Differential Operators on an Affine Variety

Q: Is D(X) always generated by functions and vector fields?

No. It is for regular X in characteristic zero, and it fails for the cusp, where ∂^2 - 2t^-1∂ preserves K[t^2,t^3] but has symbol ξ^2 , while everything in the subring generated by functions and derivations has order- 2 symbol divisible by t^2 .

Q: Why does the operator with a pole still act on the ring?

Because the ring has a gap. K[t^2,t^3] contains every power of t except t^1 , and the coefficient k(k-3) in P_2(t^k) = k(k-3)t^k-2 vanishes at k = 3 , exactly the value that would produce t^1 . The pole and the gap cancel.

Q: Is D(X) always Noetherian?

No. It is for smooth X , and for curves with quasi-homogeneous singularities such as the cusp. For the cone over a smooth plane cubic it is not Noetherian and is not even a finitely generated algebra, by the 1972 result of Bernstein, Gelfand and Gelfand.

Q: What replaces D(X) when X is singular?

In practice, one embeds X in a smooth ambient variety Y and works with D(Y) -modules supported on X . Kashiwara's equivalence identifies those with D-modules on X when X is a smooth closed subvariety; for singular X the category of D_Y -modules supported on X is taken as the definition, precisely to avoid D(X) .

Q: Do vector fields on a singular variety have to vanish at the singular points?

On the cusp, yes: every element of Der_K(K[t^2,t^3]) is f∂ with f ∈ tK[t] , so all vanish at t = 0 . This is not universal - it depends on the singularity - but it is exactly what causes the generation failure here.

Overview

The inductive definition attaches a ring $D(R)$ to every commutative $K$ -algebra, with no smoothness hypothesis anywhere. Applying it to the coordinate ring of an affine variety $X$ gives $D(X)$ , the ring of algebraic differential operators on $X$ . The question this page settles is what that ring actually looks like, and how much of the Weyl algebra picture survives.

The answer splits sharply. If $X$ is smooth and irreducible over a field of characteristic zero, everything one hopes for is true: $D(X)$ is generated by the regular functions $O(X)$ together with the vector fields $Der_K(O(X))$ ; it is a simple Noetherian domain; each filtration piece is a projective $O(X)$ -module; and the associated graded ring is the coordinate ring of the cotangent bundle. The Weyl algebra is the case $X = K^n$ .

If $X$ is singular, none of that is automatic. The smallest example is already decisive: for the cuspidal cubic, with coordinate ring $K[t^2,t^3]$ , the operator $∂^2 - 2t^-1∂$ maps the ring into itself, so it is a differential operator on the cusp, yet a short symbol computation shows it is not in the subring generated by the ring and its derivations. Further out, Bernstein, Gelfand and Gelfand exhibited a cone on which $D(X)$ is neither Noetherian nor finitely generated as an algebra.

This page states the smooth theorem precisely, gives the cusp computation in full, describes the idealiser construction that relates operators on $X$ to operators on the ambient affine space, and records what is known in the singular case without overstating it.

Definition

Throughout, $K$ is algebraically closed of characteristic zero, $X ⊂ eq K^n$ is a closed affine variety with radical ideal $J ⊂ eq S = K[x_1, … ,x_n]$ , and $O(X) = S/J$ is its coordinate ring.

Differential operators on $X$

$D(X) := D(O(X))$ , the ring of finite-order $K$ -linear endomorphisms of $O(X)$ defined by the induction $D^-1 = 0$ , $D^m = P : [P,a] ∈ D^m-1 for all a ∈ O(X)$ . Its order filtration is written $D^m(X)$ .

The idealiser and the ambient descriptionCoutinho, Ch. 3, Exercises 3.1-3.4

Let $J A_n$ be the right ideal of $A_n = A_n(K)$ generated by $J$ , and let

I(J A_n) = , P ∈ A_n ;:; P,(J A_n) ⊂ eq J A_n ,

be its idealiser, the largest subring of $A_n$ in which $J A_n$ is a two-sided ideal. Restriction of operators to $X$ induces an injective ring homomorphism

I(J A_n) / J A_n ↪ D(X).

(3.9)

Its image is the set of operators on $X$ that extend to operators on the ambient $K^n$ preserving $J$ .

Why (3.9) is injective and where surjectivity is subtle

If $P ∈ A_n$ satisfies $P(J) ⊂ eq J$ then $P$ descends to a well-defined operator $\bar{P}$ on $S/J$ of order at most $ord(P)$ , and $P = 0$ exactly when $P(S) ⊂ eq J$ , which for $P = ∑ f_β ∂^β$ forces every $f_β ∈ J$ by applying $P$ to $1$ , then to each $x_i$ , and so on. Hence the kernel is $J A_n$ and (3.9) is injective. Surjectivity is a different matter: it holds when $X$ is smooth, and on a singular $X$ it has to be checked, not assumed.

Core Concepts

Two competing intuitions

The first intuition says a differential operator on $X$ should be built from functions and vector fields, because that is what happens on a manifold. The second says a differential operator on $X$ is anything on the ambient space that respects $X$ . On a smooth variety both intuitions give the same answer, and both give $D(X)$ . On a singular variety the second is closer to correct, and even it can fall short, because operators with poles along the singular locus can still map $O(X)$ into itself.

Where the cusp goes wrong

On $K[t^2,t^3]$ the derivations are $R,t∂ + R,t^2∂$ : every vector field vanishes at the cusp point, because a vector field must not move off the curve and the curve has no smooth direction there. Products of vanishing vector fields vanish to order at least two, so every operator built from them has a symbol divisible by $t^2$ . But an operator like $∂^2 - 2t^-1∂$ has symbol $ξ^2$ , divisible by nothing. Its coefficient has a pole; the pole is invisible on $R$ because the relevant monomial $t^1$ is missing from $R$ .

Smoothness as the dividing line

Regularity of $O(X)$ is exactly the condition under which $Der_K(O(X))$ is projective of the right rank, which in turn is what makes $Sym_O(X)(Der_K O(X)) → gr,D(X)$ an isomorphism. Every good property of $D(X)$ in the smooth case is inherited from that identification: Noetherianity from Noetherianity of the graded ring, the domain property from the graded ring being a domain, simplicity from a separate but related argument.

Bad behaviour is not universal

It would be wrong to conclude that all singular varieties give pathological rings. For the cusp, $D(X)$ is still a simple Noetherian domain, finitely generated as a $K$ -algebra, and its module category is equivalent to that of $A_1$ . What fails is only the description by generators. The genuinely pathological examples, where Noetherianity itself fails, need worse singularities.

Key Equations

For a smooth affine $X$ of dimension $d$ the order filtration has projective pieces and the graded ring is the functions on the cotangent bundle:

gr,D(X) ; ≅ ; Sym_O(X)! (Der_K O(X) ) ; ≅ ; O(T^*X), T^*X = 2d.

(3.10)

The ambient description via the idealiser, for $X ⊂ eq K^n$ cut out by the radical ideal $J$ :

I(J A_n) / J A_n ↪ D(X), (I(JA_n) → D(X) ) = J A_n = P ∈ A_n : P(S) ⊂ eq J.

(3.11)

For a domain $R$ with fraction field $Q$ , operators are detected inside $D(Q)$ :

D(R) = , P ∈ D(Q) : P(R) ⊂ eq R ,.

(3.12)

For the cusp $R = K[t^2,t^3]$ the two operators of the worked example act diagonally on the monomial basis:

P_2(t^k) = k(k-3),t^k-2, P_3(t^k) = k(k-2)(k-4),t^k-3.

(3.13)

Their symbols are $ξ^2$ and $ξ^3$ , while symbols of order- $m$ elements of the subring generated by $R$ and $Der_K(R)$ all lie in $t^mK[t],ξ^m$ .

Variable Definitions

$K$: an algebraically closed field of characteristic zero
$X$: a closed affine variety in $K^n$
$S$: the ambient polynomial ring $K[x_1, … ,x_n]$
$J$: the radical ideal of $X$ in $S$
$O(X)$: the coordinate ring $S/J$
$D(X)$: the ring of differential operators of $O(X)$ over $K$
$I(JA_n)$: the idealiser of the right ideal $JA_n$ in $A_n$
$R$: in the worked example, the coordinate ring $K[t^2,t^3]$ of the cusp
$∂$: differentiation with respect to $t$ on $K(t)$
$ξ$: the symbol of $∂$ , a coordinate on the cotangent fibre
$\tilde{X}$: the normalisation of an irreducible variety $X$

Properties and Behaviour

The smooth caseCoutinho, Ch. 3 §2 remark; McConnell-Robson, Ch. 15

Let $K$ have characteristic zero and let $X$ be a smooth irreducible affine variety over $K$ of dimension $d$ , with $R = O(X)$ . Then:

$D(X)$ is generated as a $K$ -algebra by $R$ and $Der_K(R)$ ;
$D^m(X)$ is a finitely generated projective $R$ -module for every $m$ ;
$gr,D(X) ≅ Sym_R(Der_K R) ≅ O(T^*X)$ , a Noetherian domain of Krull dimension $2d$ ;
$D(X)$ is left and right Noetherian, is a domain, and is a simple ring;
$R$ is a simple $D(X)$ -module.

For $X = K^n$ this is the theorem $D(K[x_1, … ,x_n]) = A_n(K)$ ; see the Weyl algebra as a ring of differential operators.

The fraction-field descriptionCoutinho, Ch. 3, Exercise 3.7

Let $R$ be a domain, finitely generated over $K$ , with fraction field $Q$ . Every differential operator on $R$ extends uniquely to $Q$ , and restriction identifies

D^m(R) = , P ∈ D^m(Q) ;:; P(R) ⊂ eq R ,

for every $m$ . In particular, for an irreducible curve $X$ with normalisation parametrised by $t$ , every operator on $X$ is an operator with coefficients in $K(t)$ , and the only condition to check is that it preserves $O(X)$ . This is the practical route to computing $D(X)$ for curves.

Localisation and open subsets

For $f ∈ R$ non-zero, $D(R_f) ≅ R_f ⊗ _R D(R)$ , so operators on a principal open subset are operators on $X$ with denominators from $f$ . Consequently $𝒟$ sheafifies, and $D(X)$ for affine $X$ is the module of global sections of the sheaf $D_X$ .

Singular curvesSmith-Stafford 1988

Let $X$ be an irreducible affine curve over an algebraically closed field of characteristic zero, with normalisation $\tilde{X}$ . If the singularities of $X$ are quasi-homogeneous - that is, if each singular point admits local coordinates in which the defining equations are weighted homogeneous, as for $y^2 = x^3$ with weights $3$ and $2$ - then $D(X)$ is a Noetherian domain, finitely generated as a $K$ -algebra, and Morita equivalent to $D( X)$ . Since simplicity is a Morita invariant and $D( X)$ is simple for smooth irreducible $\tilde{X}$ , such a $D(X)$ is simple as well. In particular $D(K[t^2,t^3])$ is Morita equivalent to $A_1(K)$ - although, as the worked example shows, it is not generated by $R$ and $Der_K(R)$ .

The cubic coneBernstein-Gelfand-Gelfand 1972

Let $R = K[x,y,z]/(f)$ where $f$ is a non-singular plane cubic form, so that $X$ is the affine cone over a smooth elliptic curve. Then $D(X)$ is neither left nor right Noetherian, and it is not finitely generated as a $K$ -algebra. This is the standard counterexample showing that no general finiteness theorem for $D(X)$ can hold.

Examples and Special Cases

Affine space

$X = K^n$ gives $D(X) = A_n(K)$ , generated by the coordinates and the partial derivatives, simple, Noetherian, a domain, with $gr = K[x,ξ]$ . Every good property in the smooth theorem is visible here.

The torus

$X = (K^*)^n$ with $O(X) = K[x_1^ ± 1, … ,x_n^ ± 1]$ . Then $D(X)$ is the localisation of $A_n$ at the monomials, generated by the $x_i^ ± 1$ and the Euler operators $θ_i = x_i∂_i$ . It is smooth, so the theorem applies; the Euler operators are the natural generators because they are invariant under the group structure.

A smooth conic

$X : xy = 1$ in $K^2$ is smooth and isomorphic to $K^*$ , so $D(X)$ is the previous example with $n = 1$ . Nothing about the equation being non-linear causes trouble; only singularity does.

The cusp

$X : y^2 = x^3$ , $O(X) = K[t^2,t^3]$ . The generation statement fails, as computed above. Nevertheless $D(X)$ is a simple Noetherian domain, finitely generated as an algebra, and Morita equivalent to $A_1$ : the module category is the same as for the affine line, even though the rings are not isomorphic.

A smooth affine elliptic curve

$X : y^2 = x^3 - x$ in $K^2$ is smooth. The derivation $D = y∂_x + 12(3x^2-1)∂_y$ preserves the ideal, since $D(y^2 - x^3 + x) = y(3x^2-1) + (1 - 3x^2)y = 0$ , and it generates $Der_K(O(X))$ as a free module of rank $1$ - the differential $dx/y$ is regular and nowhere vanishing on $X$ . So $D(X)$ is generated by $O(X)$ and the single vector field $D$ . Non-linear equations and positive genus cause no difficulty; only singularity does.

The cubic cone

$X$ the affine cone over a smooth plane cubic, $O(X) = K[x,y,z]/(f)$ with $f$ a non-singular cubic form. Here $D(X)$ is not Noetherian and not a finitely generated $K$ -algebra. This is the Bernstein-Gelfand-Gelfand example and it is the reason no general structure theorem exists.

Worked Example

The cusp: an operator that no vector fields produce

Step 1 - the ring and its derivations
Let $R = K[t^2, t^3] ⊂ eq K[t]$ , the coordinate ring of the cuspidal cubic $y^2 = x^3$ under $x = t^2$ , $y = t^3$ . As a $K$ -space $R = K ⊕ span_Kt^k : k ≥ 2$ : every power of $t$ occurs except $t^1$ . From the page on derivations,

$Der_K(R) = , f∂ ;:; f ∈ tK[t] , = R,(t∂) + R,(t^2∂), ∂ = d/dt.$
Step 2 - two operators that preserve $R$
Work inside $D(K(t))$ , the operators with rational coefficients, and use the fraction-field description: $D(R)$ consists of those preserving $R$ . Consider

$P_2 = ∂^2 - 2t^-1∂, P_3 = ∂^3 - 3t^-1∂^2 + 3t^-2∂.$

On the monomial $t^k$ these act diagonally. For $P_2$ : $∂^2 t^k = k(k-1)t^k-2$ and $2t^-1∂ t^k = 2k,t^k-2$ , so

$P_2(t^k) = (k(k-1) - 2k )t^k-2 = k(k-3),t^k-2.$

For $P_3$ , the three terms give $k(k-1)(k-2)$ , $-3k(k-1)$ and $3k$ times $t^k-3$ , and the bracket simplifies:

$P_3(t^k) = k [(k-1)(k-2) - 3(k-1) + 3 ]t^k-3 = k(k^2 - 6k + 8),t^k-3 = k(k-2)(k-4),t^k-3.$
Step 3 - check that both preserve $R$
It suffices to check on $t^k$ for $k = 0$ and $k ≥ 2$ . For $P_2$ : $k = 0$ gives $0$ ; $k = 2$ gives $2(-1)t^0 = -2 ∈ R$ ; $k = 3$ gives $3 ⋅ 0 ⋅ t = 0$ , which is the only place a forbidden $t^1$ could have appeared, and the coefficient kills it; for $k ≥ 4$ the output is a multiple of $t^k-2$ with $k - 2 ≥ 2$ , so it lies in $R$ .

For $P_3$ : $k = 0$ gives $0$ and $k = 2$ gives $2 ⋅ 0 ⋅ (-2) = 0$ ; $k = 3$ gives $3 ⋅ 1 ⋅ (-1) = -3$ , so $P_3(t^3) = -3 ∈ R$ ; $k = 4$ gives $4 ⋅ 2 ⋅ 0 = 0$ , which is exactly where the forbidden $t^1$ would have appeared; for $k ≥ 5$ the output is a multiple of $t^k-3$ with $k - 3 ≥ 2$ . So $P_2, P_3 ∈ D(R)$ , of orders $2$ and $3$ .
Step 4 - the symbol obstruction
Let $E ⊂ eq D(R)$ be the subring generated by $R$ and $Der_K(R)$ , and compute symbols inside $gr,D(K(t)) = K(t)[ξ]$ , which contains $gr,D(R)$ . Every derivation of $R$ is $f∂$ with $f ∈ tK[t]$ , so its symbol lies in $tK[t],ξ$ . An element of $ℰ$ of order at most $2$ is an $R$ -combination of products of at most two derivations, so its degree- $2$ symbol lies in

$R ⋅ (tK[t] ) (tK[t] ),ξ^2 ⊂ eq t^2K[t],ξ^2.$

But $σ_2(P_2) = ξ^2$ , and $1 ∉ t^2K[t]$ . Hence $P_2 ∉ E$ .
Step 5 - conclusion
$D(R) ≠ E$ : the ring of differential operators on the cusp is strictly larger than the subring generated by its regular functions and its vector fields. The same argument applied in degree $3$ shows $P_3 ∉ E$ , since symbols of order- $3$ elements of $ℰ$ lie in $t^3K[t]ξ^3$ while $σ_3(P_3) = ξ^3$ .

The mechanism is visible: every vector field on the cusp vanishes at the singular point, so products of vector fields vanish to increasing order there, while genuine differential operators need not. On a smooth curve no vector field is forced to vanish anywhere, and the obstruction disappears.

Result

For $R = K[t^2,t^3]$ the operators $P_2 = ∂^2 - 2t^-1∂$ and $P_3 = ∂^3 - 3t^-1∂^2 + 3t^-2∂$ lie in $D(R)$ , because $P_2(t^k) = k(k-3)t^k-2$ and $P_3(t^k) = k(k-2)(k-4)t^k-3$ both vanish exactly when the output would be the missing monomial $t$ . Neither lies in the subring generated by $R$ and $Der_K(R)$ , whose order- $m$ symbols are divisible by $t^m$ . So the naive description of $D(X)$ fails on the simplest singular curve.

Applications and Industry Use

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

D-modules on varieties. All of the machinery of this collection - characteristic varieties, holonomicity, the six functors - is developed for smooth varieties, precisely because $D(X)$ is well behaved exactly there. Extending to singular $X$ is done by embedding into a smooth ambient space rather than by working with $D(X)$ directly.
Representation theory. For $X$ a homogeneous space $G/H$ , $D(X)$ carries an action of the Lie algebra of $G$ and its modules encode representations; the Beilinson-Bernstein localisation theorem identifies representations of a semisimple Lie algebra with D-modules on the flag variety.
Invariant theory. Comparing $D(X)^G$ with $D(X/!!/G)$ for a reductive group action is the Levasseur-Stafford problem; the quotient is usually singular, so the singular theory is unavoidable.
Non-commutative ring theory. $D(X)$ for singular $X$ supplies natural examples of simple non-Noetherian rings and of Morita equivalences between non-isomorphic rings.
Rings of differential operators in computation. Algorithms for local cohomology and for $b$ -functions of singular hypersurfaces work in the ambient Weyl algebra with the idealiser description, never in $D(X)$ directly.

Design Considerations

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

Work upstairs, not downstairs

Practically every computation involving a singular $X$ embeds $X$ in a smooth ambient space and works with $A_n$ modulo the ideal $J$ , using the idealiser description (3.11). The reason is that $A_n$ is Noetherian with a usable normal form and a Groebner theory, and $D(X)$ may have none of those. Choosing the embedding is then a modelling decision: a different embedding gives a different presentation of the same ring.

Which category of modules

On smooth $X$ , left and right $D(X)$ -modules are equivalent categories via twisting by the canonical module $ω_X$ . On singular $X$ that equivalence is unavailable, so the side must be fixed at the outset. Similarly, coherence over $D(X)$ is a good finiteness condition only when $D(X)$ is Noetherian.

Normalisation as a tool, not a fix

For curves it is tempting to replace $X$ by its normalisation $\tilde{X}$ , which is smooth. That does not compute $D(X)$ : the two rings are generally not isomorphic. What can be true, under hypotheses, is Morita equivalence, which preserves module categories but not the ring. If the object of interest is the category of D-modules, normalisation may suffice; if it is the ring, it does not.

Standards and Codes

For mathematics, the relevant standards are notation, numeric and markup standards together with reference implementations.

ISO 80000-2 governs the notation for $∂$ , $dim$ and set operations used above.
ISO/IEC 40314 (MathML 3.0) encodes the mathematics on this page.
Terminology is not uniform. Regular, smooth and non-singular coincide for varieties over an algebraically closed field of characteristic zero but differ in general; papers on $D(X)$ usually say regular because the hypothesis really is about the local rings.
Notation for the idealiser varies: $I(J)$ , $I(J)$ and $I_A(J)$ all appear. Coutinho writes $I(J)$ ; this page writes $𝕀$ to avoid collision with the ideal itself.
Macaulay2 and Singular both implement Weyl-algebra computations over $ℚ$ by default; neither provides a standardised interface to $D(X)$ for singular $X$ .

Material Selection

The material of a mathematical construction is its numeric substrate: the ground field, the coefficient ring and the representation used to store it.

Characteristic zero and algebraic closure. Smoothness and regularity coincide over a perfect field; over an imperfect field the two differ and the smooth theorem must be stated with regularity. Characteristic zero is needed for the identification with the Weyl algebra in the model case.
Reducedness. For non-reduced $O(X)$ , that is for schemes with nilpotents, $𝒟$ still makes sense but the geometric intuition fails badly; for finite-dimensional algebras it is all of $End_K$ .
Irreducibility. $𝒟$ of a disjoint union is the product of the $𝒟$ 's, so reducible $X$ with disjoint components adds nothing new. Components meeting each other, as at a node, do produce new behaviour.
Storage. An operator on $X$ is stored as a representative in $A_n$ together with $J$ ; equality testing requires reduction modulo $J A_n$ , which needs a Groebner basis of that right ideal. For monomial curves the diagonal representation by a rational function $c(k)$ is far more compact.

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.

Present $X$ as $V(J) ⊂ eq K^n$ with $J$ radical, computed by a radical-ideal algorithm if necessary.
For each order $m$ , set up the linear conditions on $P = ∑ _|β| ≤ m f_β ∂^β ∈ A_n$ expressing $P(J) ⊂ eq J$ ; this is a syzygy computation over $S$ and returns $I(JA_n) ∩ D^m$ .
Reduce modulo $J A_n ∩ D^m$ to obtain the image of (3.9) in order $m$ .
If $X$ is a curve, cross-check with the fraction-field method: parametrise the normalisation, write operators with coefficients in $K(t)$ , and impose that $O(X)$ is preserved on a generating set of monomials.
Look for stabilisation of the generating set as $m$ grows. Failure to stabilise is evidence, not proof, that $D(X)$ is not finitely generated.

Every step but the last is a finite commutative computation. The last is not guaranteed to terminate, and by the cubic cone example there is no algorithm that always returns a finite presentation of $D(X)$ . Macaulay2's Dmodules and Singular's dmod.lib and dmodapp.lib support the ambient computations; neither offers a general $D(X)$ constructor for singular $X$ , which is a limitation of the mathematics rather than of the software.

For the diagonal operators appearing on monomial curves such as the cusp, there is a shortcut worth knowing. Since $R = K[t^a_1, … ,t^a_r]$ is spanned by a set $Γ ⊂ eq N$ of exponents, every operator with coefficients in $K(t)$ that acts diagonally on $t^k$ by a rational function $c(k)$ lies in $D(R)$ if and only if $c(k) = 0$ whenever $k ∈ Γ$ and $k - m ∉ Γ$ , where $m$ is the shift. That reduces membership to a finite check on the finitely many gaps of $Γ$ .

Limits of Validity

The smooth theorem needs smoothness everywhere. A single singular point breaks the generation statement, as the cusp shows. There is no version that holds on the smooth locus and patches over the singularities.
No finiteness in general. By the cubic cone example, $D(X)$ can fail to be Noetherian and can fail to be a finitely generated algebra. Any argument assuming a finite presentation must justify it for the $X$ at hand.
Surjectivity of (3.9) is not automatic. The idealiser gives the operators that extend to the ambient space. On smooth $X$ that is everything; on singular $X$ it must be checked.
Morita equivalence is not isomorphism. $D(cusp)$ and $A_1$ have equivalent module categories but are not isomorphic rings. Transporting a statement across a Morita equivalence is legitimate only for statements about modules.
Characteristic zero. In characteristic $p$ even $D(K^n)$ is larger than the Weyl algebra, so none of the smooth theory as stated applies.

Failure Modes and Common Mistakes

Assuming functions and vector fields generate

This is the single most common error. It is a theorem for regular rings in characteristic zero and false in general. Coutinho flags it explicitly after the Weyl algebra theorem, pointing to the cusp; the worked example on this page supplies the proof. Any argument that writes an arbitrary operator on a singular $X$ as a polynomial in derivations is invalid.

Thinking a pole in the coefficients disqualifies an operator

$∂^2 - 2t^-1∂$ has a pole at $t = 0$ and is still a differential operator on $K[t^2,t^3]$ . What matters is only whether the operator maps the ring into itself; the pole is cancelled by the gap in the ring, since the coefficient $k(k-3)$ vanishes at exactly the value of $k$ that would produce the missing monomial $t$ .

Concluding that singular means pathological

For the cusp, $D(X)$ is simple, Noetherian, a domain, finitely generated, and Morita equivalent to $A_1$ . The only failure is the description by generators. Genuine pathology - non-Noetherian, non-finitely-generated - requires worse singularities such as the cone over an elliptic curve.

Replacing $X$ by its normalisation

$D(X)$ and $D( X)$ are different rings. Under hypotheses they are Morita equivalent, which is enough to transport statements about module categories but not statements about the ring - not simplicity of a specific element, not the shape of the order filtration, not the generators.

Assuming $D(X)$ determines $X$

It does not. Non-isomorphic varieties can have Morita equivalent, and in some cases isomorphic, rings of differential operators. Recovering $X$ needs the pair consisting of $D(X)$ together with its action on $O(X)$ , or equivalently the filtered ring rather than the bare ring.

Historical Notes

Grothendieck's inductive definition, published in EGA IV in 1967, made $D(X)$ available for arbitrary schemes and immediately raised the question of how it behaves on singular varieties. The answer came quickly and was discouraging: in 1972 Bernstein, Gelfand and Gelfand computed $D(X)$ for the affine cone over a smooth plane cubic and found a ring that is neither Noetherian nor finitely generated. That paper is the reason the D-module literature works almost exclusively on smooth varieties.

The positive side was assembled over the same period. That $D(X)$ is generated by functions and vector fields for regular $X$ in characteristic zero, and is then a simple Noetherian domain when $X$ is smooth and irreducible, is treated systematically in McConnell and Robson's book, Chapter 15. It generalises Coutinho's Chapter 3 theorem for affine space.

Curves were then classified. Smith and Stafford's 1988 paper analyses $D(X)$ for irreducible affine curves and shows, under a quasi-homogeneity hypothesis on the singularities, that it is a Noetherian domain, finitely generated as an algebra, and Morita equivalent to the ring of operators on the normalisation. The cusp falls on the good side of that line, which is why it is the right example to make the generation point without also dragging in non-Noetherian behaviour.

Comparison

Behaviour of $D(X)$ across five coordinate rings.
$O(X)$	Generated by $O(X)$ and $Der$ ?	Noetherian?	Finitely generated algebra?
$K[x_1, … ,x_n]$	yes	yes	yes, it is $A_n$
smooth irreducible affine	yes	yes	yes
$K[t^2,t^3]$ (cusp)	no	yes	yes
cone over a smooth plane cubic	no	no	no
$K[x]/(x^2)$	no	yes, it is $End_K(O(X))$	yes

Key Takeaways

Key points

$D(X)$ is defined for every affine variety by the same induction that defines the Weyl algebra; smoothness is not part of the definition.
For smooth irreducible $X$ over a field of characteristic zero, $D(X)$ is generated by $O(X)$ and $Der_K O(X)$ , and is a simple Noetherian domain with $gr,D(X) = O(T^*X)$ .
The idealiser gives an injection $I(JA_n)/JA_n ↪ D(X)$ whose image is the operators extending to the ambient affine space.
On the cusp $K[t^2,t^3]$ the operator $∂^2 - 2t^-1∂$ lies in $D(X)$ but not in the subring generated by functions and vector fields; the obstruction is that all vector fields vanish at the cusp point.
The cusp is still well behaved as a ring: simple, Noetherian, finitely generated, Morita equivalent to $A_1$ .
For the cone over a smooth plane cubic, $D(X)$ is neither Noetherian nor finitely generated, so no general finiteness theorem is possible.

FAQs

D(X)

always generated by functions and vector fields?

No. It is for regular $X$ in characteristic zero, and it fails for the cusp, where $∂^2 - 2t^-1∂$ preserves $K[t^2,t^3]$ but has symbol $ξ^2$ , while everything in the subring generated by functions and derivations has order- $2$ symbol divisible by $t^2$ .

Why does the operator with a pole still act on the ring?

Because the ring has a gap. $K[t^2,t^3]$ contains every power of $t$ except $t^1$ , and the coefficient $k(k-3)$ in $P_2(t^k) = k(k-3)t^k-2$ vanishes at $k = 3$ , exactly the value that would produce $t^1$ . The pole and the gap cancel.

Does every operator on

X

come from an operator on the ambient space?

Those that do form the image of the idealiser map (3.9), which is injective in general. For smooth $X$ the map is onto. For singular $X$ surjectivity is a genuine question and should not be assumed.

D(X)

always Noetherian?

No. It is for smooth $X$ , and for curves with quasi-homogeneous singularities such as the cusp. For the cone over a smooth plane cubic it is not Noetherian and is not even a finitely generated algebra, by the 1972 result of Bernstein, Gelfand and Gelfand.

D(X)

is Morita equivalent to

A_1

, are the two rings the same?

No. Morita equivalence means the module categories are equivalent. The rings are not isomorphic: one is generated by $x$ and $∂$ in the usual way, the other is not generated by its functions and derivations at all. Statements about modules transport; statements about elements do not.

What replaces

D(X)

when

X

is singular?

In practice, one embeds $X$ in a smooth ambient variety $Y$ and works with $D(Y)$ -modules supported on $X$ . Kashiwara's equivalence identifies those with D-modules on $X$ when $X$ is a smooth closed subvariety; for singular $X$ the category of $D_Y$ -modules supported on $X$ is taken as the definition, precisely to avoid $D(X)$ .

Do vector fields on a singular variety have to vanish at the singular points?

On the cusp, yes: every element of $Der_K(K[t^2,t^3])$ is $f∂$ with $f ∈ tK[t]$ , so all vanish at $t = 0$ . This is not universal - it depends on the singularity - but it is exactly what causes the generation failure here.

D(X)

simple when

X