the Koszul complex of a regular sequence in a local ring or of a homogeneous regular sequence in a graded algebra finitely generated over a field.

Let X be an aspherical space, i.e., its universal cover E is contractible. Then every singular (or simplicial) chain complex of E is a free resolution of the module Z not only over the ring Z but also over the group ring Z [π₁(X)].

One class of examples of Abelian categories without projective resolutions are the categories ${\displaystyle {\text{Coh}}(X)}$ of coherent sheaves on a scheme ${\displaystyle X}$. For example, if ${\displaystyle X=\mathbb {P} _{S}^{n}}$ is projective space, any coherent sheaf ${\displaystyle {\mathcal {M}}}$ on ${\displaystyle X}$ has a presentation given by an exact sequence

${\displaystyle \bigoplus _{i,j=0}{\mathcal {O}}_{X}(s_{i,j})\to \bigoplus _{i=0}{\mathcal {O}}_{X}(s_{i})\to {\mathcal {M}}\to 0.}$

The first two terms are not in general projective since ${\displaystyle H^{n}(\mathbb {P} _{S}^{n},{\mathcal {O}}_{X}(s))\neq 0}$ for ${\displaystyle s>0}$. But, both terms are locally free, and locally flat. Both classes of sheaves can be used in place for certain computations, replacing projective resolutions for computing some derived functors.