Holonomic Modules: Definition and First Examples

Q: Is A_n/A_n D holonomic for a single operator D when n ≥ 2?

No. For 0 ≠ D ∈ A_n the quotient has dimension exactly 2n-1 , which exceeds n whenever n ≥ 2 . A single equation is never enough in more than one variable; one needs a system whose relations cut the dimension all the way down to n .

Q: Coutinho writes m(M) for multiplicity, but this collection writes e(M). Which is standard?

Both appear in the literature. e is the more common choice in commutative algebra, where multiplicity is the classical Samuel multiplicity, and it is the convention used throughout this collection. When reading Coutinho, translate m to e ; nothing else changes.

Overview

Bernstein's inequality confines the dimension of a non-zero finitely generated $A_n$ -module to the interval $n ≤ d(M) ≤ 2n$ . Both ends of that interval are occupied, and the two ends behave completely differently. At the top sits $A_n$ itself, a ring with a rich and complicated module theory. At the bottom sits a class of modules so rigid that almost every finiteness statement one could hope for is true of them. Those are the holonomic modules, and this page defines them and assembles the first supply of examples.

The definition is as short as it could be: a finitely generated left $A_n$ -module $M$ is holonomic if $M = 0$ or $d(M) = n$ . Everything interesting is a consequence of the inequality, not of the definition. Because $n$ is the minimum, a submodule or a quotient of a holonomic module cannot have smaller dimension, so it is holonomic too. Because multiplicity is additive when dimensions agree, and because multiplicities are positive integers, chains of submodules cannot be long. That single observation is the seed of finite length, of artinianness, and eventually of cyclicity.

Analysts met these modules first, under a different name. A system of linear partial differential equations with polynomial coefficients is called maximally overdetermined when it imposes the largest number of independent conditions that can be imposed without forcing the solution space to collapse; translated into algebra, that is exactly holonomicity. For $n = 1$ the translation is completely concrete: every non-zero linear ordinary differential operator with polynomial coefficients gives a holonomic $A_1$ -module.

One warning belongs up front. In dimension one, "holonomic" and "finitely generated torsion" are the same condition. In dimension two and above they are not: there are finitely generated torsion $A_n$ -modules of dimension $2n-1$ , and $2n-1 > n$ as soon as $n ≥ 2$ . The worked example below exhibits one and computes its Hilbert polynomial exactly.

Definition

Throughout, $K$ is a field of characteristic zero, $A_n = A_n(K)$ is the $n$ -th Weyl algebra with generators $x_1, … ,x_n,∂_1, … ,∂_n$ , and $B_i$ is the Bernstein filtration, $B_i$ being the $K$ -span of the monomials $x^α ∂^β$ with $| α | + | β | ≤ i$ . Modules are left modules unless stated otherwise.

Holonomic moduleCoutinho (10.1)

Let $M$ be a finitely generated left $A_n$ -module. Then $M$ is holonomic if either $M = 0$ , or $M ≠ 0$ and

d(M) = n,

(10.1)

where $d(M)$ is the dimension of $M$ , that is the degree of the Hilbert polynomial of $M$ with respect to any good filtration relative to $B$ .

Why the definition is not vacuous

Two facts make the definition worth stating. First, $d(M)$ does not depend on the good filtration used to compute it, so (10.1) is a property of $M$ and not of a presentation. Second, by Bernstein's inequality no non-zero finitely generated $A_n$ -module has dimension below $n$ ; holonomic modules are therefore the modules of minimal dimension, which is why they inherit so much structure. Without the inequality the condition $d(M) = n$ would single out an arbitrary slice of the range rather than an extreme.

Conventions

The zero module is declared holonomic by fiat, so that the class is closed under quotients and kernels without case analysis. Some authors write $d(0) = - ∈ fty$ ; the effect is the same. Finite generation is part of the definition and cannot be dropped, because a module without a good filtration has no Hilbert polynomial and hence no dimension. Coutinho writes the multiplicity as $m(M)$ ; this collection writes $e(M)$ , following the convention fixed on the multiplicity page.

Core Concepts

Minimal dimension means maximal constraint

Dimension measures how fast a module grows: $_K Γ_i$ behaves like $i^d(M)$ for a good filtration $Γ$ . A module of dimension $2n$ grows as fast as $A_n$ itself, which means the relations imposed on it are negligible in the large. A module of dimension $n$ grows as slowly as anything can, which means the relations are as heavy as they can be without killing the module. Holonomicity is therefore a statement about how constrained a module is, and constraint is exactly what one wants when trying to prove finiteness.

The differential-equations picture

Given a system of linear partial differential equations with polynomial coefficients, one forms $M = A_n^r/N$ where $N$ encodes the equations. The solutions of the system in a function space P2 are the elements of $Hom_A_n(M, F)$ . Adding equations shrinks $M$ and shrinks the solution space. Holonomicity is the point at which the system is as overdetermined as it can be while still having a non-zero module attached, which is why PDE theorists call such systems maximally overdetermined. For $n = 1$ this reproduces the familiar fact that a single non-trivial ordinary differential equation of order $k$ has a solution space of dimension at most $k$ .

The geometric picture

Read through the order filtration instead, $d(M)$ is the dimension of the characteristic variety $Ch(M)$ inside the $2n$ -dimensional cotangent space. Gabber's theorem says $Ch(M)$ is involutive, and an involutive subvariety of a symplectic space of dimension $2n$ has dimension at least $n$ . Holonomic modules are precisely those whose characteristic variety is Lagrangian — as small as involutivity permits. That description is the one that survives when the Weyl algebra is replaced by the sheaf of differential operators on a smooth variety.

What it does not mean

Holonomic does not mean simple, and simple does not mean holonomic. $K[x] ⊕ K[x]$ is holonomic over $A_1$ and visibly not simple. Conversely, for $n ≥ 2$ there exist simple $A_n$ -modules of dimension strictly greater than $n$ ; see non-holonomic irreducible modules. The two conditions coincide only under an extra hypothesis, namely multiplicity one.

Key Equations

For a good filtration $Γ$ of a non-zero finitely generated $M$ , the Hilbert polynomial has the form

χ_Γ(i) = (e(M))/(d(M)!), i^d(M) + (lower order),

(10.2)

so holonomicity is the assertion that $χ_Γ = n$ .

The two extremes are computed directly. Filtering $A_n$ by itself gives

_K B_i = 2n+i2n = i^2n(2n)! + O(i^2n-1),

(10.3)

so $d(A_n) = 2n$ and $e(A_n) = 1$ : the Weyl algebra is not holonomic for any $n ≥ 1$ .

Filtering $K[X] = K[x_1, … ,x_n]$ by the image of $B_i$ on the generator $1$ leaves only the multiplication monomials, because $∂_j ⋅ 1 = 0$ :

Γ_i = B_i ⋅ 1 = , f ∈ K[X] : f ≤ i ,, _K Γ_i = n+in,

(10.4)

a polynomial of degree $n$ with leading coefficient $1/n!$ , so $d(K[X]) = n$ and $e(K[X]) = 1$ .

The intermediate case that separates torsion from holonomic is $M = A_n/A_n∂_n$ . The induced filtration $Γ_i = (B_i + A_n∂_n)/A_n∂_n$ has as basis the images of the canonical monomials $x^α ∂^β$ with $β_n = 0$ and $| α | + | β | ≤ i$ , which are monomials in the $2n-1$ letters $x_1, … ,x_n,∂_1, … ,∂_n-1$ :

_K Γ_i = 2n-1+i2n-1, d(A_n/A_n∂_n) = 2n-1.

(10.5)

So $A_n/A_n∂_n$ is holonomic exactly when $2n - 1 = n$ , that is when $n = 1$ . For $n ≥ 2$ it is a finitely generated torsion module that is not holonomic.

Variable Definitions

$K$: the ground field, of characteristic zero
$A_n$: the $n$ -th Weyl algebra over $K$
$X$ , $K[X]$: the variables $x_1, … ,x_n$ and the polynomial ring in them
$B_i$: the $i$ -th piece of the Bernstein filtration of $A_n$
$Γ_i$: the $i$ -th piece of a good filtration of the module under discussion
$χ_Γ(i)$: the Hilbert polynomial: the polynomial agreeing with $_K Γ_i$ for $i ≫ 0$
$d(M)$: the dimension of $M$ , that is $χ_Γ$
$e(M)$: the multiplicity of $M$ , that is $d(M)!$ times the leading coefficient of $χ_Γ$
$α, β$: multi-indices in $N^n$ , with $| α | = α_1 + … + α_n$
$Ch(M)$: the characteristic variety of $M$ , computed with the order filtration

Properties and Behaviour

Submodules, quotients and finite sumsCoutinho (10.1.1)

Let $n ≥ 1$ and let $M$ be a holonomic $A_n$ -module.

Every submodule $N ⊂ eq M$ and every quotient $M/N$ is holonomic.
A finite sum of holonomic modules is holonomic.

The details, and the extension to short exact sequences, are on the closure properties page.

Proof

By additivity in exact sequences, $d(M) = d(N), d(M/N)$ , so both $d(N) ≤ n$ and $d(M/N) ≤ n$ . If $N ≠ 0$ then Bernstein's inequality forces $d(N) ≥ n$ , hence $d(N) = n$ ; the same applies to $M/N$ . For finite sums, $d(M_1 ⊕ … ⊕ M_k) = _i d(M_i) = n$ , and a finite sum of submodules of a module is a quotient of the direct sum.

Holonomic implies torsionCoutinho (10.1.3)

Let $n ≥ 1$ . Every holonomic $A_n$ -module is a torsion module: for each $u ∈ M$ there is a non-zero $D ∈ A_n$ with $D u = 0$ .

Proof

Let $0 ≠ u ∈ M$ and let $φ : A_n → M$ be $φ(a) = a u$ . Its image is a non-zero submodule of $M$ , hence holonomic, so $d(imφ) = n$ . Applying additivity to $0 → φ → A_n → imφ → 0$ gives

2n = d(A_n) = , d( φ), n ,.

For $n ≥ 1$ we have $n < 2n$ , so $d( φ) = 2n$ and in particular $φ ≠ 0$ . Any non-zero $D ∈ φ$ annihilates $u$ . If $u = 0$ there is nothing to prove.

In one variable, torsion and holonomic agreeCoutinho (10.1.2)

A finitely generated $A_1$ -module is holonomic if and only if it is a torsion module.

One direction is the proposition above. For the other, let $M$ be generated by $u_1, … ,u_r$ and choose non-zero $θ_i ∈ A_1$ with $θ_i u_i = 0$ . Then $A_1 u_i$ is a quotient of $A_1/A_1θ_i$ , which is holonomic because $d(A_1/A_1θ_i) ≤ 2 ⋅ 1 - 1 = 1$ by Coutinho (9.3.5) and $≥ 1$ by Bernstein's inequality. A finite sum of holonomic modules is holonomic.

Cyclic $A_1$ -modules are almost always holonomic

If $I$ is a non-zero left ideal of $A_1$ , then $A_1/I$ is holonomic. Indeed $d(A_1/I) ≤ 1$ by Coutinho (9.3.5), and if $I ≠ A_1$ then $A_1/I ≠ 0$ and Bernstein's inequality gives $d(A_1/I) = 1$ ; if $I = A_1$ the quotient is zero, which is holonomic by convention. Only $I = 0$ is excluded, and $A_1/0 = A_1$ has dimension $2$ .

The degenerate case $n = 0$

$A_0 = K$ , and a finitely generated $K$ -module is a finite-dimensional vector space, whose Hilbert function is eventually the constant $_K M$ . So $d(M) = 0 = n$ for every such module: over $A_0$ every finitely generated module is holonomic, with $e(M) = _K M$ . The statement "holonomic implies torsion" fails here only because $K$ has no non-zero non-units, which is why the proposition is stated for $n ≥ 1$ .

Examples and Special Cases

The polynomial ring

$K[X] ≅ A_n/(A_n∂_1 + … + A_n∂_n)$ is holonomic with $d = n$ and $e = 1$ , by (10.4). Because its multiplicity is $1$ it is simple. It is the smallest holonomic module in the strongest sense: nothing has smaller multiplicity except $0$ .

Any ordinary differential operator

For $0 ≠ θ ∈ A_1$ , the module $A_1/A_1θ$ is holonomic with $d = 1$ and $e = θ$ , the Bernstein degree of $θ$ . Concretely, $A_1/A_1∂ ≅ K[x]$ has $e = 1$ ; $A_1/A_1 x$ is the Dirac delta module, also with $e = 1$ ; and $A_1/A_1(x∂ - λ)$ has $e = 2$ .

Localisations of the polynomial ring

For $0 ≠ p ∈ K[X]$ , the ring $K[X][1/p]$ of rational functions with denominators a power of $p$ is an $A_n$ -module, and it is holonomic with $e ≤ ( p + 1)^n$ . This is the most important family of examples in the chapter and is proved on its own page; it is what makes the Bernstein-Sato polynomial exist.

A quotient of ideals

If $0 ≠ J ⊂ eq I$ are left ideals of $A_1$ , then $I/J$ is holonomic. The quick route is to observe that $A_1/J$ is holonomic by the corollary above, and $I/J$ is a submodule of it. The same argument fails for $J = 0$ : $I/0 = I$ is a non-zero left ideal of $A_1$ and has dimension $2$ .

The Weyl algebra itself

$A_n$ is finitely generated over itself (by $1$ ) and has $d(A_n) = 2n$ , so it is not holonomic for any $n ≥ 1$ . Neither is any non-zero left ideal of $A_n$ , nor any free module of positive rank. Being cyclic is no evidence of holonomicity.

The full field of rational functions

$K(X) = K(x_1, … ,x_n)$ carries an $A_n$ -action extending the one on $K[X]$ , with $∂_i$ acting by the quotient rule. It is not finitely generated over $A_n$ , so it has no dimension and is not holonomic. Only the submodules $K[X][1/p]$ , one denominator at a time, are.

Worked Example

A torsion module that is not holonomic: $A_2/A_2∂_2$

Step 1 - the module and its filtration
Take $n = 2$ and $M = A_2/A_2∂_2$ , generated by the class $\bar{1}$ . Filter $M$ by the image of the Bernstein filtration, $Γ_i = (B_i + A_2∂_2)/A_2∂_2$ . This is a good filtration, being the one induced by the single generator $\bar{1}$ .

The canonical monomials $x_1^a_1x_2^a_2∂_1^b_1∂_2^b_2$ form a $K$ -basis of $A_2$ , and those with $b_2 ≥ 1$ form a basis of the left ideal $A_2∂_2$ . So the classes of the monomials with $b_2 = 0$ form a basis of $M$ , and $Γ_i$ has as basis the classes of the monomials in the three letters $x_1, x_2, ∂_1$ of total degree at most $i$ .
Step 2 - count, and check the small values by hand
Counting monomials of degree at most $i$ in $3$ variables:

$_K Γ_i = i+33 = ((i+1)(i+2)(i+3))/(6).$

Check the first three values directly. For $i = 0$ the only class is $\bar{1}$ , so $= 1$ , and $33 = 1$ . For $i = 1$ the classes are $1, x_1, x_2, ∂_1$ (the class of $∂_2$ is zero), so $= 4$ , and $43 = 4$ . For $i = 2$ we add $x_1^2, x_1x_2, x_2^2, x_1∂_1, x_2∂_1, ∂_1^2$ , giving $10$ , and $53 = 10$ . The formula is right.
Step 3 - read off the dimension
$χ_Γ(i) = 16 (i+1)(i+2)(i+3)$ is a cubic, so

$d(M) = 3 = 2n - 1, e(M) = 3! ⋅ 16 = 1.$

Since $n = 2$ and $3 > 2$ , the module $M$ is not holonomic. It also sits strictly inside the allowed band: $2 ≤ 3 ≤ 4$ , consistent with Bernstein's inequality but at neither endpoint.
Step 4 - but every element is torsion
Every class in $M$ has a representative of the form $∑ _j=0^J c_j x_2^,j$ with $c_j ∈ A_1 = K ⟨ x_1, ∂_1 ⟩$ , because the canonical monomials surviving in $M$ contain no $∂_2$ . Now use the commutation identity

$∂_2^,a x_2^,j = ∑ _t = 0^ (a,j) at, (j!)/((j-t)!); x_2^,j-t, ∂_2^,a-t.$

If $a = J+1 > j$ then every term has $∂_2$ -exponent $a - t ≥ a - j ≥ 1$ , so $∂_2^,J+1 x_2^,j ∈ A_2∂_2$ for every $j ≤ J$ . Since $∂_2$ commutes with $x_1$ and $∂_1$ , we get $∂_2^,J+1 ( ∑ _j c_j x_2^,j ) ∈ A_2∂_2$ , that is, $∂_2^,J+1$ annihilates the class. Every element of $M$ is a torsion element.
Step 5 - contrast with $n = 1$
Repeat the computation with $n = 1$ : $M' = A_1/A_1∂ ≅ K[x]$ , $Γ_i$ has basis the classes of $x^a$ with $a ≤ i$ , $_K Γ_i = i+1 = i+11$ , so $d(M') = 1 = 2n-1 = n$ . Here the two coincide, which is exactly why torsion and holonomic are the same condition over $A_1$ and not over $A_n$ for $n ≥ 2$ .

Result

$M = A_2/A_2∂_2$ has Hilbert polynomial $i+33$ , hence $d(M) = 3$ and $e(M) = 1$ . It is finitely generated and every element is annihilated by a power of $∂_2$ , so it is a torsion module, yet $d(M) = 3 > 2 = n$ , so it is not holonomic. Torsion is strictly weaker than holonomic once $n ≥ 2$ .

Applications and Industry Use

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

Ordinary differential equations. Every non-zero linear ODE with polynomial coefficients gives a holonomic $A_1$ -module, and conversely every holonomic $A_1$ -module is cyclic, hence of the form $A_1/I$ . The correspondence between equations and modules is therefore essentially complete in one variable.
Special functions. A function annihilated by a holonomic module of operators is determined by finitely many initial conditions and finitely many recurrences. This underlies the encyclopaedic treatment of holonomic functions and the algorithms that manipulate them.
Automatic proof of identities. Zeilberger's method and creative telescoping work because the class of holonomic objects is closed under the operations that appear in a combinatorial identity: sums, products and definite summation or integration.
Analytic continuation. The holonomicity of $K[X][1/p]$ is what produces the b-function of $p$ , and the b-function is what continues $| p | ^s$ meromorphically in $s$ .
Representation theory. Modules attached to highest weight representations, and to $𝒟$ -modules on flag varieties, are holonomic; finite length is what makes character computations finite.
Singularity theory. The roots of the b-function of a singular polynomial are invariants of the singularity, and they exist only because the localisation is holonomic.

Design Considerations

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

Which filtration defines the dimension

This page defines $d(M)$ through the Bernstein filtration, because its pieces are finite dimensional over $K$ and the counting is direct. The order filtration gives the same number but through a relative Hilbert function over $K[X]$ , and it is the filtration one wants when the characteristic variety is in play. Choose the Bernstein filtration for existence proofs and elementary bounds; choose the order filtration for geometry and for statements that must generalise beyond affine space.

Which invariants to carry along

Dimension alone takes only $n+1$ values and is too coarse to detect anything inside the holonomic class. Multiplicity refines it, is additive in exact sequences, and bounds the length; for any argument that walks through a chain of submodules, carry the pair $(d, e)$ . For arguments that must survive a change of variety or a functor, carry $Ch(M)$ .

Presenting the module

Since holonomic modules are cyclic, they can always be written $A_n/I$ for a single left ideal $I$ . That is the compact representation and the one computer algebra systems prefer. It is not always the natural one: $K[X][1/p]$ is easier to reason about as a set of rational functions than as $A_n/Ann(1/p)$ , and the annihilator can be expensive to compute.

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.

Deciding holonomicity of an explicitly presented module is a Gröbner basis computation in the Weyl algebra followed by a commutative Hilbert polynomial computation.

Present $M$ as $A_n^r/N$ with $N$ given by explicit generators.
Compute a Gröbner basis of $N$ for a term order refining the Bernstein (or order) filtration.
Take initial terms to obtain a graded module over a commutative polynomial ring in $2n$ variables.
Compute its Hilbert polynomial; the degree is $d(M)$ and $d(M)!$ times the leading coefficient is $e(M)$ .
Compare $d(M)$ with $n$ .

The cost is dominated by step 2, which is doubly exponential in $2n$ in the worst case. In practice $n ≤ 4$ or $5$ with modest degrees is routine. Macaulay2's Dmodules package exposes this as a holonomicity test, Singular's dmod.lib and bfun.lib cover the same ground, and SageMath's ore_algebra handles the one-variable and Ore-algebra cases efficiently. Bernstein's inequality doubles as a correctness check on any implementation: a reported dimension below $n$ for a non-zero module is a bug.

Limits of Validity

The definition depends on three hypotheses, and each one is doing work.

Characteristic zero. Everything rests on Bernstein's inequality, which is false in characteristic $p$ : there $A_n$ is a finite module over its centre and possesses non-zero finitely generated modules of dimension $0$ . "Minimal dimension" would then be $0$ , not $n$ , and the class defined by $d(M) = n$ would have none of the finiteness properties. See the positive characteristic page.
Finite generation. A module with no good filtration has no Hilbert polynomial. $K(X)$ is the standard example: it is a torsion $A_n$ -module, but it is not finitely generated and "holonomic" simply does not apply to it.
Left versus right. The definition above is for left modules. The transposition anti-automorphism preserves Bernstein degree and exchanges left and right modules, so the right-module theory is a mirror image; but the two must not be mixed inside one argument.
$n$ is fixed. Holonomicity is relative to a Weyl algebra. A module can be holonomic over $A_1$ and, regarded through an inclusion $A_1 ⊂ eq A_2$ , fail to be finitely generated over $A_2$ at all. Always say which $n$ is meant.

What the definition does not give you for free

Holonomicity is a statement about a single number, the degree of a Hilbert polynomial. That number does not by itself bound multiplicity, does not identify the composition factors, and does not determine the characteristic variety; it only says the variety has the minimal possible dimension. The finer invariants have to be computed separately, and the comparison page sets out what each of them controls.

Failure Modes and Common Mistakes

Believing torsion implies holonomic

It does for $n = 1$ and it fails for every $n ≥ 2$ . The worked example above gives a finitely generated torsion $A_2$ -module of dimension $3$ . The equivalence in one variable is a coincidence of the arithmetic $2n - 1 = n$ at $n = 1$ , not a structural fact, and it is one of the most common places where an argument valid for ordinary differential equations is misapplied to partial ones.

Confusing holonomic with simple

Neither implies the other. $K[x] ⊕ K[x]$ is holonomic of multiplicity $2$ and is not simple. For $n ≥ 2$ there are simple $A_n$ -modules of dimension greater than $n$ , so simple does not imply holonomic either. What is true is the one-way implication: a holonomic module of multiplicity $1$ is simple, because a proper non-zero submodule would force a strictly positive multiplicity to be subtracted from $1$ .

Forgetting to check finite generation before computing a dimension

It is tempting to write down a filtration on a module, compute the growth of $_K Γ_i$ , and conclude that the dimension is the degree of the answer. That step is only valid when the filtration is good, which presupposes finite generation. The correct tool for an a priori infinitely generated module is the polynomial-bound criterion, Coutinho (10.3.1), which deduces finite generation from a growth bound rather than assuming it. It is the engine behind the localisation theorem.

Reading $d(M) = n$ as " $M$ is $n$ -dimensional over $K$ "

Every non-zero $A_n$ -module is infinite dimensional over $K$ , since $A_n$ has no non-zero finite-dimensional representations in characteristic zero. The number $d(M)$ is a growth exponent, not a vector space dimension. $K[x]$ is holonomic over $A_1$ and is countably infinite dimensional over $K$ .

Assuming few generators means small dimension

$A_n$ is generated by one element and has the largest dimension available. Conversely every holonomic module is cyclic. The number of generators carries no information about $d(M)$ ; the size of the annihilator does.

Historical Notes

The class was isolated by I. N. Bernstein in 1971-72 while solving a problem of Gelfand on the meromorphic continuation of $p^s$ . His inequality showed that dimension $n$ was the floor, and the modules attaining it turned out to have the finiteness properties his argument needed. He did not call them holonomic; the name came from the Japanese school.

M. Sato, M. Kashiwara and T. Kawai, working analytically on microlocal analysis in the early 1970s, arrived at the same class from the direction of maximally overdetermined systems. The word holonomic is borrowed from classical mechanics, where a holonomic constraint is one expressible as an equation among the coordinates rather than among the velocities; the analogy is that the characteristic variety is cut out as tightly as the symplectic geometry allows.

The geometric characterisation came later. O. Gabber proved in 1981 that characteristic varieties are involutive, so that $d(M) ≥ n$ holds for coherent modules over rings of differential operators on smooth varieties, and holonomic became synonymous with Lagrangian characteristic variety. That is the formulation that survives in the modern theory and in the Riemann-Hilbert correspondence.

Comparison

Where the standard modules sit in the band $n ≤ d(M) ≤ 2n$ .
Module over $A_n$	$d(M)$	$e(M)$	Torsion?	Holonomic?
$A_n$	$2n$	$1$	no	no (for $n ≥ 1$ )
a non-zero left ideal $I ⊂ eq A_n$	$2n$	$≥ 1$	no	no (for $n ≥ 1$ )
$A_n/A_nθ$ , $θ ≠ 0$	$2n-1$	$deg θ$	yes	only if $n = 1$
$K[x_1, … ,x_n]$	$n$	$1$	yes	yes
$K[X][1/p]$ , $p ≠ 0$	$n$	$≤ ( p + 1)^n$	yes	yes
$A_1/A_1(x∂ - λ)$ over $A_1$	$1$	$2$	yes	yes
$K(x_1, … ,x_n)$	undefined	undefined	yes	no: not finitely generated
$0$	conventionally $- ∈ fty$	$0$	yes	yes, by convention

Key Takeaways

Key points

A finitely generated $A_n$ -module is holonomic when it is zero or has $d(M) = n$ , the minimum permitted by Bernstein's inequality.
Submodules, quotients and finite sums of holonomic modules are holonomic; the class is closed under everything that cannot raise dimension.
For $n ≥ 1$ , holonomic modules are torsion modules; for $n = 1$ the converse also holds, and finitely generated torsion is the same as holonomic.
For $n ≥ 2$ the converse fails: $A_n/A_n∂_n$ is torsion with $d = 2n-1 > n$ .
Standard holonomic modules are $K[x_1, … ,x_n]$ , every $A_1/I$ with $I ≠ 0$ , and every localisation $K[X][1/p]$ .
$A_n$ itself, and every non-zero left ideal of it, has dimension $2n$ and is not holonomic for $n ≥ 1$ .
Holonomic and simple are independent conditions; multiplicity $1$ is what forces simplicity.

FAQs

Why is the zero module called holonomic?

So that the class is closed under quotients and submodules without exceptions. If $N = M$ then $M/N = 0$ , and one wants to say "quotients of holonomic modules are holonomic" without a caveat. The convention $d(0) = - ∈ fty$ has the same effect and is used interchangeably.

Is a holonomic module finitely generated by definition or as a consequence?

By definition, on this page and in Coutinho. But there is a criterion, Coutinho (10.3.1), which starts from a module with an arbitrary filtration satisfying a polynomial growth bound and deduces finite generation along with holonomicity. That is how $K[X][1/p]$ is proved holonomic, since finite generation is not obvious there.

Are all holonomic modules cyclic?

Yes, and this is a genuine theorem, not a triviality: it needs the simplicity of $A_n$ , the fact that $A_n$ is not left artinian, and the finite length of holonomic modules. See the cyclicity page.

How do I tell whether a given module is holonomic in practice?

Compute a Gröbner basis of the defining submodule in the Weyl algebra, take initial terms, and compute the Hilbert polynomial of the resulting commutative graded module. Its degree is $d(M)$ . Most computer algebra systems with D-module support expose this as a single call.

Does holonomic mean the solution space is finite dimensional?

In the classical settings where a solution space makes sense, yes in spirit: for $n = 1$ , $Hom_A_1(A_1/A_1θ, F)$ is the space of solutions of $θ u = 0$ in $F$ , and for reasonable $F$ that is finite dimensional. In general the precise statement is about the finite-dimensionality of the solution complex, which needs the theory of holonomic modules with regular singularities; do not assume the naive version.

A_n/A_n D

holonomic for a single operator

D

when

n ≥ 2

No. For $0 ≠ D ∈ A_n$ the quotient has dimension exactly $2n-1$ , which exceeds $n$ whenever $n ≥ 2$ . A single equation is never enough in more than one variable; one needs a system whose relations cut the dimension all the way down to $n$ .

Coutinho writes

m(M)

for multiplicity, but this collection writes

e(M)

. Which is standard?

Both appear in the literature. $e$ is the more common choice in commutative algebra, where multiplicity is the classical Samuel multiplicity, and it is the convention used throughout this collection. When reading Coutinho, translate $m$ to $e$ ; nothing else changes.

Why is the class named after a term from classical mechanics?

In mechanics a constraint is holonomic when it can be written as an equation among the coordinates alone. The analogy is loose but suggestive: a holonomic module is one whose characteristic variety is cut out as tightly as the symplectic structure permits, namely Lagrangian. The name entered the subject through the Japanese school of microlocal analysis.

Does the definition change if I use the order filtration?

No. The dimension computed from the order filtration equals the dimension computed from the Bernstein filtration, so the class of holonomic modules is the same. The multiplicities need not agree, however, so a multiplicity quoted without naming the filtration is ambiguous.

References

S. C. Coutinho, A Primer of Algebraic D-modules, London Mathematical Society Student Texts 33, Cambridge University Press, 1995 - Ch. 10 §1, results (10.1.1), (10.1.2), (10.1.3); Ch. 9 §3, results (9.3.2), (9.3.4), (9.3.5); Ch. 9 §4, Bernstein's inequality (9.4.2).
I. N. Bernstein, Modules over a ring of differential operators. Study of the fundamental solutions of equations with constant coefficients, Functional Analysis and its Applications 5 (1971), 89-101.
I. N. Bernstein, The analytic continuation of generalized functions with respect to a parameter, Functional Analysis and its Applications 6 (1972), 273-285.
J.-E. Björk, Rings of Differential Operators, North-Holland Mathematical Library 21, North-Holland, 1979 - Ch. 1 and Ch. 3, for holonomic modules over the Weyl algebra.
M. Sato, T. Kawai and M. Kashiwara, Microfunctions and pseudo-differential equations, Lecture Notes in Mathematics 287, Springer, 1973 - the analytic origin of maximally overdetermined systems.
O. Gabber, The integrability of the characteristic variety, American Journal of Mathematics 103 (1981), 445-468.
R. Hotta, K. Takeuchi and T. Tanisaki, D-modules, Perverse Sheaves, and Representation Theory, Progress in Mathematics 236, Birkhäuser, 2008 - Ch. 3, for the geometric definition via Lagrangian characteristic varieties.
A. Leykin and H. Tsai, Dmodules: functions for computations with D-modules, a package for Macaulay2 - holonomicity tests and holonomic rank.
ISO 80000-2:2019, Quantities and units - Part 2: Mathematics, International Organization for Standardization.

AI Suggested Questions

Compute $d$ and $e$ for $A_1/A_1(x^2∂^2 + x∂ + 1)$ and confirm it is holonomic.
Show directly that $A_3/A_3∂_3$ has dimension $5$ and is a torsion module.
Prove that a holonomic module of multiplicity $1$ is simple, and find a holonomic module of multiplicity $2$ that is not.
Explain why $K(x)$ is a torsion $A_1$ -module that is not holonomic, and identify exactly which hypothesis fails.
For which non-zero left ideals $I$ of $A_2$ is $A_2/I$ holonomic? Give an example and a non-example.
Work out the Hilbert polynomial of $K[x_1,x_2][1/x_1]$ with respect to the natural filtration and read off its multiplicity.
Describe how the definition of holonomicity is restated when the Weyl algebra is replaced by the ring of differential operators on a smooth affine variety.