Algebraic objects

We start with the definition of an inverse system (or projective system) of groups and homomorphisms. Let ${\displaystyle (I,\leq )}$ be a directed poset (not all authors require I to be directed). Let (A_i)_i∈I be a family of groups and suppose we have a family of homomorphisms ${\displaystyle f_{ij}:A_{j}\to A_{i}}$ for all ${\displaystyle i\leq j}$ (note the order) with the following properties:

1. ${\displaystyle f_{ii}}$ is the identity on ${\displaystyle A_{i}}$,
2. ${\displaystyle f_{ik}=f_{ij}\circ f_{jk}\quad {\text{for all }}i\leq j\leq k.}$

Then the pair ${\displaystyle ((A_{i})_{i\in I},(f_{ij})_{i\leq j\in I})}$ is called an inverse system of groups and morphisms over ${\displaystyle I}$, and the morphisms ${\displaystyle f_{ij}}$ are called the transition morphisms of the system.

We define the inverse limit of the inverse system ${\displaystyle ((A_{i})_{i\in I},(f_{ij})_{i\leq j\in I})}$ as a particular subgroup of the direct product of the ${\displaystyle A_{i}}$'s:

${\displaystyle A=\varprojlim _{i\in I}{A_{i}}=\left\{\left.{\vec {a}}\in \prod _{i\in I}A_{i}\;\right|\;a_{i}=f_{ij}(a_{j}){\text{ for all }}i\leq j{\text{ in }}I\right\}.}$

The inverse limit ${\displaystyle A}$ comes equipped with natural projections π_i: A → A_i which pick out the ith component of the direct product for each ${\displaystyle i}$ in ${\displaystyle I}$. The inverse limit and the natural projections satisfy a universal property described in the next section.

This same construction may be carried out if the ${\displaystyle A_{i}}$'s are sets,^[1] semigroups,^[1] topological spaces,^[1] rings, modules (over a fixed ring), algebras (over a fixed ring), etc., and the homomorphisms are morphisms in the corresponding category. The inverse limit will also belong to that category.

General definition

The inverse limit can be defined abstractly in an arbitrary category by means of a universal property. Let ${\textstyle (X_{i},f_{ij})}$ be an inverse system of objects and morphisms in a category C (same definition as above). The inverse limit of this system is an object X in C together with morphisms π_i: X → X_i (called projections) satisfying π_i = ${\displaystyle f_{ij}}$ ∘ π_j for all i ≤ j. The pair (X, π_i) must be universal in the sense that for any other such pair (Y, ψ_i) there exists a unique morphism u: Y → X such that the diagram

commutes for all i ≤ j. The inverse limit is often denoted

${\displaystyle X=\varprojlim X_{i}}$

with the inverse system ${\textstyle (X_{i},f_{ij})}$ being understood.

In some categories, the inverse limit of certain inverse systems does not exist. If it does, however, it is unique in a strong sense: given any two inverse limits X and X' of an inverse system, there exists a unique isomorphism X′ → X commuting with the projection maps.

Inverse systems and inverse limits in a category C admit an alternative description in terms of functors. Any partially ordered set I can be considered as a small category where the morphisms consist of arrows i → j if and only if i ≤ j. An inverse system is then just a contravariant functor I → C. Let ${\displaystyle C^{I^{\mathrm {op} }}}$ be the category of these functors (with natural transformations as morphisms). An object X of C can be considered a trivial inverse system, where all objects are equal to X and all arrow are the identity of X. This defines a "trivial functor" from C to ${\displaystyle C^{I^{\mathrm {op} }}.}$ The inverse limit, if it exists, is defined as a right adjoint of this trivial functor.

Mittag-Leffler condition

If the ranges of the morphisms of an inverse system of abelian groups (A_i, f_ij) are stationary, that is, for every k there exists j ≥ k such that for all i ≥ j :${\displaystyle f_{kj}(A_{j})=f_{ki}(A_{i})}$ one says that the system satisfies the Mittag-Leffler condition.

The name "Mittag-Leffler" for this condition was given by Bourbaki in their chapter on uniform structures for a similar result about inverse limits of complete Hausdorff uniform spaces. Mittag-Leffler used a similar argument in the proof of Mittag-Leffler's theorem.

The following situations are examples where the Mittag-Leffler condition is satisfied:

• a system in which the morphisms f_ij are surjective
• a system of finite-dimensional vector spaces or finite abelian groups or modules of finite length or Artinian modules.

An example where ${\displaystyle \varprojlim {}^{1}}$ is non-zero is obtained by taking I to be the non-negative integers, letting A_i = pⁱZ, B_i = Z, and C_i = B_i / A_i = Z/pⁱZ. Then

${\displaystyle \varprojlim {}^{1}A_{i}=\mathbf {Z} _{p}/\mathbf {Z} }$

where Z_p denotes the p-adic integers.

Further results

More generally, if C is an arbitrary abelian category that has enough injectives, then so does C^I, and the right derived functors of the inverse limit functor can thus be defined. The nth right derived functor is denoted

${\displaystyle R^{n}\varprojlim :C^{I}\rightarrow C.}$

In the case where C satisfies Grothendieck's axiom (AB4*), Jan-Erik Roos generalized the functor lim¹ on Ab^I to series of functors limⁿ such that

${\displaystyle \varprojlim {}^{n}\cong R^{n}\varprojlim .}$

It was thought for almost 40 years that Roos had proved (in Sur les foncteurs dérivés de lim. Applications. ) that lim¹ A_i = 0 for (A_i, f_ij) an inverse system with surjective transition morphisms and I the set of non-negative integers (such inverse systems are often called "Mittag-Leffler sequences"). However, in 2002, Amnon Neeman and Pierre Deligne constructed an example of such a system in a category satisfying (AB4) (in addition to (AB4*)) with lim¹ A_i ≠ 0. Roos has since shown (in "Derived functors of inverse limits revisited") that his result is correct if C has a set of generators (in addition to satisfying (AB3) and (AB4*)).

Barry Mitchell has shown (in "The cohomological dimension of a directed set") that if I has cardinality ${\displaystyle \aleph _{d}}$ (the dth infinite cardinal), then Rⁿlim is zero for all n ≥ d + 2. This applies to the I-indexed diagrams in the category of R-modules, with R a commutative ring; it is not necessarily true in an arbitrary abelian category (see Roos' "Derived functors of inverse limits revisited" for examples of abelian categories in which limⁿ, on diagrams indexed by a countable set, is nonzero for n > 1).