Computing a Direct Limit: Worked Constructions

Q: Is H_0 the same as the formal power series ring?

No. H_0 ≅ Cz consists of series with positive radius of convergence and sits strictly inside C[[z]] . Both are local rings and both are discrete valuation rings, which is why the distinction is easy to lose; but C[[z]] is an inverse limit of truncations, not a direct limit of function spaces.

Q: Why does localisation need no hypothesis on f?

Because the definition of M[1/f] already builds in the condition that u/f^,i is zero when some power of f kills u . That is exactly condition (2) of the criterion. When f is a non-zero-divisor nothing dies and the limit is a union; otherwise the f -torsion is discarded, possibly leaving 0 .

Q: Is H_0 finitely generated over A_1(C)?

No. The germs 1/(1-az) for distinct a are linearly independent, so H_0 has uncountable dimension over C , whereas a finitely generated module over the countably infinite dimensional algebra A_1(C) has countable dimension. This is why the theory of dimension and multiplicity developed later applies to the modules of equations, not to the modules solutions live in.

Overview

The construction of a direct limit presents $M_i$ as the disjoint union of the stages modulo eventual agreement. That description proves the limit exists and tells you what its elements are, but it is almost useless for identifying a particular limit. Nobody computing $H(ε)$ works with equivalence classes of pairs; they say "this is the ring of convergent power series" and then prove it.

The proof always has the same shape. Guess the answer $L$ . Produce compatible homomorphisms $ψ_i : M_i → L$ . Invoke the universal property to get a single map $ψ : M_i → L$ . Then check exactly two things: that the images of the $ψ_i$ exhaust $L$ , and that anything killed by some $ψ_i$ was already going to die inside the system. Those two conditions are necessary and sufficient, and they are usually easy to verify because they are conditions about one stage at a time.

This page states that criterion precisely, proves it, and then applies it four times: to an increasing union, to localisation, to the rationals as a limit over a directed set that is not a chain, and — the computation Chapter 6 needs — to germs of holomorphic functions at the origin. The last of these is the input to the construction of microfunctions, and the same criterion identifies the transition maps in that system as injective, which is what licenses checking non-vanishing at a single radius.

One theme runs through every example: the limit routinely has properties that no stage has. The germs at the origin form a local ring; no ring $H(ε)$ of functions holomorphic on a disc is local. Passing to the limit is where the invertibility, the exactness and the extra solutions come from.

Core Concepts

Before computing anything, four questions settle what kind of object is being built. Answering them in order removes most of the mistakes.

1. What is the index set, and which way do the arrows run?

The direction matters more than the notation. In every family indexed by shrinking neighbourhoods, the later stage is the smaller neighbourhood, because that is the direction in which the restriction maps point. On this page $ε' ≤ ε$ always means $D(ε') ⊂ eq D(ε)$ , and $ε^{'}$ is the later stage. Coutinho writes the order the other way round; the mathematics is identical, but a system read with the arrows reversed is an inverse limit and a different module.

2. Are the transition maps injective?

If they are, each stage embeds in the limit, the limit is the increasing union of the images, and an element is zero exactly when it was zero where it started. If they are not, elements can die later, and every equality in the limit is a claim that two things become equal at some finite stage. This single question decides how hard the second half of the recognition criterion will be.

3. Over which ring are the transition maps linear?

The limit is a module over whatever ring the transition maps respect, and no more. The system that computes $K[x][1/x]$ has transition maps given by multiplication by $x$ , which are $K[x]$ -linear but emphatically not $A_1$ -linear, since $∂(xf) = xf' + f ≠ x ∂ f$ . That system therefore computes a $K[x]$ -module. The Weyl algebra structure on $K[x][1/x]$ is real, but it comes from the quotient rule, not from the limit.

4. Is there a cofinal sequence?

An uncountable index set is a nuisance. If a countable subfamily is cofinal — every index bounded above by a member of it — the limit over the subfamily is the same module, and the limit becomes a countable increasing union or a countable colimit that induction can handle. For discs about the origin, the radii $1/n$ are cofinal in all positive radii, which is why germs can be discussed one sequence at a time.

Construction and Proof

Throughout, $M_i, ϕ_ij$ is a direct system of left $R$ -modules over a directed set $I$ , with canonical maps $ϕ_i : M_i → M_i$ .

Recognition criterion

Let $L$ be a left $R$ -module and let $ψ_i : M_i → L$ be homomorphisms with $ψ_j ϕ_ij = ψ_i$ whenever $i ≤ j$ . Let $ψ : M_i → L$ be the unique homomorphism with $ψ ϕ_i = ψ_i$ , supplied by the universal property. Then $ψ$ is an isomorphism if and only if both of the following hold.

Exhaustion. $L = ⋃ _i ∈ I ψ_i(M_i)$ .
No premature death. For every $i ∈ I$ and every $u ∈ M_i$ with $ψ_i(u) = 0$ , there exists $j ≥ i$ with $ϕ_ij(u) = 0$ .

Proof

Suppose (1) and (2). Every element of $M_i$ is $ϕ_i(u)$ for a single $i$ and $u ∈ M_i$ , so $ψ(ϕ_i(u)) = ψ_i(u)$ , and (1) says these values cover $L$ : $ψ$ is surjective. For injectivity, let $ϕ_i(u)$ lie in $ker ψ$ , so $ψ_i(u) = 0$ . By (2) there is $j ≥ i$ with $ϕ_ij(u) = 0$ , whence $ϕ_i(u) = ϕ_j(ϕ_ij(u)) = 0$ .

Conversely, if $ψ$ is an isomorphism then (1) holds because every element of the limit comes from a stage, and (2) holds because $ψ_i(u) = 0$ forces $ϕ_i(u) = 0$ , and an element of $M_i$ dies in the limit only by being killed by some $ϕ_ij$ .

Condition (2) is the one that is forgotten. Compatible maps that are collectively surjective are cheap; the pitfalls section below exhibits a system with a compatible surjection onto $Z/3Z$ whose limit is $Z[1/2]$ .

Injective transition maps

If every $ϕ_ij$ is injective, condition (2) reduces to the injectivity of each $ψ_i$ . In that case the criterion reads: the $ψ_i$ are injective and their images exhaust $L$ , so $L$ is the increasing union of copies of the stages. Every computation on this page except the localisation of a ring with zero divisors is of this kind.

Cofinal subsystems

If $I' ⊂ eq I$ is cofinal and directed, the maps $ϕ_i$ for $i ∈ I'$ satisfy both conditions of the criterion with $L = _i ∈ I M_i$ : exhaustion because any $ϕ_i(u)$ with $i ∈ I$ can be pushed to some $i' ∈ I'$ with $i' ≥ i$ , and (2) because a $ϕ_ij$ that kills $u$ can be followed by a further map into $I^{'}$ . Hence $_I' M_i ≅ _I M_i$ .

Key Equations

The criterion in symbols, for compatible maps $ψ_i : M_i → L$ :

L = ⋃ _i ∈ I ψ_i(M_i) ψ_i = ⋃ _j ≥ i ϕ_ij ( ∀ i) ⟺ ψ is an isomorphism.

(6.1)

The germ computation, over the directed set of radii ordered by $ε' ≤ ε$ when $D(ε') ⊂ eq D(ε)$ :

H_0 ;=; _ε > 0 H(ε) ; ≅ ; Cz ;=; ∑ _n ≥ 0 a_n z^n : _n | a_n | ^1/n < ∈ fty .

(6.2)

Localisation at a non-zero-divisor $f$ of a commutative ring $R$ , as a limit over $(N, ≤ )$ with every stage $M$ and every transition map multiplication by a power of $f$ :

M[1/f] ; ≅ ; ( M ⟶ f M ⟶ f M → ⋅ s ), ϕ_i(u) ⟷ uf^,i.

(6.3)

The rationals, over the positive integers ordered by divisibility, with $ϕ_mn$ multiplication by $n/m$ when $m ∣ n$ :

Q ; ≅ ; _(N_>0, ∣ ) Z, ϕ_m(a) ⟷ (a)/(m).

(6.4)

The interchange that makes all of this relevant to differential equations: for $M$ finitely presented over $A_n$ — which by the presentation-matrix description every system provides —

Hom_A_n (M, _i S_i ) ; ≅ ; _i Hom_A_n(M, S_i).

(6.5)

Variable Definitions

$I$: the directed index set of a direct system
$M_i$ , $ϕ_ij$: the module at stage $i$ and the transition map $M_i → M_j$ for $i ≤ j$
$ϕ_i$: the canonical map from stage $i$ into the direct limit
$L$ , $ψ_i$: a proposed answer and the compatible maps $M_i → L$ tested by the recognition criterion
$D(ε)$: the open disc of radius $ε$ about the origin of $ℂ$
$H(ε)$: the holomorphic functions on $D(ε)$ , a left $A_1(C)$ -module and a $ℂ$ -algebra
$H_0$: the module of germs of holomorphic functions at the origin
$ℂ {z}$: the ring of power series in $z$ with strictly positive radius of convergence
$ord(f)$: the order of vanishing of a germ at the origin, that is the index of its first non-zero Taylor coefficient
$M_ε$: the stage at radius $ε$ of the system whose limit is the module of microfunctions

Properties and Behaviour

What survives to the limit

Ring structure. If every $M_i$ is a $K$ -algebra and every $ϕ_ij$ a $K$ -algebra map, the limit is a $K$ -algebra, because products of two elements can be formed at a common stage.
Module structure over a fixed ring. If the $ϕ_ij$ are $R$ -linear for a ring $R$ acting on all stages compatibly, the limit is an $R$ -module. Nothing more is inherited; the ring must be fixed in advance.
Exactness. A directed limit of short exact sequences is short exact, so submodules, quotients and kernels may be computed stage by stage.
Torsion-freeness and flatness. A directed limit of torsion-free modules is torsion-free, and a directed limit of flat modules is flat; both follow from the fact that a putative counterexample would be witnessed at a single stage.

What does not survive

Finite generation. Every stage of the germ system is a module over a ring of functions, and the limit is not finitely generated over $A_1(C)$ ; likewise $ℚ$ is a limit of cyclic $ℤ$ -modules and is not finitely generated.
Being non-zero. A limit of non-zero modules can vanish, as the $K[x]/(x^2)$ localisation shows.
Noetherianity. $ℂ {z}$ happens to be Noetherian, but a limit of Noetherian rings need not be: $K[x_1, x_2, … ]$ is a limit of polynomial rings in finitely many variables.

Solutions in a limit come from a stageCoutinho (6.2), with (6.1.2)

Let $M = A_n/J$ be the module of a system of differential equations, which is cyclic and finitely presented, and let $S = S_i$ . Combining (6.5) with the description of solutions as homomorphisms, every solution of the system in $S$ is the image of a solution in some $S_i$ , and two solutions in $S_i$ give the same solution in $S$ exactly when they agree in some $S_j$ with $j ≥ i$ . For germs: a germ solution of a linear ODE with polynomial coefficients is an honest holomorphic solution on a disc of positive radius.

Examples and Special Cases

A filtration as a direct limit

Let $B_0 ⊂ eq B_1 ⊂ eq ⋅ s$ be the Bernstein filtration of $A_1$ , so $B_m$ is the $K$ -span of the monomials $x^a ∂^b$ with $a + b ≤ m$ , of dimension $m+22$ . Take $R = K$ , $I = N$ and the inclusions as transition maps. The maps $ψ_m : B_m ↪ A_1$ are compatible; they are injective, and they exhaust $A_1$ because every operator has finite Bernstein degree. By the criterion, $A_1 = B_m$ as a $K$ -vector space.

This is the trivial case, and it is worth naming precisely because it shows what the general construction buys: nothing at all when the transition maps are inclusions. Every exhaustive filtration is a direct limit, and the interesting examples are the ones where the transition maps are something else.

The rationals over a directed set that is not a chain

Order $N_>0$ by divisibility: $m ≤ n$ means $m ∣ n$ . This is directed — take $n = lcm$ — but it is not a chain, since $2$ and $3$ are incomparable. Put $M_m = Z$ for every $m$ and let $ϕ_mn : Z → Z$ be multiplication by $n/m$ when $m ∣ n$ . Compatibility is the identity $(k/n)(n/m) = k/m$ .

Take $L = Q$ and $ψ_m(a) = a/m$ . Compatibility: $ψ_n(ϕ_mn(a)) = (n/m)a/n = a/m$ . Exhaustion: any rational is $a/m$ . And $ψ_m(a) = 0$ forces $a = 0$ , so condition (2) is immediate. Hence $Q ≅ Z$ over the divisibility order. Restricting the index set to the powers of a single prime $p$ is not cofinal — $3$ divides no power of $2$ — and gives $Z[1/p]$ instead; restricting to the factorials $m!$ is cofinal and gives $ℚ$ again.

Localisation

Let $R$ be commutative, $M$ an $R$ -module and $f ∈ R$ . Index by $ℕ$ , set $M_i = M$ and let $ϕ_ij$ be multiplication by $f^,j-i$ . With $L = M[1/f]$ and $ψ_i(u) = u/f^,i$ , exhaustion is the definition of $M[1/f]$ , and condition (2) says: if $u/f^,i = 0$ then $f^,ku = 0$ for some $k$ — which is exactly the definition of equality in a localisation. So (6.3) holds with no hypothesis on $f$ at all; when $f$ is a zero divisor, condition (2) is doing real work, and the elements that die are precisely the $f$ -torsion. See localisation of rings and modules.

The extreme case is instructive: over $R = K[x]/(x^2)$ with $M = R$ and $f = x$ , every element is killed by $x^2$ , so every element dies at the second step and $M[1/x] = 0$ even though every stage is a two-dimensional vector space.

Germs of holomorphic functions

The system used throughout Chapter 6: stages $H(ε)$ , transition maps restriction, later stage = smaller radius. The limit is $H_0$ , the germs at the origin, and the worked example below identifies it with $ℂ {z}$ . Restriction is injective by the identity theorem, so every $H(ε)$ embeds in $H_0$ and the limit is a genuine increasing union.

The transition maps are $A_1(C)$ -linear, since restricting commutes with multiplying by $z$ and with differentiating. So unlike the localisation example, this system computes a limit in the category of $A_1(C)$ -modules and the answer carries a Weyl algebra action.

The microfunction system

The stages are $M_ε = H( D(ε)) / π^*H(D(ε))$ , where $D(ε) = z ∈ C : ℜ(z) < ε$ covers the punctured disc via $π(z) = e^z$ , and the transition maps are induced by restriction. Here the stages are quotients, so injectivity of the transition maps is a claim that has to be proved rather than read off; it is proved on the microfunctions page, and it is what allows a microfunction to be shown non-zero by exhibiting one radius at which its representative is not in the image of $π^*$ .

This is the example that motivates the whole apparatus. The limit contains a solution of $x u = 0$ — the Dirac delta — which no stage of any system of honest functions contains.

Worked Example

Germs at the origin are the convergent power series

Step 1 - the system, stated so the arrows point the right way
Index set $I = R_>0$ , ordered by $ε' ≤ ε$ if and only if $ε' ≤ ε$ as real numbers, so that later stages are smaller discs. Given $ε_1, ε_2$ the radius $ε_1,ε_2$ is a later stage than both, so $I$ is directed. Stage $ε$ is $H(ε)$ , the holomorphic functions on $D(ε)$ , and the transition map $ρ_εε' : H(ε) → H(ε')$ is restriction. Restriction of a restriction is a restriction, so the compatibility conditions hold.

Each $H(ε)$ is a left $A_1(C)$ -module with $z$ acting by multiplication and $∂$ by $d/dz$ , and restriction respects both actions, so this is a direct system of $A_1(C)$ -modules.
Step 2 - the candidate and the compatible maps
Let $L = Cz$ be the set of power series $∑ _n ≥ 0 a_n z^n$ with a strictly positive radius of convergence, and let $ψ_ε : H(ε) → Cz$ send $f$ to its Taylor series at the origin. A function holomorphic on $D(ε)$ has Taylor series with radius of convergence at least $ε > 0$ , so $ψ_ε$ lands in $L$ .

Compatibility is the statement that the Taylor coefficients at $0$ of $f$ and of $f | _D(ε')$ are the same numbers, which they are: $a_n = f^(n)(0)/n!$ depends only on the germ. Hence $ψ_ε'ρ_εε' = ψ_ε$ , and the universal property gives $ψ : H_0 → Cz$ .
Step 3 - exhaustion
Let $∑ a_n z^n ∈ Cz$ have radius of convergence $ρ > 0$ . Pick any $ε$ with $0 < ε < ρ$ . The series converges uniformly on compact subsets of $D(ε)$ and so defines $f ∈ H(ε)$ whose Taylor series is the given one. Therefore $∑ a_n z^n = ψ_ε(f)$ , and condition (1) of the criterion holds.

Note where the hypothesis $ρ > 0$ is used: it is exactly what guarantees that some stage of the system sees the series. A formal series with $ρ = 0$ , such as $∑ n!,z^n$ , appears at no stage and is not a germ.
Step 4 - no premature death
Suppose $f ∈ H(ε)$ has $ψ_ε(f) = 0$ , that is, all Taylor coefficients of $f$ at $0$ vanish. Since $D(ε)$ is connected and the zero set of $f$ has an accumulation point at $0$ of infinite order, the identity theorem gives $f = 0$ on all of $D(ε)$ . So $u = f$ is already zero at its own stage, and condition (2) holds in the strongest possible way. By the recognition criterion $ψ$ is an isomorphism of $A_1(C)$ -modules and of $ℂ$ -algebras, proving (6.2).

The same argument shows each $ρ_εε'$ is injective, so each $H(ε)$ embeds in $H_0$ and the limit is the increasing union $⋃ _ε > 0 H(ε)$ inside $ℂ {z}$ .
Step 5 - a property the limit has and no stage has
In $ℂ {z}$ a series is invertible exactly when $a_0 ≠ 0$ : if $a_0 ≠ 0$ the geometric-series inversion converges on a small enough disc, and if $a_0 = 0$ then every multiple has zero constant term. So $H_0$ is a local ring with maximal ideal $z ℂ {z}$ , and since every non-zero germ is $z^k$ times a unit, it is a discrete valuation ring with valuation $ord$ .

Check the arithmetic on the smallest interesting case. The polynomial $1 + z$ is entire, hence present at every stage. Its inverse is

$(1)/(1+z) = ∑ _n ≥ 0 (-1)^n z^n, radius of convergence 1.$

Multiplying out: the constant term is $1$ , and for $k ≥ 1$ the coefficient of $z^k$ in $(1+z) ∑ (-1)^n z^n$ is $(-1)^k + (-1)^k-1 = 0$ . So the product is $1$ . But this inverse exists only at the stages $ε ≤ 1$ ; the element $1 + z$ is not invertible in $H(2)$ , since $1 + z$ vanishes at $z = -1 ∈ D(2)$ . Invertibility is created by passing to a later stage, which is precisely the phenomenon the direct limit records.

One consequence worth recording: the germs $1/(1 - az)$ for distinct $a ∈ C$ are linearly independent over $ℂ$ , so $H_0$ has uncountable dimension as a $ℂ$ -vector space. A finitely generated $A_1(C)$ -module has countable dimension, because $A_1(C)$ does. Hence $H_0$ is not finitely generated over the Weyl algebra.

Result

$H_0 = _ε > 0 H(ε) ≅ Cz$ , an isomorphism of $ℂ$ -algebras and of left $A_1(C)$ -modules. The limit is a discrete valuation ring with maximal ideal $z ℂ {z}$ and residue field $ℂ$ ; it is an increasing union of the stages, since all transition maps are injective; and it is not finitely generated over $A_1(C)$ .

Applications and Industry Use

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

Solutions near a point. Germs of holomorphic functions are the natural target module for local solutions of an ODE with polynomial coefficients; because the germ module is a limit, a germ solution of a system is an actual solution on a disc, which is what makes local existence theorems usable inside the algebra.
Constructing generalised solutions. Hyperfunctions and microfunctions are defined as direct limits of quotients, and there is no other elementary way to produce a module in which $x u = 0$ has a non-zero solution.
Localisation. Inverting a polynomial is a direct limit; the D-module $K[x][1/f]$ obtained this way is the object whose holonomicity gives the Bernstein-Sato polynomial.
Reduction to finite data. Writing a module as the limit of its finitely generated submodules transfers statements from arbitrary modules to Noetherian ones, and is the standard first move in proofs about very large modules.
Sheaves and stalks. Every stalk in algebraic geometry or analysis is a direct limit over neighbourhoods, computed by exactly the criterion above; $H_0$ is the stalk at the origin of the sheaf of holomorphic functions.

Design Considerations

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

Choose the index set for the proof, not for the object

Germs may be computed over all positive radii or over the sequence $1/n$ ; the answers agree by cofinality. Use the uncountable version when the statement should be manifestly independent of choices, and the sequence when the proof is an induction. The same applies to neighbourhoods of a point in any topological setting: any neighbourhood basis is cofinal in all neighbourhoods.

Decide early which ring the limit will be a module over

This is the choice that most often goes wrong, and it is invisible in the notation. Multiplication by $x$ is $K[x]$ -linear and not $A_1$ -linear; restriction of holomorphic functions is $A_1(C)$ -linear; the maps induced on the microfunction stages are $A_1(C)$ -linear because $π^*$ is. Write down which ring acts before writing down the limit, and check the transition maps against it.

Prefer the universal property to the explicit quotient

The explicit model — pairs modulo eventual agreement — is the right object for proving general theorems such as exactness. For identifying a specific limit it is almost always worse than the recognition criterion, because it forces a discussion of representatives where the criterion asks two questions about one stage at a time. The exception is when no candidate $L$ presents itself, which is exactly the situation for microfunctions, where the limit is the definition of the object.

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.

An element of a direct limit is represented by a pair: an index and an element of that stage. Three operations are then needed, in increasing order of difficulty.

Addition. Find a common later stage, push both representatives forward, add there. Cheap whenever the index set has computable upper bounds.
Equality. Reduce to testing whether a difference is zero, which is the next item.
Zero testing. Decide whether some $ϕ_ij$ kills the representative. This is the only step that can fail to be decidable, and everything practical turns on it.

For localisation, zero testing is clearing denominators: $u/f^,i = 0$ if and only if $f^,ku = 0$ for some $k$ , and over a Noetherian ring the annihilator chain stabilises, so a bound on $k$ exists and the test terminates. For germs, no faithful finite representation exists at all: a germ is an infinite object, and equality of germs is equality of infinitely many Taylor coefficients.

The standard workaround in computer algebra is to represent a germ not by its coefficients but by a differential system it satisfies together with finitely many initial values — that is, by a cyclic module $A_1/J$ and a point of the solution space. Germs representable this way are the holonomic ones, equality is then decided by comparing annihilating ideals and finitely many initial coefficients, and the whole of the Zeilberger machinery is built on this representation. Germs outside that class, and in particular arbitrary elements of $ℂ {z}$ , are simply not objects a computer algebra system can hold.

Practical implementations of the localisation side include Macaulay2 and Singular for commutative localisation and saturation, and the D-module packages Dmodules and dmod.lib for the Weyl algebra localisations $K[x][1/f]$ , where the relevant finiteness is holonomicity rather than any property of the limit as such.

Limits of Validity

The criterion needs a candidate. It decides whether a proposed $L$ is the limit; it does not produce one. When no candidate is available the explicit construction is the definition, and that is the case for microfunctions.
Directedness is used throughout. Exhaustion by single stages and the stage-by-stage description of the kernel both fail over an index set that is not directed, and with them the criterion.
Zero testing may be undecidable. Condition (2) is an existential statement about later stages. Nothing guarantees it can be checked effectively, and for germs it cannot.
The limit forgets the system. Non-isomorphic systems have isomorphic limits: $ℚ$ arises from the divisibility system and from the factorial chain, and $H_0$ from every cofinal family of radii. Any invariant of a stage that is needed later must be carried along explicitly.

Failure Modes and Common Mistakes

Checking exhaustion and calling it a day

Compatible maps that jointly surject need not identify the limit. Take $M_i = Z$ for $i ∈ N$ with $ϕ_ij$ multiplication by $2^,j-i$ , whose limit is $Z[1/2]$ . Define $ψ_i : Z → Z/3Z$ by $ψ_i(a) = 2^-i a 3$ , using that $2$ is invertible modulo $3$ . These are compatible, since $ψ_i+1(2a) = 2^-(i+1)2a = 2^-ia$ , and each is surjective. But $Z[1/2] ≅ Z/3Z$ . Condition (2) fails: $ψ_0(3) = 0$ while $3$ is killed by no transition map.

Assuming the limit is a module over the biggest ring in sight

The system $K[x] ⟶ x K[x] ⟶ x ⋅ s$ has $K[x][1/x]$ as its limit, and $K[x][1/x]$ carries an action of $A_1$ . It does not follow, and it is not true, that this system is a direct system of $A_1$ -modules: multiplication by $x$ does not commute with $∂$ . The Weyl algebra structure on the localisation comes from extending the derivation by the quotient rule, an independent fact about localisation. If the transition maps are only $K[x]$ -linear, the criterion establishes only an isomorphism of $K[x]$ -modules.

Using a subfamily that is not cofinal

Restricting the germ system to radii $ε ≥ 1$ produces a directed subfamily whose limit is $H(1)$ , not $H_0$ : with smaller radii as later stages, no $ε ≥ 1$ bounds $ε = 1/2$ from above. Restricting the divisibility system to powers of $2$ gives $Z[1/2]$ rather than $ℚ$ for the same reason. Cofinality has to be checked in the order actually being used, which is why fixing the direction of the arrows first is not pedantry.

Treating a formal power series as a germ

$C[[z]]$ is not the limit of the $H(ε)$ ; it is an inverse limit, of the truncations $C[z]/(z^n)$ . The germ module is the strictly smaller ring $ℂ {z}$ of series with positive radius of convergence. The distinction is not cosmetic: $∑ n!,z^n$ is a perfectly good formal solution of $z^2u' + (z-1)u + 1 = 0$ and is not a germ of any holomorphic function, so a differential equation can have formal solutions and no local analytic ones.

Confusing the limit of quotients with the quotient of limits done in the wrong order

For the microfunction system the stages are already quotients, $M_ε = H( D(ε))/π^*H(D(ε))$ . Because direct limits are exact, taking the limit of the quotients agrees with taking the quotient of the limits — but this is a theorem being used, not a triviality, and it fails for inverse limits. Whenever a stage is a quotient, say which fact is being invoked.

Comparison

The systems computed on this page, with the data that determines each answer.
System	Index set	Transition map	Injective?	Limit
$B_m ⊂ eq A_1$	$(N, ≤ )$	inclusion	yes	$A_1$ as a $K$ -space
$ℤ$ at every index	$(N_>0, ∣ )$	multiplication by $n/m$	yes	$ℚ$
$M$ at every index	$(N, ≤ )$	multiplication by $f$	iff $f$ is a non-zero-divisor on $M$	$M[1/f]$
$K[x]/(x^2)$	$(N, ≤ )$	multiplication by $x$	no	$0$
$H(ε)$	radii, smaller = later	restriction	yes	$ℂ {z}$
$M_ε$	radii, smaller = later	induced by restriction	yes, but it needs proof	microfunctions $ℳ$

Reading the table across: the limit is determined by the transition maps far more than by the stages. Rows three and four have the same shape and the same kind of stage; the difference between $M[1/f]$ and $0$ is entirely whether $f$ kills anything.

Key Takeaways

Key points

To identify a direct limit, guess the answer $L$ , build compatible maps $ψ_i : M_i → L$ , and check exhaustion together with the condition that anything killed by a $ψ_i$ is already killed by some transition map.
Both conditions are necessary; a compatible family of surjections proves nothing on its own.
When all transition maps are injective the limit is the increasing union of the stages, and the second condition reduces to injectivity of each $ψ_i$ .
$_ε > 0 H(ε) ≅ Cz$ , the convergent power series: exhaustion is convergence on a small disc, the second condition is the identity theorem.
The limit is a module only over the ring whose action the transition maps respect; multiplication by $x$ is not $A_1$ -linear.
The limit routinely acquires properties no stage has - $H_0$ is a discrete valuation ring, and the microfunction limit contains a solution of $x u = 0$ .
A cofinal subfamily gives the same limit; a subfamily that is not cofinal generally does not.

FAQs

Why not just compute with the explicit construction?

Because it forces every statement to be about equivalence classes of pairs. The recognition criterion replaces that by two questions asked one stage at a time, and the stages are ordinary modules one already understands. The explicit model remains the right tool for proving general theorems such as exactness.

H_0

the same as the formal power series ring?

No. $H_0 ≅ Cz$ consists of series with positive radius of convergence and sits strictly inside $C[[z]]$ . Both are local rings and both are discrete valuation rings, which is why the distinction is easy to lose; but $C[[z]]$ is an inverse limit of truncations, not a direct limit of function spaces.

Does the germ module depend on which family of radii is used?

Not as long as the family is cofinal, meaning that arbitrarily small radii occur in it. The radii $1/n$ give the same limit as all positive radii. A family bounded away from $0$ gives the functions on the smallest disc in the family instead.

Why does localisation need no hypothesis on

f

Because the definition of $M[1/f]$ already builds in the condition that $u/f^,i$ is zero when some power of $f$ kills $u$ . That is exactly condition (2) of the criterion. When $f$ is a non-zero-divisor nothing dies and the limit is a union; otherwise the $f$ -torsion is discarded, possibly leaving $0$ .

Can I compute a direct limit on a machine?

Only if zero testing is effective. Representing an element as an index plus a stage element is always possible, and addition needs only an upper bound in the index set; but deciding whether an element dies later is an existential search. For localisation over a Noetherian ring it terminates. For germs it is not a finite problem at all, and one works with annihilating ideals instead of coefficients.

H_0

finitely generated over

A_1(C)

No. The germs $1/(1-az)$ for distinct $a$ are linearly independent, so $H_0$ has uncountable dimension over $ℂ$ , whereas a finitely generated module over the countably infinite dimensional algebra $A_1(C)$ has countable dimension. This is why the theory of dimension and multiplicity developed later applies to the modules of equations, not to the modules solutions live in.

Where does the recognition criterion get used in this chapter?

Twice. It identifies the germ module, which is the stage-level input to the microfunction construction; and it is the argument pattern behind the proof that the transition maps of the microfunction system are injective, which is what makes a microfunction detectable at a single radius.

Does a solution of a differential system in a limit module really come from one stage?

Yes, provided the module of the system is finitely presented over $A_n$ , which it always is. Then $Hom$ out of it commutes with the limit, by (6.5), so a solution in $S_i$ is the image of a solution in some $S_i$ . Without finite presentation the interchange can fail.

References

S. C. Coutinho, A Primer of Algebraic D-modules, London Mathematical Society Student Texts 33, Cambridge University Press, 1995 - Ch. 6 §2, where the germ system is used as the running example of a direct limit.
M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, 1969 - Ch. 2 exercises 14-19 for direct limits, Ch. 3 for localisation as the universal solution to inverting an element.
J. J. Rotman, An Introduction to Homological Algebra, second edition, Springer, 2009 - Ch. 5, for filtered colimits, the universal property and the interchange with Hom.
R. C. Gunning and H. Rossi, Analytic Functions of Several Complex Variables, Prentice-Hall, 1965 - Ch. 1, for the ring of germs of holomorphic functions and its local structure.
L. V. Ahlfors, Complex Analysis, third edition, McGraw-Hill, 1979 - Ch. 5, for the identity theorem and radius of convergence, the two analytic inputs to the worked example.
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, Journal of the Faculty of Science, University of Tokyo 8 (1959), 139-193 - the construction of generalised functions as a limit of quotients of spaces of holomorphic functions.
ISO 80000-2:2019, Quantities and units - Part 2: Mathematics, International Organization for Standardization, for the notation used for limits, number systems and operators.

AI Suggested Questions

Apply the recognition criterion to show that every module is the direct limit of its finitely generated submodules.
Compute the direct limit of the system $Z/p → Z/p^2 → ⋅ s$ with the maps induced by multiplication by $p$ , and identify it.
Give a direct system of $ℤ$ -modules with a compatible family of surjections onto $Z/5Z$ whose limit is not $Z/5Z$ .
Show that $ℂ {z}$ is a discrete valuation ring and identify its field of fractions.
Verify that restriction maps between spaces of holomorphic functions are homomorphisms of $A_1(C)$ -modules, and say exactly which axioms are used.
Decide whether the germs at the origin of solutions of $z^2u' + (z-1)u + 1 = 0$ form a non-zero vector space, and reconcile the answer with the formal solution $∑ n!,z^n$ .
Exhibit a cofinal subfamily of the divisibility order on $N_>0$ that is a chain, and one that is not.