Direct Limits of Modules

Q: Can the direct limit be zero when all the modules are non-zero?

Yes. Take all stages equal to K[x]/(x^2) with transition maps multiplication by x ; every element is killed within two steps, so the limit vanishes. The limit depends on the maps at least as much as on the modules.

Overview

The large modules in which generalised solutions of differential equations live - germs of holomorphic functions, hyperfunctions, microfunctions - are not given by generators and relations. They are given as limits of approximations: functions holomorphic on a disc of radius $ε$ , for smaller and smaller $ε$ . The algebraic device that turns such a family into a single module is the direct limit, and this page constructs it.

The input is a directed set $I$ of indices, a module $M_i$ for each index, and a transition map $ϕ_ij : M_i → M_j$ whenever $i ≤ j$ , compatible with composition. The output is a module $M_i$ whose elements are elements of the individual $M_i$ , two of them being identified as soon as they agree after being pushed far enough along the system.

Two facts about the result are used constantly and are worth stating before anything else. Every element of the limit comes from some single stage: there is no need to take infinite sums. And an element coming from a stage is zero in the limit precisely when it is killed at some later stage - not merely when it looks small. Together these say that the limit remembers nothing except eventual behaviour.

The construction is more than a convenience. Direct limits over directed sets are exact: they commute with kernels as well as with cokernels, which distinguishes them sharply from inverse limits and from colimits over unstructured index sets. That exactness is why localisation, germs and stalks behave so well, and it is the reason the module of microfunctions can be handled by pure algebra.

Definition

Directed set

A set $I$ with a relation $≤$ is pre-ordered if $≤$ is reflexive and transitive; antisymmetry is not assumed. A pre-ordered set is directed if for all $i,j ∈ I$ there is $k ∈ I$ with $i ≤ k$ and $j ≤ k$ .

Direct system of modulesCoutinho, Ch. 6 §2

Let $R$ be a ring and $I$ a directed set. A direct system of left $R$ -modules over $I$ consists of a module $M_i$ for each $i ∈ I$ and a homomorphism $ϕ_ij : M_i → M_j$ for each pair $i ≤ j$ , such that

ϕ_ii = id_M_i ϕ_jk,ϕ_ij = ϕ_ik whenever i ≤ j ≤ k.

(6.20)

Direct limit

Let $U = (u,i) : i ∈ I, u ∈ M_i$ be the disjoint union of the $M_i$ . Declare

(u,i) ∼ (v,j) ⟺ there is k ≥ i,j with ϕ_ik(u) = ϕ_jk(v).

(6.21)

This is an equivalence relation, and the quotient set $M_i = U/! ∼$ becomes a left $R$ -module under

(u,i) + (v,j) = (ϕ_ik(u) + ϕ_jk(v), k ) for any k ≥ i,j, a ⋅ (u,i) = (au, i).

(6.22)

Write $ϕ_i : M_i → M_i$ for the map sending $u$ to the class of $(u,i)$ ; these satisfy $ϕ_jϕ_ij = ϕ_i$ for $i ≤ j$ .

Two opposite conventions for the ordering

Coutinho sets the definitions up with the order reversed: his directed sets require a common lower bound, and his transition maps run from larger to smaller indices. Applied to discs of radius $ε$ this is natural, since the interesting direction is $ε$ decreasing. It is the same notion of direct limit, taken over the opposite pre-order, and no statement changes. This page uses the convention that transition maps go from smaller to larger indices, so for germs one orders radii by reverse inclusion: $ε ≤ ε'$ means $D(ε) ⊃ eq D(ε')$ . When comparing sources, check which way the arrows point before comparing formulas.

Core Concepts

Eventual behaviour is all that survives

Reading (6.21) aloud: two elements of a direct system are the same in the limit if they can be made equal by waiting long enough. All the identifications are of that form, and no others are imposed. This is why direct limits model localisation, germs, and every other construction whose objects are defined only near a point or only up to some finite stage.

Why directedness is not a technicality

Directedness is used twice in the construction and both times for the same reason: any two stages can be compared inside a third. Drop it and three things fail at once. The relation (6.21) stops being transitive; addition has no canonical stage at which to be performed; and the description of elements in (6.24) collapses, since a general colimit element is a finite sum of contributions from different indices rather than a single one. The direct sum over a two-element index set with no relations is the smallest example: its elements are pairs, not elements of one summand.

Injective transition maps: the limit is a union

If every $ϕ_ij$ is injective then so is every $ϕ_i$ , by (6.24), and the limit is the increasing union of the images of the $M_i$ . This is the intuitive picture, and it is the case for germs of holomorphic functions, where restriction is injective by the identity theorem. When the transition maps are not injective, elements genuinely disappear, and the limit can be zero even when every $M_i$ is non-zero.

Cofinality

If $I' ⊂ eq I$ is cofinal - every $i ∈ I$ satisfies $i ≤ i'$ for some $i' ∈ I'$ - then the limit over $I^{'}$ agrees with the limit over $I$ . This is what allows an uncountable index set to be replaced by a sequence: for germs at the origin one may take the radii $1/n$ , $n ∈ N$ , instead of all positive reals, which makes the limit a countable increasing union.

Construction and Proof

Three things have to be verified: that (6.21) is an equivalence relation, that the operations (6.22) are well defined, and that the result has the expected universal property. Only the first two use directedness, and both use it in the same way.

(6.21) is an equivalence relation

Reflexivity is $ϕ_ii = id$ , and symmetry is immediate. For transitivity, suppose $(u,i) ∼ (v,j)$ via $k$ and $(v,j) ∼ (w,l)$ via $k^{'}$ , so that $ϕ_ik(u) = ϕ_jk(v)$ and $ϕ_jk'(v) = ϕ_lk'(w)$ . Choose $r ≥ k,k'$ , which exists by directedness. Then

ϕ_ir(u) = ϕ_krϕ_ik(u) = ϕ_krϕ_jk(v) = ϕ_jr(v) = ϕ_k'rϕ_jk'(v) = ϕ_k'rϕ_lk'(w) = ϕ_lr(w),

so $(u,i) ∼ (w,l)$ via $r$ . Without directedness there need be no such $r$ and the relation generated by (6.21) is strictly larger than (6.21) itself.

The sum does not depend on the choices

Let $k$ and $k^{'}$ both be upper bounds of $i$ and $j$ , and choose $r ≥ k,k'$ . Applying $ϕ_kr$ to the first candidate sum and $ϕ_k'r$ to the second gives

ϕ_kr (ϕ_ik(u)+ϕ_jk(v) ) = ϕ_ir(u)+ϕ_jr(v) = ϕ_k'r (ϕ_ik'(u)+ϕ_jk'(v) ),

so the two candidates are equivalent. Replacing $(u,i)$ by an equivalent $(u',i')$ is handled the same way, by pushing everything past a common upper bound. Scalar multiplication needs no choice at all, and the module axioms then follow from those in each $M_i$ , since any finite collection of elements can be transported to a common stage.

The two facts one actually uses

Let $L = M_i$ with canonical maps $ϕ_i$ . Then:

every element of $L$ is $ϕ_i(u)$ for some $i ∈ I$ and $u ∈ M_i$ ;
$ϕ_i(u) = 0$ if and only if $ϕ_ij(u) = 0$ for some $j ≥ i$ .

The first is immediate from the construction; note that it is a statement about a single stage, not about finite sums, and it is exactly what fails for colimits over non-directed index sets. The second follows by comparing $(u,i)$ with $(0,i)$ in (6.21).

Universal property

Given a module $N$ and homomorphisms $ψ_i : M_i → N$ with $ψ_jϕ_ij = ψ_i$ for all $i ≤ j$ , there is a unique homomorphism $ψ : M_i → N$ with $ψϕ_i = ψ_i$ for all $i$ . It is defined by $ψ(ϕ_i(u)) = ψ_i(u)$ , which is forced, and is well defined precisely because of the compatibility hypothesis. Consequently the direct limit is unique up to a unique isomorphism, and it may be characterised by this property without reference to the construction.

Key Equations

The system, the limit and the canonical maps:

L = _i ∈ I M_i, ϕ_i : M_i → L, ϕ_j,ϕ_ij = ϕ_i (i ≤ j).

(6.23)

Kernels and images of the canonical maps are described stage by stage:

ϕ_i = ⋃ _j ≥ i ϕ_ij, L = ⋃ _i ∈ I ϕ_i(M_i).

(6.24)

Exactness of the operation, for a directed system of short exact sequences:

0 → M_i' → M_i → M_i'' → 0 exact for all i ⟹ 0 → M_i' → M_i → M_i'' → 0 exact.

(6.25)

Interchange with $Hom$ , valid when the source is finitely presented - which over a Noetherian ring means finitely generated, so it applies to every module of a differential system:

Hom_R (M, _i S_i ) ; ≅ ; _i Hom_R(M, S_i) (M finitely presented).

(6.26)

Variable Definitions

$R$: the ring of scalars; in this chapter usually $A_1(C)$ or a polynomial ring
$I$: the directed index set
$M_i$: the module at stage $i$
$ϕ_ij$: the transition map $M_i → M_j$ , defined when $i ≤ j$
$𝒰$: the disjoint union of the $M_i$ , whose elements are pairs $(u,i)$
$M_i$: the direct limit, that is $𝒰$ modulo eventual agreement
$ϕ_i$: the canonical map $M_i → M_i$
$D(ε)$: the open disc of radius $ε$ about $0$ in $ℂ$
$H(ε)$: the holomorphic functions on $D(ε)$ , a left $A_1(C)$ -module
$H_0$: the module of germs of holomorphic functions at the origin

Properties and Behaviour

Exactness

Direct limits over a directed index set are exact. Given a directed system of short exact sequences as in (6.25), with all squares commuting, the limit sequence is short exact.

Proof sketch: right exactness holds for every colimit. For injectivity on the left, suppose $u' ∈ M_i'$ has $ϕ_i(u') = 0$ in $M_i$ . By (6.24) there is $j ≥ i$ with $ϕ_ij(u') = 0$ in $M_j$ ; since $M_j' → M_j$ is injective and the square commutes, the image of $u^{'}$ in $M_j'$ is already $0$ , so $u^{'}$ dies in $M_i'$ . The whole argument consists of moving to a stage where the statement becomes a statement about ordinary modules, which is exactly what directedness makes possible.

What direct limits commute with

kernels, cokernels, images and finite direct sums - by exactness;
tensor products: $( M_i) ⊗ _R N ≅ (M_i ⊗ _R N)$ , since tensoring is itself a colimit;
$Hom$ out of a finitely presented module, as in (6.26);
other direct limits: a directed limit of directed limits is a directed limit.

Every module is a direct limit of its finitely generated submodules

Order the finitely generated submodules of a module $N$ by inclusion. Any two are contained in the submodule they jointly generate, which is again finitely generated, so the set is directed, and the inclusions form a direct system whose limit is $N$ . This is the standard way to reduce a statement about arbitrary modules to one about finitely generated modules, and it is used for the large target modules of solutions, none of which is finitely generated.

Solutions in a limit come from a finite stage

Let $M$ be the module of a system of differential equations, so $M$ is finitely presented over $A_n$ , and let $S = S_i$ be a target module built as a direct limit. Then by (6.26)

Hom_A_n(M, S) ≅ _i Hom_A_n(M, S_i),

so every generalised solution in $S$ is represented by a solution at some finite stage $S_i$ , and two such represent the same solution exactly when they agree at a later stage. For germs of holomorphic functions this says a germ solution is a genuine solution on some disc of positive radius; see solutions as homomorphisms.

Examples and Special Cases

Increasing unions

Let $N_1 ⊂ eq N_2 ⊂ eq ⋅ s$ be submodules of a fixed module, indexed by $ℕ$ with the inclusions as transition maps. The limit is $⋃ _k N_k$ . Every increasing union is a direct limit, and this is the case where the general construction adds nothing new.

Germs of holomorphic functions

For $ε > 0$ let $H(ε)$ be the holomorphic functions on the disc $D(ε)$ , a left $A_1(C)$ -module with $z$ acting by multiplication and $∂$ by differentiation. Order the radii so that smaller discs are later stages, and take restriction as the transition map; restriction is $A_1(C)$ -linear and, the discs being connected, injective.

The limit $H_0 = H(ε)$ is the module of germs of holomorphic functions at the origin. Two pairs $(f,ε)$ and $(g,ε')$ are equal in it exactly when $f$ and $g$ agree on some disc around $0$ , which by the identity theorem happens exactly when they have the same Taylor series at $0$ . So $H_0$ is the ring of convergent power series $ℂ {z}$ , viewed as an $A_1(C)$ -module.

Microfunctions

The module of microfunctions in one variable is by construction a direct limit of quotients $M_ε$ formed from holomorphic functions on a universal cover of a punctured disc. The Dirac delta appears as the class of an explicit function in one of the $M_ε$ , and the corollary above is what guarantees that the resulting solution of $xf = 0$ is already visible at a finite radius.

Localisation

For a commutative ring $R$ , an element $f ∈ R$ and an $R$ -module $N$ , the system $N ⟶ f N ⟶ f N → ⋅ s$ indexed by $ℕ$ has limit $N[1/f]$ . The worked example below carries this out for $N = R = K[x]$ and $f = x$ . Note the transition maps are $R$ -linear but not $A_1$ -linear, so this is a colimit of $K[x]$ -modules; the $A_1$ -structure on $K[x,1/x]$ is not inherited from the system.

A limit of non-zero modules that vanishes

Take $R = K[x]/(x^2)$ , all $M_i = R$ for $i ∈ N$ , and transition maps multiplication by $x$ . Every $M_i$ is a two-dimensional $K$ -vector space, but $ϕ_i,i+2$ is multiplication by $x^2 = 0$ , so every element dies within two steps and the limit is $0$ . Non-vanishing at every stage says nothing about the limit.

Worked Example

The localisation $K[x][1/x]$ as a direct limit

Step 1 - the system
Take $R = K[x]$ , index set $I = N$ with its usual order, $M_i = K[x]$ for every $i$ , and $ϕ_ij : K[x] → K[x]$ multiplication by $x^,j-i$ for $i ≤ j$ .

Check the axioms (6.20): $ϕ_ii$ is multiplication by $x^0 = 1$ , the identity; and $ϕ_jkϕ_ij$ is multiplication by $x^k-jx^j-i = x^k-i = ϕ_ik$ . The index set $ℕ$ is directed, since $max {i, j}$ bounds both.
Step 2 - the candidate map
Define $ψ_i : K[x] → K[x,1/x]$ by $ψ_i(f) = f/x^i$ . These are compatible: for $i ≤ j$ ,

$ψ_j(ϕ_ij(f)) = x^,j-ifx^,j = fx^,i = ψ_i(f).$

By the universal property there is a unique $K[x]$ -linear $ψ : M_i → K[x,1/x]$ with $ψ(ϕ_i(f)) = f/x^i$ .
Step 3 - surjective and injective
Surjective: a general element of $K[x,1/x]$ is $g/x^i$ with $g ∈ K[x]$ and $i ≥ 0$ , which is $ψ(ϕ_i(g))$ .

Injective: suppose $ψ(ϕ_i(f)) = 0$ , that is $f/x^i = 0$ , hence $f = 0$ ; then $ϕ_i(f) = 0$ already. For a general element the same computation applies after moving to a single stage, which (6.24) permits. Note where the hypothesis enters: $x$ is not a zero divisor in $K[x]$ , so no element dies along the system, every $ϕ_i$ is injective, and the limit is the increasing union of the $x^-iK[x]$ .
Step 4 - arithmetic in the limit, checked against the answer
Add $ϕ_2(f)$ and $ϕ_3(g)$ . The recipe (6.22) says: choose an upper bound, here $k = 3$ , push both forward, and stay at stage $3$ :

$ϕ_2(f) + ϕ_3(g) = ϕ_3 (ϕ_23(f) + g ) = ϕ_3(xf + g).$

Under $ψ$ this is $(xf+g)/x^3 = f/x^2 + g/x^3$ , which is the sum computed directly in $K[x,1/x]$ . Taking $f = 1$ and $g = x^2$ : the left side is $ϕ_3(x + x^2)$ and the right side is $1/x^2 + 1/x = (x+x^2)/x^3$ . The two agree.
Step 5 - what changes if the multiplier is a zero divisor
Run the same system over $R = K[x]/(x^2)$ with all $M_i = R$ and the same transition maps. Now $ϕ_i,i+2 = 0$ , so by (6.24) every element of every stage is zero in the limit and $M_i = 0$ , even though $_K M_i = 2$ for all $i$ . The limit of the system depends on the maps, not on the modules.

Result

$(K[x] ⟶ x K[x] ⟶ x ⋅ s ) ≅ K[x][1/x] = K[x,1/x]$ , with $ϕ_i(f)$ corresponding to $f/x^i$ ; the isomorphism is one of $K[x]$ -modules, and the transition maps are not $A_1$ -linear, so it says nothing on its own about the Weyl algebra structure. Over $K[x]/(x^2)$ the same shape of system has limit $0$ .

Applications and Industry Use

In a mathematics topic, this section covers downstream use inside mathematics, computing and engineering rather than a manufactured product.

Germs and stalks. The stalk of a sheaf at a point is by definition a direct limit over the neighbourhoods of the point. Everything local in geometry or analysis is a direct limit, and the exactness of the limit is why exactness of sequences of sheaves can be checked stalk by stalk.
Generalised solutions. Hyperfunctions and microfunctions are constructed as direct limits of quotients of modules of holomorphic functions, which is how objects with no pointwise values are still handled by algebra.
Localisation. Inverting an element or a multiplicative set is a direct limit, and the exactness of localisation - the fact that it preserves kernels - is the exactness of direct limits in disguise. Compare localisation and localisation as a tensor product.
Reduction to the finitely generated case. Writing an arbitrary module as the limit of its finitely generated submodules converts statements about huge modules into statements about small ones, which is how many finiteness arguments in this collection are set up.
Approximation in computation. A computable object presented as a direct limit is manipulated at a finite stage; the corollary that solutions of a finitely presented system come from a finite stage is precisely the statement that such computations are legitimate.

Design Considerations

For a mathematical object, design considerations are the modelling choices: which ring, which filtration, which category to work in.

Choosing the index set

The index set is part of the data, and there is usually a choice. Radii in $R_>0$ or radii $1/n$ in $ℕ$ give the same germs, because the second family is cofinal in the first. Preferring a countable cofinal subsystem makes the limit a countable increasing union, which is easier to reason about and to implement. Preferring the full family avoids having to prove cofinality. Neither choice changes the answer.

Pre-order or partial order

Antisymmetry is deliberately not required. Directed pre-orders arise naturally - for example when indexing by open sets, where two different sets can contain one another only in one direction but distinct indices may carry isomorphic modules - and nothing in the construction uses it.

Direct limit or inverse limit

The two constructions look symmetric and behave very differently. Direct limits are exact, have a clean description of elements, and model localisation. Inverse limits are only left exact, their elements are compatible families rather than individual elements, and they model completion. If the objects being approximated are defined near a point, the limit is direct; if they are defined by successively finer approximations of a single object, it is inverse.

Computational Notes

Read this as the manufacturing section of the template: how the object is actually built by machine, at what cost, and where the computation stops being decidable.

A direct limit is not a finite object and cannot be stored as one. What is computable is a stage together with a rule for moving forward.

Represent an element of $M_i$ by a pair: an index $i$ and an element of $M_i$ . This is always possible by (6.24).
To add two elements, find a common upper bound of the two indices and push both forward - the operation of (6.22).
To test whether an element is zero, one must decide whether it dies at some later stage. This is the only step that can be undecidable, and it is decidable exactly when the system provides an effective bound on how long an element can survive.
When the transition maps are injective, equality testing reduces to equality at a common stage, and the limit is as computable as the stages are.
When the system stabilises - all transition maps beyond some index are isomorphisms - the limit is that stage, and everything is finite. Noetherian hypotheses often force stabilisation, and it is worth checking for it before building any machinery.

For the localisation example the representation is exactly the familiar one: a numerator and an exponent, with equality decided by clearing denominators. For germs of holomorphic functions no faithful finite representation exists, which is why symbolic systems work with the annihilating ideal of a germ rather than with the germ itself.

Limits of Validity

Directedness cannot be dropped. Over a non-directed index set the colimit still exists, but (6.24) fails, the limit need not be exact, and the elementary description of elements is lost.
The limit forgets the stages. Non-isomorphic systems can have isomorphic limits, and no invariant of the limit recovers the individual $M_i$ . If information about a particular stage is needed, it must be carried separately.
No topology or convergence is involved. Despite the name, nothing converges. A direct limit is a purely algebraic colimit, and it says nothing about the analytic limits that motivate the examples.
Structure need not be inherited. If the transition maps are linear only over a subring, the limit is a module only over that subring, and any further structure on it - such as an action of the Weyl algebra - has to be constructed separately, as the localisation example shows.

Failure Modes and Common Mistakes

Assuming the stages embed into the limit

They do only when the transition maps are injective. In general $ϕ_i = ⋃ _j ≥ i ϕ_ij$ , which can be all of $M_i$ . The system over $K[x]/(x^2)$ with transition maps multiplication by $x$ has every stage non-zero and limit $0$ .

Confusing the direct limit with the direct sum or the product

The direct sum is the colimit over a discrete, hence non-directed, index set; its elements are finite tuples, not elements of a single stage. The direct product is not a colimit at all. Only for directed systems does the description every element comes from one stage hold.

Reversing the arrows by accident

Coutinho's convention orders indices so that transition maps go downward, which is convenient for shrinking discs; most other sources order them upward. A direct limit taken with the arrows accidentally reversed is an inverse limit, which is a different module with different exactness properties - for the germ example it would produce compatible families of functions defined on every disc of the family at once, that is functions holomorphic on the union of all of them, rather than germs.

Expecting Hom to commute with the limit in the first variable

$Hom_R(M, S_i) ≅ Hom_R(M,S_i)$ requires $M$ to be finitely presented. In the other variable the statement is false in the same form: $Hom_R( M_i, N)$ is an inverse limit, not a direct one, because $Hom$ is contravariant there.

Historical Notes

Inductive limits, as they were first called, entered mathematics through topology in the 1930s, in the work of Pontryagin and of Steenrod on homology of increasing families of complexes, and were codified in the axiomatic treatment of Eilenberg and Steenrod. Sheaf theory made them unavoidable: the stalk of a sheaf is a direct limit over neighbourhoods, and Leray's construction of sheaf cohomology in the 1940s depends on their exactness.

The categorical formulation as a filtered colimit and the recognition of exactness as a property that a category may or may not have - Grothendieck's axiom AB5 in the 1957 Tôhoku paper - put the construction in its final form. In analysis, Sato's definition of hyperfunctions around 1959 is a direct limit of quotients of spaces of holomorphic functions, and the microfunctions of Sato, Kashiwara and Kawai are built the same way. That is the use to which the construction is put in the next sections of this chapter.

Comparison

Direct limits against inverse limits and against unstructured colimits.
	Direct limit over a directed set	Inverse limit	Colimit over a discrete set
Arrows	$M_i → M_j$ for $i ≤ j$	$M_j → M_i$ for $i ≤ j$	none
Elements	one element of one stage	a compatible family over all stages	finite tuples
Exactness	exact	only left exact	exact, but no stage description
Typical example	germs, localisation	formal power series, completion	direct sum
Commutes with	kernels, cokernels, tensor	kernels only	cokernels
Zero limit possible?	yes, if elements die	yes, if the system has no threads	only if all summands are zero

Key Takeaways

Key points

A direct system is a family $M_i$ over a directed index set with compatible transition maps $ϕ_ij : M_i → M_j$ for $i ≤ j$ .
The direct limit is the disjoint union of the $M_i$ modulo the relation that identifies two elements as soon as they become equal at a later stage.
Every element of the limit comes from a single stage, and an element from stage $i$ is zero exactly when some $ϕ_ij$ kills it.
Directedness is used to prove transitivity of the relation and to define addition; without it both fail.
Direct limits over directed sets are exact and commute with tensor products, and with $Hom$ out of a finitely presented module.
Germs of holomorphic functions, localisations and the module of microfunctions are all direct limits; the transition maps decide everything, and a limit of non-zero modules can be zero.

FAQs

Why must the index set be directed?

Because two elements coming from different stages must be comparable somewhere in order to be added, and because transitivity of the identification (6.21) needs a common later stage. Without directedness the colimit still exists but loses the description of its elements and its exactness.

Can the direct limit be zero when all the modules are non-zero?

Yes. Take all stages equal to $K[x]/(x^2)$ with transition maps multiplication by $x$ ; every element is killed within two steps, so the limit vanishes. The limit depends on the maps at least as much as on the modules.

Is the direct limit the same as the union?

Only when the transition maps are injective, in which case each stage embeds and the limit is the increasing union of the images. In general the limit is a quotient of the disjoint union, not a union.

How does this differ from an inverse limit?

The arrows run the other way, elements are compatible families rather than single elements, and the operation is only left exact. Formal power series are an inverse limit of truncations; germs of convergent series are a direct limit of functions on shrinking discs. The two are not dual in any way that makes their properties match.

Does the limit depend on the choice of the index set?

Not if the change is to a cofinal subset. Restricting the family of radii $ε > 0$ to the sequence $1/n$ gives the same germs, because every radius is eventually bounded by some $1/n$ in the relevant direction.

Why is exactness such an important property here?

Because the modules of generalised solutions are defined as limits of quotients. Exactness is what allows a short exact sequence of stages - for instance holomorphic functions on a punctured disc modulo those on the disc - to pass to the limit without correction terms, and it is what makes stalkwise checking of exactness legitimate in sheaf theory.

If a differential system has a solution in a limit module, where does the solution live?

At some finite stage. Because the module of a system is finitely presented over $A_n$ , (6.26) applies, and every homomorphism into the limit factors through some $S_i$ . A germ solution is therefore an honest solution on some neighbourhood of positive size.

Is the direct limit unique?

Up to a unique isomorphism compatible with the canonical maps, yes, since it is defined by a universal property. The explicit construction as a quotient of a disjoint union is one realisation of it, convenient because it makes the two facts about elements visible.

References

S. C. Coutinho, A Primer of Algebraic D-modules, London Mathematical Society Student Texts 33, Cambridge University Press, 1995 - Ch. 6 §2, with the germ example and the verification that the operations are well defined; Ch. 6 §4, Exercises 4.5-4.7, for hyperfunctions.
A. Grothendieck, Sur quelques points d'algèbre homologique, Tôhoku Mathematical Journal 9 (1957), 119-221 - the axiom AB5, exactness of filtered colimits.
S. Eilenberg and N. Steenrod, Foundations of Algebraic Topology, Princeton University Press, 1952 - Ch. VIII, the classical treatment of direct systems and limits.
M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, 1969 - Ch. 2, exercises 14-19, for direct limits and their exactness in the commutative setting.
J. J. Rotman, An Introduction to Homological Algebra, second edition, Springer, 2009 - Ch. 5, for filtered colimits, the interchange with Hom, and counterexamples.
S. Mac Lane, Categories for the Working Mathematician, second edition, Springer, 1998 - Ch. IX, filtered colimits and their exactness.
M. Sato, Theory of hyperfunctions I, II, Journal of the Faculty of Science, University of Tokyo 8 (1959-60), 139-193 and 387-437 - hyperfunctions defined as a direct limit.
M. Kashiwara, T. Kawai and T. Kimura, Foundations of Algebraic Analysis, Princeton University Press, 1986 - Ch. 1, for microfunctions built by the same procedure.

AI Suggested Questions

Prove that a direct limit over a directed set commutes with tensor products, using only the two facts about elements.
Give a direct system with injective transition maps whose limit is not finitely generated, and explain why that is typical.
Show that a cofinal subsystem has the same direct limit, and identify where directedness is used.
Construct a colimit over a non-directed index set which is not exact.
Realise $K[x][1/f]$ for an arbitrary non-zero polynomial $f$ as a direct limit and describe the transition maps.
Verify that restriction maps between spaces of holomorphic functions are homomorphisms of Weyl algebra modules.
Explain why $Hom$ out of a module that is not finitely presented can fail to commute with a direct limit, with an example.