Solutions as Module Homomorphisms

Q: Can the solution space be infinite-dimensional?

Yes. The operator x_1∂_2 - x_2∂_1 in two variables annihilates every polynomial in x_1^2+x_2^2 , so its polynomial solution space is infinite-dimensional. Finiteness is a consequence of holonomicity, not of the correspondence itself.

Q: What does Ext^1 mean concretely?

For a single operator it is the cokernel of the action of P on the target: the space of inhomogeneous right-hand sides g for which Pf = g has no solution, modulo those for which it does. For P = x and target K[x] , that space is one-dimensional and is detected by the single condition g(0) = 0 .

Q: How does this relate to the classical count of solutions of an ODE?

For an ordinary equation of order k with non-vanishing leading coefficient near a point, the analytic solution space near that point has dimension exactly k , and the module-theoretic invariant reproducing k is the holonomic rank. Polynomial solutions are a subspace of that and can be much smaller.

Overview

A system of linear differential equations with polynomial coefficients has already been converted into a single module $M = A_n/J$ on the previous page. That conversion deliberately discarded the functions. This page recovers them.

The recovery statement is short: for any left $A_n$ -module $S$ , the solutions of the system with values in $S$ are in natural bijection with the module homomorphisms $M \to S$ . The bijection sends a homomorphism to its value on the canonical generator, and a solution $s$ to the unique homomorphism sending the generator to $s$ . It is linear over the ground field, so solution spaces are literally $Hom$ spaces.

Three things follow, and they are why the reformulation is worth making. First, the question where does this equation have solutions becomes the question which $S$ to choose, and $S$ ranges over a category rather than over an ad hoc list of function spaces; distributions, hyperfunctions and microfunctions enter as target modules on the same footing as polynomials. Second, the assignment $M ↦ Hom_A_n(M,S)$ is a contravariant functor, so maps between systems produce maps between solution spaces automatically, in the reversed direction. Third, that functor is left exact but not exact, and its first derived functor measures precisely the failure of an inhomogeneous equation to be solvable.

What the theorem does not do is produce solutions. It says nothing about whether $Hom_A_n(M,S)$ is non-zero, or finite-dimensional. Both can fail, and the examples below show that both do.

Definition

Throughout $K$ has characteristic zero, $K[X] = K[x_1, … ,x_n]$ is the polynomial module, and $M = A_n/J$ with $J = ∑ _i A_nP_i$ , with canonical generator $u = 1 + J$ .

Solution space in a moduleCoutinho, Ch. 6 §1

Let $S$ be a left $A_n$ -module, not necessarily finitely generated, and let $M$ be a finitely generated left $A_n$ -module. The solution space of $M$ in $S$ is

Sol_S(M) = Hom_A_n(M,S),

(6.8)

the set of all homomorphisms of left $A_n$ -modules from $M$ to $S$ . It is a vector space over $K$ under pointwise addition and scalar multiplication.

Solutions are homomorphismsCoutinho (6.1.2)

Let $M = A_n/J$ be the module of the system $P_1f = … = P_mf = 0$ , and let $S$ be any left $A_n$ -module. Then the map

Hom_A_n(M,S) ⟶ , s ∈ S : P_1s = … = P_ms = 0 ,, φ ⟼ φ(u),

(6.9)

is an isomorphism of $K$ -vector spaces. In particular, taking $S = K[X]$ , the space of polynomial solutions of the system is isomorphic to $Hom_A_n(M,K[X])$ .

Note

The finiteness assumptions are asymmetric on purpose. $M$ is finitely generated because it comes from a finite system; $S$ is not, and typically cannot be. The module $C^ ∈ fty(U)$ over $A_n(R)$ is not finitely generated, as Coutinho's Exercise 4.4 shows using Baire's theorem, and neither is the module of germs of holomorphic functions.

Core Concepts

The target module is a modelling decision

Classically one asks for solutions and then argues about what kind of object a solution is allowed to be. Here the kind is fixed first, by naming a left $A_n$ -module $S$ , and the answer is then determined. Enlarging $S$ can only enlarge the solution space, since an inclusion $S ↪ S^{'}$ of $A_n$ -modules induces an injection of $Hom$ spaces. Every classical extension of the notion of solution - weak solutions, distributions, hyperfunctions - is an instance of enlarging $S$ .

Why generalised solutions are forced on us

The equation $xf = 0$ in one variable has no non-zero classical solution of any kind, because a function vanishing away from a point vanishes. Its module $A_1/A_1x$ is nevertheless non-zero. The mismatch is resolved only by choosing a target in which an object concentrated at the origin exists; the Dirac delta is such an object, and it is constructed algebraically as a microfunction later in the chapter.

Solutions of the system, not of the module

It is worth keeping the two descriptions apart. The set $s ∈ S : P_is = 0$ depends on the chosen list of operators; the space $Hom_A_n(M,S)$ depends only on $M$ . The theorem says they agree, and therefore that the solution set is an invariant of the module - equivalent systems have identical solution spaces in every target simultaneously.

Construction and Proof

The proof of the theorem is the universal property of a quotient, applied twice.

Proof of the theorem

The map is well defined. If $φ : M \to S$ is $A_n$ -linear then $P_iφ(u) = φ(P_iu) = φ(0) = 0$ , so $φ(u)$ does lie in the stated solution set. Linearity in $φ$ is immediate from the pointwise operations.

Injectivity. $M = A_nu$ is generated by $u$ , so a homomorphism is determined by its value there: if $φ(u) = ψ(u)$ then $φ(Qu) = Qφ(u) = Qψ(u) = ψ(Qu)$ for every $Q ∈ A_n$ .

Surjectivity. Let $s ∈ S$ satisfy $P_is = 0$ for all $i$ . Define $φ : A_n → S$ by $φ(Q) = Qs$ . This is $A_n$ -linear because $φ(RQ) = (RQ)s = R(Qs)$ . Its kernel is a left ideal containing each $P_i$ , hence containing $J$ , so $\tilde{φ}$ factors as $A_n → A_n/J = M ⟶ φ S$ with $φ(u) = s$ .

Why the correspondence is only $K$ -linear

$Hom_A_n(M,S)$ carries no natural $A_n$ -module structure, and not even a $K[X]$ -module structure. The obvious candidate fails: for $g ∈ K[X]$ and $φ$ a homomorphism, the map $g φ$ defined by $m ↦ g,φ(m)$ is not $A_n$ -linear, because

(gφ)(∂_i m) = g,∂_i φ(m) ∂_i,(gφ)(m) = g,∂_iφ(m) + (∂ g)/(∂ x_i),φ(m),

and the two agree for all $m$ only when $∂ g/∂ x_i$ annihilates the image. This is not a defect of the construction: it reflects the fact that the solution set of a differential equation is closed under addition and constant scalars but not under multiplication by functions.

What does act is smaller and larger at once. The ring $End_A_n(S)$ acts on the left of $Hom_A_n(M,S)$ by composition, and $End_A_n(M)$ acts on the right. If $S$ happens to be an $(A_n,R)$ -bimodule for some ring $R$ , then $Hom_A_n(M,S)$ is a right $R$ -module; the case $S = A_n$ gives the dual module and is the entry point to duality theory.

Functoriality

A homomorphism $θ : M \to N$ induces $θ^* : Hom_A_n(N,S) → Hom_A_n(M,S)$ by $φ \mapsto φ θ$ , so $Sol_S$ is a contravariant functor. Concretely, if $N = A_n/J'$ with $J ⊂ eq J'$ , so that $N$ is the module of a larger system, then $M$ surjects onto $N$ and the induced map on solution spaces is the inclusion of the solutions of the larger system into those of the smaller one. Arrows between systems and arrows between solution spaces run in opposite directions.

Key Equations

For a cyclic module the whole theory reduces to evaluation at the generator:

Hom_A_n(A_n/J,; S) ; ≅ ; , s ∈ S : Js = 0 ,, φ ↦ φ(1+J).

(6.10)

For a single operator in one variable the free resolution is two terms long, because $A_1$ is a domain and right multiplication by $P ≠ 0$ is injective:

0 ⟶ A_1 ⟶ ⋅ ,P A_1 ⟶ A_1/A_1P ⟶ 0.

(6.11)

Applying $Hom_A_1(-,S)$ and identifying $Hom_A_1(A_1,S)$ with $S$ turns the resolution into a two-term complex $S \to S$ given by the action of $P$ itself, so that

Hom_A_1(A_1/A_1P,,S) = (P : S → S), Ext^1_A_1(A_1/A_1P,,S) = coker(P : S → S),

(6.12)

and all higher $Ext$ groups vanish. The kernel is the solution space of the homogeneous equation; the cokernel is the obstruction space for the inhomogeneous equation $Pf = g$ , which is solvable in $S$ for every $g$ precisely when that cokernel is zero.

Left exactness in general: for a short exact sequence $0 → M' → M → M'' → 0$ of $A_n$ -modules,

0 → Hom_A_n(M'',S) → Hom_A_n(M,S) → Hom_A_n(M',S) → Ext^1_A_n(M'',S) → ⋅ s

(6.13)

Variable Definitions

$A_n$: the $n$ -th Weyl algebra over a field $K$ of characteristic zero
$J$: the left ideal $∑ _i A_nP_i$ generated by the operators of the system
$M$: the module $A_n/J$ of the system, with canonical generator $u = 1+J$
$S$: the target module in which solutions are sought
$Hom_A_n(M,S)$: the space of $A_n$ -linear maps $M \to S$ , that is the solution space
$Ext^1_A_n(M,S)$: the first derived functor of Hom, the obstruction space for inhomogeneous problems
$K[X]$: the polynomial ring $K[x_1, … ,x_n]$ as a left $A_n$ -module
$C^ ∈ fty(U)$: the smooth real functions on an open set $U ⊂ eq R^n$ , a left $A_n(R)$ -module
$H(U)$: the holomorphic functions on an open set $U ⊂ eq C$ , a left $A_1(C)$ -module
$ℳ$: the module of microfunctions in one variable

Properties and Behaviour

Left exactness, and its failure to be exactness

For fixed $S$ the functor $Hom_A_n(-,S)$ is additive, contravariant and left exact: it converts a short exact sequence into the exact sequence (6.13). It is exact for every $S$ if and only if $S$ is an injective $A_n$ -module, which the usual function modules are not.

An explicit failure of exactness

Take $n = 1$ , $S = K[x]$ and the exact sequence $0 → A_1x → A_1 → A_1/A_1x → 0$ . Since $A_1$ is a domain, $A_1x ≅ A_1$ as a left module, so applying $Hom_A_1(-,K[x])$ gives the complex

0 → 0 → K[x] ⟶ ⋅ ,x K[x],

whose right-hand map is multiplication by $x$ : injective, with image $xK[x]$ . The cokernel is one-dimensional, so $Ext^1_A_1(A_1/A_1x,,K[x]) ≅ K$ . Concretely: $xf = g$ has a polynomial solution exactly when $g(0) = 0$ , and the single condition $g(0) = 0$ is the obstruction group.

Ordinary equations have finite-dimensional solution spacesCoutinho, Ch. 6 §4, Exercise 4.2

Let $0 ≠ P ∈ A_1(C)$ have order $k$ in $∂$ . Then the space of polynomial solutions of $Pf = 0$ has dimension at most $k$ ; in particular it is finite-dimensional.

Reason: if $k = 0$ then $P$ is a non-zero polynomial and the only polynomial solution is $0$ . If $k ≥ 1$ , write $P = ∑ _i ≤ k f_i(x)∂^i$ with $f_k ≠ 0$ and pick $a ∈ C$ with $f_k(a) ≠ 0$ . Near $a$ the equation is an ordinary linear differential equation with analytic coefficients and non-vanishing leading coefficient, so its analytic solution space there has dimension exactly $k$ . Polynomial solutions inject into that space, since a polynomial vanishing to infinite order at $a$ is zero.

Partial equations need notCoutinho, Ch. 6 §4, Exercise 4.1

Let $P = x_1∂_2 - x_2∂_1 ∈ A_2$ . The polynomials $(x_1^2+x_2^2)^k$ , $k ≥ 0$ , are all solutions of $Pf = 0$ : applying the operator gives $2k(x_1x_2 - x_2x_1)(x_1^2+x_2^2)^k-1 = 0$ . They have distinct degrees, so they are linearly independent and the polynomial solution space is infinite-dimensional. In fact, in characteristic zero it is exactly $K[x_1^2+x_2^2]$ , the invariants of the infinitesimal rotation.

This is consistent with the dimension theory: $A_2/A_2P$ has $d(M) = 3 > 2$ , so it is not holonomic and no finiteness theorem applies.

The finiteness theorem that does hold

If $M$ and $S$ are both holonomic $A_n$ -modules then $Hom_A_n(M,S)$ , and indeed every $Ext^i_A_n(M,S)$ , is a finite-dimensional $K$ -vector space. This is proved in Björk and in Hotta-Takeuchi-Tanisaki rather than in the primer, and it is the correct generalisation of the one-variable statement above, since $K[X]$ is holonomic and $A_1/A_1P$ is holonomic for every non-zero $P$ . Nothing of the kind is available when $M$ fails to be holonomic.

Examples and Special Cases

A catalogue of targets

$S = K[X]$ , the polynomial module: the classical polynomial solutions.
$S = K[[x_1, … ,x_n]]$ , formal power series: formal solutions, always at least as many as convergent ones.
$S = H(U)$ for $U ⊂ eq C$ open: holomorphic solutions, the natural target in one complex variable.
$S = C^ ∈ fty(U)$ for $U ⊂ eq R^n$ open, over $A_n(R)$ : smooth solutions.
$S$ a module of distributions, hyperfunctions or microfunctions: generalised solutions.
$S = K[X][1/f]$ : solutions with poles along $f$ , the setting of the Bernstein-Sato polynomial.
$S = A_n$ itself: not a solution space in any classical sense, but the dual module, whose derived version drives duality theory.

Constants and de Rham cohomology

Take $M = K[X] = A_n/ ∑ _i A_n∂_i$ . Then (6.10) gives $Hom_A_n(K[X],S) = s ∈ S : ∂_i s = 0 for all i$ , the constants of $S$ . For $S = K[X]$ this is $K$ , so $End_A_n(K[X]) = K$ .

For $n = 1$ and $S = K[x,1/x]$ the constants are again $K$ , but now $Ext^1_A_1(K[x],K[x,1/x]) = coker(∂ : K[x,1/x] → K[x,1/x])$ is one-dimensional, spanned by the class of $1/x$ , which has no antiderivative in $K[x,1/x]$ . That one dimension is the residue, and the computation is the algebraic de Rham cohomology of the punctured line.

A solution space that a bigger target creates

$M = A_1/A_1x$ has $Hom_A_1(M,S) = (x : S → S)$ . For $S = K[x]$ or $S = K[x,1/x]$ this kernel is zero, because both are domains. For $S = K[x,1/x]/K[x]$ it is one-dimensional. The worked example below carries out the computation.

Worked Example

Where does $xf = 0$ have a solution?

Step 1 - the module and the general shape of the answer
The system is the single equation $xf = 0$ in one variable, so $J = A_1x$ and $M = A_1/A_1x$ , a non-zero module with $K$ -basis the classes of $∂^b$ , $b ≥ 0$ . By (6.12), for every left $A_1$ -module $S$ ,

$Hom_A_1(M,S) = (x : S → S), Ext^1_A_1(M,S) = coker(x : S → S).$
Step 2 - polynomials and rational functions with poles at 0
$S = K[x]$ : multiplication by $x$ is injective, so the solution space is $0$ . The cokernel is $K[x]/xK[x] ≅ K$ , one-dimensional.

$S = K[x,1/x]$ : multiplication by $x$ is now bijective, being invertible with inverse multiplication by $1/x$ . So both the solution space and the obstruction space are $0$ . Enlarging the target from $K[x]$ to $K[x,1/x]$ has removed the obstruction without creating a solution.
Step 3 - the quotient module, where the solution appears
Take $S = K[x,1/x]/K[x]$ , a left $A_1$ -module because $K[x]$ is a submodule of $K[x,1/x]$ . Its elements are classes of finite sums $∑ _j ≥ 1 c_j x^-j$ , and the classes of $x^-1, x^-2, …$ form a $K$ -basis.

Compute the action of $x$ on such a class:

$x [ ∑ _j ≥ 1 c_j x^-j ] = [c_1 + ∑ _j ≥ 2 c_j x^-(j-1) ] = [ ∑ _j ≥ 2 c_j x^-(j-1) ],$

since the constant $c_1$ lies in $K[x]$ and dies in the quotient. The class is killed exactly when $c_2 = c_3 = … = 0$ . Hence $(x) = K ⋅ [x^-1]$ is one-dimensional.
Step 4 - identify the solution
So $xf = 0$ has, up to scalars, exactly one solution in $K[x,1/x]/K[x]$ , namely the class of $1/x$ . Its annihilator in $A_1$ is $A_1x$ : it is killed by $x$ , and no operator of lower complexity kills it, so $A_1 ⋅ [x^-1] ≅ A_1/A_1x = M$ . In the analytic setting the same computation produces the Dirac delta, which appears as the class of $1/2π i z$ ; see the delta page and microfunctions.

Check the identification by hand: $∂^b[x^-1] = (-1)^b b!,[x^-(b+1)]$ , so the classes $∂^b[x^-1]$ for $b ≥ 0$ are, up to non-zero scalars, the basis $[x^-(b+1)]$ of $S$ . Thus the cyclic submodule generated by $[x^-1]$ is all of $S$ , and $S ≅ M$ .
Step 5 - read off what changed
The module $M$ was the same in all three computations. The solution space went $0$ , $0$ , $K$ ; the obstruction space went $K$ , $0$ , $0$ . Nothing about the equation changed - only the category in which one agreed to look.

Result

For $M = A_1/A_1x$ : $Hom(M,K[x]) = 0$ with $Ext^1 ≅ K$ ; $Hom(M,K[x,1/x]) = 0$ with $Ext^1 = 0$ ; and $Hom(M, K[x,1/x]/K[x]) ≅ K$ , spanned by the class of $1/x$ , which is the algebraic Dirac delta. Moreover $K[x,1/x]/K[x] ≅ M$ as $A_1$ -modules.

Applications and Industry Use

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

Deciding solvability by computer. Since solution spaces are $Hom$ spaces, algorithms for polynomial and rational solutions of a system are algorithms computing $Hom_A_n(M,S)$ for structured $S$ , and they run on the presentation of $M$ rather than on the equations as written.
Holonomic rank and asymptotics. For a holonomic system, the dimension of the solution space in formal or convergent power series at a generic point equals the holonomic rank, which is computed from the module. This is how one predicts the number of independent special functions attached to a system before finding any of them.
Integral transforms. The Fourier transform is an automorphism of $A_n$ , so it transports solution spaces between modules; a hard equation can be traded for an easy one by transporting the module rather than the functions.
Riemann-Hilbert. Assigning to a holonomic module its solution complex in a suitable analytic target is the functor at the heart of the Riemann-Hilbert correspondence; the derived version, not just $Hom$ , is what makes the correspondence an equivalence.
Engineering models. In the behavioural approach to linear systems, the solution set of a system of operator equations in a chosen signal space is exactly a $Hom$ space, and the choice of signal space - smooth, distributional, or discrete - is made explicitly for the same reasons as here.

Design Considerations

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

Choosing $S$ is the main design decision on this page, and the choices are not interchangeable.

How the target module changes the answer for the same system.
Target $S$	What solutions mean	Typical difficulty
$K[X]$	polynomial solutions	often zero; purely algebraic
$K[[x]]$	formal solutions	largest; convergence not addressed
$H(U)$	holomorphic solutions	sensitive to the shape of $U$
$C^ ∈ fty(U)$	smooth real solutions	not finitely generated; analysis required
distributions	weak solutions	existence theorems available
$ℳ$ , microfunctions	singularity-level solutions	algebraically constructed as a direct limit

A second decision is whether to work with $Hom$ alone or with the whole derived functor. For a single equation, $Hom$ and $Ext^1$ together already answer both the homogeneous and inhomogeneous questions, and stopping at $Hom$ discards the second. For systems, the full solution complex is the object that behaves well under the operations of the theory.

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.

Solution spaces are computed from a presentation of $M$ , never from the equations in raw form.

Compute a Gröbner basis of $J$ in $A_n$ ; this gives a normal form for elements of $M$ and, by taking leading terms, its dimension and multiplicity.
For polynomial solutions, bound the degree of a possible solution using the module invariants, then solve the resulting finite linear system over $K$ .
For rational solutions, first compute the singular locus from the characteristic variety, bound the pole orders, and reduce again to linear algebra.
For power series solutions at a point, compute the indicial equation from a Gröbner deformation and read off the exponents.
For $Ext$ groups, build a free resolution of $M$ over $A_n$ and apply $Hom(-,S)$ termwise; the resolution has length at most $n$ for holonomic $M$ .

The Dmodules package of Macaulay2 implements holonomic rank, restriction and $Ext$ computations of exactly this shape; dmod.lib in Singular and ore_algebra in SageMath cover overlapping ground. Note the practical asymmetry: computing $M$ is routine, while computing $Hom_A_n(M,S)$ requires $S$ to be given by finite data, which excludes $C^ ∈ fty(U)$ and all genuinely analytic targets.

Limits of Validity

No existence is asserted. The theorem is a bijection between two possibly empty sets. It gives no reason for a solution to exist in a given target, and none of the classical existence theorems follow from it.
No finite dimension in general. $Hom_A_n(M,S)$ can be infinite-dimensional over $K$ , already for one equation in two variables. Finiteness requires holonomicity of both $M$ and $S$ , and that is a theorem beyond the primer.
Homogeneous systems only. Inhomogeneous problems are not solution spaces of a module; they are questions about $Ext^1$ , or about a coset inside $S$ .
No topology. All of this is algebra. Statements about approximate solutions, stability under perturbation, or convergence of formal solutions are invisible to $Hom$ and must be imported from analysis.

Failure Modes and Common Mistakes

Treating the solution space as a module over the functions

Solutions can be added and scaled by constants, not multiplied by polynomials: if $Pf = 0$ then $P(gf) ≠ 0$ in general, because $P$ does not commute with $g$ . Correspondingly $Hom_A_n(M,S)$ is only a $K$ -vector space. Code that stores solution spaces as modules over the coefficient ring will silently produce nonsense.

Confusing $Hom_A_n(M,S)$ with $Hom_K(M,S)$

The subscript is the whole content. $K$ -linear maps $M \to S$ form a huge space with no relation to the differential equation; the $A_n$ -linear ones are the solutions. When $M$ is presented by a matrix, this is the difference between solving a linear system over $K$ and solving one over $A_n$ .

Assuming a bigger target always gives more solutions

It does when the enlargement is an inclusion of $A_n$ -modules, since $Hom$ is left exact. It does not for quotients: passing from $K[x,1/x]$ to $K[x,1/x]/K[x]$ is not an enlargement, yet the solution space of $xf = 0$ grows from $0$ to $K$ . The correct statement mentions the map, not the size.

Expecting the correspondence to be exact

Passing from a module to its solution space converts a surjection into an injection but need not convert an injection into a surjection. Deducing that every solution of a subsystem extends to a solution of the full system is the standard form of this error, and the discrepancy is precisely $Ext^1$ .

Historical Notes

Reading a solution as a homomorphism goes back to the constant-coefficient theory, where a system corresponds to a module over a polynomial ring and solutions in a signal space are homomorphisms into it; this is the algebraic core of the Ehrenpreis-Palamodov fundamental principle of the 1960s. The variable-coefficient version is due to Kashiwara, whose 1970 thesis made the solution complex, rather than the solution space, the object of study, and proved the finiteness theorems that make it usable.

The insistence that the derived functor and not merely $Hom$ carries the information is what later became the Riemann-Hilbert correspondence, established by Kashiwara and independently by Mebkhout around 1980. The elementary statement presented here, restricted to $Hom$ and to the Weyl algebra, is the entry point to that circle of ideas and is the form in which Coutinho's primer introduces it.

Comparison

Homogeneous and inhomogeneous questions in module language, for a single operator $P$ acting on a target $S$ .
Question	Module-theoretic form	Answer for $P = x$ , $S = K[x]$
Which $f$ satisfy $Pf = 0$ ?	$Hom_A_1(A_1/A_1P, S) = (P\|_S)$	$0$
Which $g$ admit $Pf = g$ ?	image of $P\|_S$	$xK[x]$
What obstructs $Pf = g$ ?	$Ext^1_A_1(A_1/A_1P,S) = coker(P\|_S)$	$K$ , detected by $g(0)$
Higher obstructions?	$Ext^i$ for $i ≥ 2$	all zero, the resolution has length one

Key Takeaways

Key points

For $M = A_n/J$ and any left $A_n$ -module $S$ , evaluation at the canonical generator is a $K$ -linear bijection from $Hom_A_n(M,S)$ to the set of solutions of the system in $S$ .
Solution spaces are vector spaces over $K$ only; they are not modules over $A_n$ or over the polynomial ring, because solutions are not closed under multiplication by functions.
$Sol_S = Hom_A_n(-,S)$ is contravariant and left exact; adding equations shrinks the module and shrinks the solution space.
For one operator, $Hom(A_1/A_1P,S) = (P|_S)$ and $Ext^1(A_1/A_1P,S) = coker(P|_S)$ : homogeneous solutions and inhomogeneous obstructions in one formula.
Solution spaces can be zero, finite-dimensional, or infinite-dimensional; one ordinary equation of order $k$ has at most $k$ independent polynomial solutions, while one partial equation may have infinitely many.
Generalised solutions are not a new theory but a new target module, which is why distributions, hyperfunctions and microfunctions fit into the same statement.

FAQs

Why is the solution space only a vector space and not a module?

Because multiplying a solution by a function destroys the equation: $P(gf)$ involves derivatives of $g$ and does not vanish. Formally, $g φ$ fails to commute with the action of $∂_i$ unless $∂ g/∂ x_i$ acts as zero. Only the constants survive, and the constants of $A_n$ are $K$ .

Does the theorem require

S

to be finitely generated?

No, and it must not. The interesting targets are not finitely generated: $C^ ∈ fty(U)$ is not, nor is the module of germs of holomorphic functions. Only the module of the system is required to be finitely generated, which it is automatically.

Can the solution space be infinite-dimensional?

Yes. The operator $x_1∂_2 - x_2∂_1$ in two variables annihilates every polynomial in $x_1^2+x_2^2$ , so its polynomial solution space is infinite-dimensional. Finiteness is a consequence of holonomicity, not of the correspondence itself.

What does

Ext^1

mean concretely?

For a single operator it is the cokernel of the action of $P$ on the target: the space of inhomogeneous right-hand sides $g$ for which $Pf = g$ has no solution, modulo those for which it does. For $P = x$ and target $K[x]$ , that space is one-dimensional and is detected by the single condition $g(0) = 0$ .

If the module is non-zero, must there be a solution somewhere?

There is always a tautological one: the canonical generator is a solution in $M$ itself, corresponding to the identity homomorphism. Whether there is a solution in a target of independent interest is a different question, and $xf = 0$ shows the answer can be no for every classical target.

How does this relate to the classical count of solutions of an ODE?

For an ordinary equation of order $k$ with non-vanishing leading coefficient near a point, the analytic solution space near that point has dimension exactly $k$ , and the module-theoretic invariant reproducing $k$ is the holonomic rank. Polynomial solutions are a subspace of that and can be much smaller.

Why insist on a functor rather than just the isomorphism?

Because the operations of the theory - restricting to a subvariety, integrating along a fibre, changing coordinates - are functors on modules, and functoriality is what transports them to solution spaces. Without it, each such operation would have to be re-derived for each function space.

Is there a version for right modules?

Yes, symmetrically, and the transposition anti-automorphism converts between them. Right modules are the natural home for densities, so integration-type statements are usually phrased there; see side-changing functors.

References

S. C. Coutinho, A Primer of Algebraic D-modules, London Mathematical Society Student Texts 33, Cambridge University Press, 1995 - Ch. 6 §1, Theorem (6.1.2), and Exercises 4.1-4.4 of that chapter.
M. Kashiwara, Algebraic Study of Systems of Partial Differential Equations, master's thesis, University of Tokyo 1970; Mémoires de la Société Mathématique de France 63, 1995 - solution complexes and finiteness.
J.-E. Björk, Rings of Differential Operators, North-Holland, 1979 - Ch. 1 and Ch. 5, for finiteness of Hom and Ext between holonomic modules.
R. Hotta, K. Takeuchi and T. Tanisaki, D-modules, Perverse Sheaves, and Representation Theory, Progress in Mathematics 236, Birkhäuser, 2008 - Ch. 4 and Ch. 7, for the solution functor and Riemann-Hilbert.
M. Saito, B. Sturmfels and N. Takayama, Gröbner Deformations of Hypergeometric Differential Equations, Springer, 2000 - algorithms for polynomial, rational and series solutions.
L. Ehrenpreis, Fourier Analysis in Several Complex Variables, Wiley-Interscience, 1970 - solutions as homomorphisms in the constant-coefficient case.
J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, revised edition, American Mathematical Society, 2001 - Ch. 7, for Hom and Ext over Noetherian noncommutative rings.
Macaulay2 Dmodules package documentation - holonomicRank, DrestrictionComplex and Ext computations over the Weyl algebra.

AI Suggested Questions

Compute $Hom_A_1(A_1/A_1(∂ - 1), S)$ for $S = K[x]$ , $S = K[[x]]$ and $S = H(C)$ .
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Prove that an inclusion of target modules induces an injection of solution spaces, and give an example where it is not a bijection.
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