Resolution (algebra)

In abstract algebra, a resolution is a sequence of modules (or more generally, objects in an abelian category) connected by homomorphisms, with the property that the sequence is exact. Resolutions are fundamental tools for studying the structure of modules and computing homological invariants.

More formally, let M be a module over a ring R. A resolution of M is an exact sequence of R-modules:

... → X_n → X_n-1 → ... → X₁ → X₀ → M → 0

where each X_i is an R-module and the arrows represent R-module homomorphisms. The exactness condition means that for each i, the image of the homomorphism X_i+1 → X_i is equal to the kernel of the homomorphism X_i → X_i-1, and the image of X₀ → M is M.

Several important types of resolutions exist, distinguished by properties of the modules X_i:

• Free Resolution: A resolution where each X_i is a free R-module. Every module has a free resolution.
• Projective Resolution: A resolution where each X_i is a projective R-module. A projective module is a direct summand of a free module. Every module has a projective resolution. Projective resolutions are important because they allow us to compute derived functors like Tor and Ext.
• Injective Resolution: An injective resolution is the dual concept. It is an exact sequence of the form:

0 → M → Y₀ → Y₁ → ... → Y_n → ...

where each Y_i is an injective R-module. An injective module Y is defined by the property that if A is a submodule of B, then any homomorphism A → Y can be extended to a homomorphism B → Y. Every module has an injective resolution.

• Flat Resolution: A resolution where each X_i is a flat R-module. A module F is flat if the tensor product functor - ⊗_R F preserves exact sequences.

Resolutions are used to define and compute important homological invariants of modules, such as Tor modules, Ext modules, and global dimension. The choice of resolution (free, projective, injective) depends on the context and the desired properties for computation. While the resolution itself is not unique, the resulting homological invariants obtained from the resolution are independent of the specific resolution chosen, up to isomorphism. This is a fundamental property of homological algebra. They provide tools for examining the structural properties of modules and their relationships to other modules in the category.