The Weyl Algebra in Positive Characteristic

Q: Which of R_1 and R_2 deserves the name "the Weyl algebra in characteristic p"?

By near-universal convention, R_2 : the algebra defined by the Weyl relations. It is the one with a basis of canonical monomials, the one that is a domain, and the one whose Azumaya structure is used in applications. R_1 is best regarded as its quotient of vanishing p -curvature. If a source says "the Weyl algebra over Z_p " without qualification, it almost certainly means R_2 .

Q: Does the failure of simplicity mean characteristic p has no simple ring of differential operators?

No. The full ring of differential operators on k[x] in Grothendieck's sense, generated by the divided-power operators ∂^[m] with ∂^[m](x^j) = jmx^j-m , is simple and acts faithfully. It is not finitely generated as an algebra, and it strictly contains R_1 . So the correct statement is that the algebra generated by x_i and ∂_i fails to be simple, not that differential operators fail to be simple.

Q: Why exactly does the degree argument for simplicity break?

The descent replaces D by [x_i, D] or [∂_i, D] and needs the result to be non-zero. On coefficient polynomials these brackets are partial derivatives, and the coefficient that appears is an exponent α_i or β_i regarded as an element of the field. When p ∣ α_i that coefficient is zero, the bracket vanishes without lowering to a constant, and the descent halts. The element x_1^p is exactly a fixed point of the whole process.

Q: Is R_2 Noetherian?

Yes. It is a finitely generated module over the Noetherian commutative ring Z , so it is Noetherian on both sides by the standard argument. It is also Noetherian for the reason used in characteristic zero, namely that its associated graded ring for the degree filtration is a polynomial ring in 2n variables.

Q: How large can a finite-dimensional module be?

Any dimension divisible by p^n is achievable, as direct sums and extensions of the p^n -dimensional simples. What cannot happen is a non-zero module of dimension not divisible by p : the trace argument still gives p ∣ _k M , and the Azumaya structure sharpens this to p^n ∣ _k M over an algebraically closed field.

Q: Does anything of holonomic theory survive?

Not by direct translation. The class of modules of minimal dimension becomes the finite-dimensional modules, which are not closed under the operations that make holonomicity useful, and the Bernstein-Sato polynomial has no characteristic- p analogue in the naive sense - its roots are rational numbers whose denominators interact badly with p . Arithmetic D -module theory builds a substitute finiteness notion from Frobenius structures instead.

Q: Is A_1(Z) relevant to this discussion?

Yes, as the bridge. A_1(Z) is a domain but not simple - each prime p generates a proper two-sided ideal - and reducing modulo p gives exactly R_2 over Z_p . Arguments that reduce a characteristic-zero problem modulo many primes, such as the Dixmier-Jacobian equivalence, live in A_n(Z) and use this reduction systematically.

Overview

Every page in this collection assumes the ground field $K$ has characteristic zero. That hypothesis is not decoration. In characteristic $p$ the Weyl algebra stops being the object the theory is about, and it stops in two different ways depending on which of the two standard definitions you use.

Define $A_n$ as the algebra of operators on $K[x_1, … ,x_n]$ generated by the coordinates and the partial derivatives, and you get a ring $R_1$ in which $∂_i^,p = 0$ . It contains nilpotent elements, so it is not a domain, and the canonical monomials $x^α∂^β$ are no longer linearly independent, so the degree is not even well defined.

Define $A_n$ instead by generators and relations - the free algebra on $2n$ symbols modulo $[∂_i, x_j] = δ_ij$ and the commuting relations - and you get a ring $R_2$ that is a domain, with a perfectly good degree, but that is not simple: each $x_i^,p$ and each $∂_i^,p$ is central, so there is a large supply of proper two-sided ideals. In characteristic zero these two constructions agree; in characteristic $p$ they are genuinely different rings, and $R_1$ is a proper quotient of $R_2$ .

Neither ring supports the theory. The central failure is that both have non-zero finite-dimensional modules - $K[x]/(x^p)$ is one - so the dimension theory that rests on simplicity collapses: there are modules of dimension $0$ , and Bernstein's inequality is false. This page states precisely what survives, what fails, and what the characteristic- $p$ objects are good for, because they are far from useless: $p$ -curvature is a serious tool in arithmetic geometry and was the route by which the Dixmier and Jacobian conjectures were shown to be equivalent.

Definition

Let $p$ be a prime and let $k$ be a field of characteristic $p$ ; for concreteness read $k = Z_p$ , the field with $p$ elements. We compare two constructions, both of which reduce to $A_n$ when the characteristic is zero.

The operator ring $R_1$

$R_1$ is the $k$ -subalgebra of $End_k(k[x_1, … ,x_n])$ generated by multiplication by $x_1, … ,x_n$ and by the formal partial derivatives $∂_1, … ,∂_n$ . This is the definition used in the operator definition of the Weyl algebra, read over $k$ .

The abstract ring $R_2$

$R_2$ is the quotient of the free $k$ -algebra on $2n$ symbols $z_1, … ,z_2n$ by the two-sided ideal generated by $[z_i+n, z_i] - 1$ for $1 ≤ i ≤ n$ and $[z_i,z_j]$ for all other pairs. This is the generators-and-relations definition, read over $k$ . By construction the monomials $z^α z^β$ in the two halves of the generators form a $k$ -basis.

The two rings differCoutinho, Ch. 2 §3

Let $char,k = p > 0$ . Then:

$∂_i^,p = 0$ in $R_1$ . In particular $R_1$ contains non-zero nilpotent elements and is not a domain.
$R_2$ is a domain, and the degree of an operator is additive on $R_2$ exactly as in characteristic zero.
$R_2$ is not simple: each $z_i^,p$ is central, and the two-sided ideal it generates is proper.
The natural surjection $R_2 ↠ R_1$ has kernel the two-sided ideal generated by $∂_1^,p, … ,∂_n^,p$ . In characteristic zero the corresponding map is an isomorphism.

A note on the source

Coutinho treats this in Ch. 2 §3 for $n = 1$ and leaves the structural facts about $R_2$ as Exercise 4.11, where the centre is identified. We give the general- $n$ statements here. The centre of $R_2$ is the polynomial ring on the $p$ -th powers of all $2n$ generators; a reader working from a scanned copy should be aware that this is easy to misread as involving small exponents rather than $p$ -th powers.

Core Concepts

Everything on this page follows from one identity and one consequence of it.

The $p$ -th power of a derivation is a derivation, and here it is central

In any ring of characteristic $p$ , if $δ$ is a derivation then so is $δ^p$ - the intermediate binomial coefficients in the Leibniz expansion are all divisible by $p$ . Applying this to the inner derivation $ad,∂_i$ , whose effect on coefficient polynomials is $∂/∂ x_i$ , gives $(ad,∂_i)^p = ad(∂_i^,p)$ , and $(∂/∂ x_i)^p = 0$ on polynomials because $∂^p(x^k)/∂ x^p$ involves the falling factorial $k(k-1) ⋅ s(k-p+1)$ , a product of $p$ consecutive integers, always divisible by $p$ .

Hence $ad(∂_i^,p) = 0$ : the element $∂_i^,p$ commutes with everything. The same argument on the other side makes $x_i^,p$ central.

Central elements are the enemy of simplicity

As soon as an element $c$ is central and not a unit, $R c$ is a proper non-zero two-sided ideal. So the moment $x_1^,p$ becomes central, simplicity is gone. This is not a subtle failure at the edge of the theory; it is a complete structural change. Instead of a simple algebra with centre $K$ , one has a finite algebra over a large commutative base.

The operator picture forgets the centre in one direction

In $R_1$ , the central element $∂_i^,p$ is not merely non-invertible: it is zero. That is what produces nilpotents. Meanwhile $x_i^,p$ is still central and non-zero, so $R_1$ is not simple either. Between them, $R_1$ fails both of the two theorems of this chapter, while $R_2$ fails only one.

Construction and Proof

Proof that $∂_i^,p = 0$ in $R_1$

It suffices to evaluate on a monomial basis of $k[x_1, … ,x_n]$ , and since $∂_i$ acts only in the $i$ -th variable it suffices to take $n = 1$ . For $k < p$ , $∂^,p(x^k) = 0$ because the derivative order exceeds the degree. For $k ≥ p$ ,

∂^,p(x^k) = (k!)/((k-p)!),x^k-p = k(k-1) ⋅ s(k-p+1),x^k-p.

The coefficient is a product of $p$ consecutive integers, hence divisible by $p!$ and in particular by $p$ , so it vanishes in $k$ . Therefore $∂^,p$ is the zero operator while $∂ ≠ 0$ , and $R_1$ has a non-zero nilpotent.

$R_2$ is a domainCoutinho, Ch. 2 Exercise 4.11(1)

The monomials $x^α∂^β$ are a $k$ -basis of $R_2$ by construction, so the degree is well defined. The reordering estimate and the proof of additivity of degree given on the degree page use only the relation $[∂_i,x_j] = δ_ij$ and the fact that $k[x,ξ]$ is a domain; neither requires characteristic zero. Hence $(DD') = (D)+ (D')$ and $R_2$ is a domain.

This is worth stating explicitly because it is easy to assume the whole of Chapter 2 fails in characteristic $p$ . The domain theorem does not.

$R_2$ is not simpleCoutinho, Ch. 2 §3

By (2.18) the element $c = x_1^,p$ is central. It is not a unit, since $(c) = p > 0$ and the units of $R_2$ are the non-zero scalars by the same degree argument as in characteristic zero. Therefore $R_2 c$ is a two-sided ideal, non-zero and proper. $R_2$ is not simple.

In fact the failure is as bad as possible: since $R_2$ is a free module of rank $p^2n$ over the polynomial ring $Z$ , every ideal $a ⊂ eq Z$ produces a two-sided ideal $aR_2$ , and these are pairwise distinct. The lattice of two-sided ideals is at least as rich as that of a polynomial ring in $2n$ variables.

The centre and the rankCoutinho, Ch. 2 Exercise 4.11(2), (3)

$Z(R_2) = k[x_1^,p, … ,x_n^,p,∂_1^,p, … ,∂_n^,p]$ and $R_2$ is free of rank $p^2n$ over it, with basis (2.20).

Sketch. Containment of the Frobenius powers in the centre is (2.18). Conversely, the computation of the adjoint action on canonical coefficients used in the simplicity proof shows $D$ is central if and only if all $2n$ partial derivatives of its coefficient polynomial $P_D$ vanish. In characteristic $p$ the polynomials killed by every partial derivative are exactly the polynomials in the $p$ -th powers of the variables, which gives the stated centre. Freeness and the rank follow because the basis $x^α∂^β$ of $R_2$ splits uniquely as a Frobenius power times a monomial with all exponents below $p$ .

Identification of $R_1$

The surjection $π : R_2 → R_1$ sending generators to the corresponding operators kills each $∂_i^,p$ . Its kernel is exactly the two-sided ideal generated by those elements. Indeed, modulo that ideal the monomials $x^α∂^β$ with all $β_i < p$ form a basis, and those monomials are linearly independent as operators on $k[x]$ : apply a relation $∑ _β f_β(x)∂^β = 0$ to suitable monomials $x^γ$ with $γ_i < p$ and read off the coefficients using $∂^β(x^γ) = γ!,δ_βγ$ for $|β| ≤ |γ|$ with all exponents below $p$ , where the factorials are invertible.

So $R_1 ≅ R_2 / (∂_1^,p, … ,∂_n^,p)$ : it is the fibre of $R_2$ over the locus where the $∂$ -side of the centre vanishes. $R_1$ is not simple either, since $x_1^,p$ remains central and non-invertible in the quotient.

What $R_2$ actually is: an Azumaya algebra

Beyond the Primer, the accepted structural description is that $R_2$ is an Azumaya algebra of degree $p^n$ over its centre $Z$ . Concretely, after base change to an algebraic closure and specialisation at a maximal ideal $m ⊂ Z$ , one has $R_2 ⊗ _Z Z/m ≅ M_p^n( k)$ . So $R_2$ is a family of $p^n × p^n$ matrix algebras parametrised by an affine $2n$ -space, and its simple modules all have dimension $p^n$ over $\bar{k}$ . This is quoted, not proved here.

Key Equations

The vanishing that starts everything, for any $k ≥ 0$ and $char = p$ :

∂^,p(x^k) = k(k-1) ⋅ s(k-p+1),x^k-p = 0,

(2.17)

since $p$ consecutive integers always contain a multiple of $p$ ; and $∂^,p(x^k) = 0$ trivially for $k < p$ .

The resulting central elements, in $R_2$ and for every $i$ :

[,x_i^,p,; ∂_j,] = -,p,δ_ij,x_i^,p-1 = 0, [,∂_i^,p,; x_j,] = p,δ_ij,∂_i^,p-1 = 0.

(2.18)

Hence the centre and the module structure over it:

Z(R_2) = k[,x_1^,p, … ,x_n^,p,; ∂_1^,p, … ,∂_n^,p,], rank_Z,R_2 = p^2n.

(2.19)

A $Z$ -basis is given by the $p^2n$ monomials with all exponents below $p$ :

, x^α ∂^β ;:; 0 ≤ α_i, β_i ≤ p-1 ,.

(2.20)

Finally, the module that shows dimension theory breaks:

V = k[x_1, … ,x_n] / (x_1^,p, … ,x_n^,p ), _k V = p^,n < ∈ fty.

(2.21)

The derivatives are well defined on $V$ precisely because $∂_i(x_i^,p) = p,x_i^,p-1 = 0$ .

Variable Definitions

$p$: a prime, the characteristic of the ground field
$k$: a field of characteristic $p$ ; read $Z_p$ if a concrete choice is wanted
$R_1$: the algebra of operators on $k[x_1, … ,x_n]$ generated by the coordinates and the partial derivatives
$R_2$: the algebra on $2n$ generators defined by the Weyl relations over $k$
$Z$: the centre of $R_2$ , a polynomial ring in the $2n$ Frobenius powers
$V$: the finite-dimensional module $k[x]/(x_1^,p, … ,x_n^,p)$ , of dimension $p^n$
$ψ$: the $p$ -curvature of a module: the family of operators by which the central elements $∂_i^,p$ act
$ad y$: the adjoint action $D ↦ [y,D]$

Properties and Behaviour

Every module carries a $p$ -curvature

Let $M$ be a module over $R_2$ . Since each $∂_i^,p$ is central, it acts on $M$ by an endomorphism commuting with the whole action. The resulting family $ψ_i =$ (action of $∂_i^,p$ ) is the $p$ -curvature of $M$ . It is an invariant with no characteristic-zero analogue at all, because in characteristic zero $∂_i^,p$ is not central.

Modules with vanishing $p$ -curvature are precisely those killed by the ideal generated by the $∂_i^,p$ , that is the $R_1$ -modules. By Cartier's theorem these are the ones that arise by pulling back along the Frobenius map - they are "constant" in a precise sense.

$R_2$ satisfies a polynomial identity

Because $R_2$ is a finite module over its centre, it satisfies polynomial identities and has PI degree $p^n$ . Every simple $R_2$ -module over an algebraically closed field has dimension exactly $p^n$ . This is a complete change of representation-theoretic character: $A_n$ in characteristic zero satisfies no polynomial identity and has no finite-dimensional modules at all.

What is still true about ideals

The two-sided ideals of $R_2$ are not arbitrary. Since $R_2$ is Azumaya over $Z$ , the map $a ↦ aR_2$ is a bijection between ideals of $Z$ and two-sided ideals of $R_2$ . So the two-sided ideal theory of $R_2$ is exactly the ideal theory of a polynomial ring in $2n$ variables - completely understood, but completely different from simplicity.

Worked Example

The Weyl algebra over $Z_3$ acting on a three-dimensional space

Step 1 - the nilpotent derivative
Take $p = 3$ , $n = 1$ , $k = Z_3$ . On $k[x]$ we have $∂^3(x^k) = k(k-1)(k-2)x^k-3$ , and among any three consecutive integers one is divisible by $3$ , so $∂^3 = 0$ as an operator. Yet $∂ ≠ 0$ (it sends $x$ to $1$ ) and $∂^2 ≠ 0$ (it sends $x^2$ to $2$ ). So in $R_1$ the element $∂$ is nilpotent of order exactly $3$ , and $∂ ⋅ ∂^2 = 0$ exhibits zero divisors.
Step 2 - build the three-dimensional module
Let $V = k[x]/(x^3)$ , with basis $1, x, x^2$ . Multiplication by $x$ and differentiation both descend to $V$ : the second because $∂(x^3) = 3x^2 = 0$ in $k$ . In the given basis the matrices are

$X = pmatrix 0 & 0 & 0 1 & 0 & 0 0 & 1 & 0 pmatrix, P = pmatrix 0 & 1 & 0 0 & 0 & 2 0 & 0 & 0 pmatrix,$

reading columns as images of the basis vectors: $X$ sends $1 ↦ x ↦ x^2 ↦ 0$ , and $P$ sends $1 ↦ 0$ , $x ↦ 1$ , $x^2 ↦ 2 x$ .
Step 3 - check the Weyl relation holds
Multiplying out,

$PX = pmatrix 1 & 0 & 0 0 & 2 & 0 0 & 0 & 0 pmatrix, XP = pmatrix 0 & 0 & 0 0 & 1 & 0 0 & 0 & 2 pmatrix,$

$PX - XP = pmatrix 1 & 0 & 0 0 & 1 & 0 0 & 0 & -2 pmatrix = I, since -2 = 1 in Z_3.$

So $[P, X] = I$ : a genuine $3$ -dimensional representation of the Weyl relations. Over a field of characteristic zero no such matrices exist at any size.
Step 4 - see why the trace obstruction disappears
The classical argument against finite-dimensional representations takes traces: $tr(PX - XP) = 0$ always, while $tr(I) = V$ . In characteristic zero this forces $V = 0$ . Here $V = 3 = 0$ in $Z_3$ , so the equation $0 = 3$ is satisfied and there is no contradiction. Every finite-dimensional module in characteristic $p$ must therefore have dimension divisible by $p$ - and indeed $V = p^n = 3$ . See the characteristic-zero non-existence theorem for the contrast.
Step 5 - read off the structural failures
$V$ is a simple module: from any non-zero vector, differentiating enough times reaches a non-zero scalar and then multiplying by $x$ recovers the whole space. So $R_1$ and $R_2$ both have a simple module of dimension $3 = p^n$ .

Three consequences follow at once. The annihilator of $V$ in $R_2$ is a non-zero two-sided ideal - it contains $x^3$ and $∂^3$ - so $R_2$ is not simple. The Hilbert function of $V$ under any filtration is eventually constant, so $d(V) = 0 < n = 1$ and Bernstein's inequality is false here. And $V$ shows the theory of holonomic modules cannot be transplanted: in characteristic $p$ the minimal dimension is $0$ , not $n$ .

Result

Over $Z_3$ the matrices $X$ and $P$ above satisfy $[P,X] = I$ on a $3$ -dimensional space, because $tr(I) = 3 = 0$ . The module $V = Z_3[x]/(x^3)$ is simple of dimension $p^n = 3$ , its annihilator is a proper non-zero two-sided ideal, and its dimension as a filtered module is $0$ . So in characteristic $p$ : no simplicity, no Bernstein inequality, and finite-dimensional representations exist.

Applications and Industry Use

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

Characteristic- $p$ Weyl algebras are not a curiosity. Three places where they do real work:

Reduction modulo $p$ of differential equations. The $p$ -curvature of the reduction of a linear differential equation modulo $p$ controls whether the equation has algebraic solutions. The Grothendieck-Katz $p$ -curvature conjecture - that a connection has a full set of algebraic solutions if and only if its $p$ -curvature vanishes for almost all $p$ - is a central open problem, proved only in special cases.
The Dixmier and Jacobian conjectures. Tsuchimoto, and independently Belov-Kanel and Kontsevich, used reduction modulo $p$ and the Azumaya structure of $R_2$ to prove that the Dixmier conjecture for $A_n$ and the Jacobian conjecture in $2n$ variables are equivalent. The centre of $R_2$ is what makes this possible: an endomorphism of $R_2$ induces a map on the centre, and that map is a polynomial map of affine $2n$ -space.
Arithmetic $D$ -modules. Berthelot's theory of arithmetic $D$ -modules, and the theory of crystals, are built on rings of divided-power differential operators in characteristic $p$ and their lifts. These are the correct characteristic- $p$ substitutes for $A_n$ , and they recover simplicity.

In computation, reduction modulo a prime is a standard accelerator for Gröbner basis calculations. For commutative rings the modular image is harmless if the prime avoids finitely many bad values. For $A_n$ the situation is worse: the ring itself changes character below $p$ , degrees can collapse, and any modular computation in the Weyl algebra must use primes larger than every exponent appearing, and be certified over $ℚ$ afterwards.

Limits of Validity

Care is needed about what this page does and does not claim.

It does not say the characteristic- $p$ objects are pathological or uninteresting. $R_2$ is a well-understood Azumaya algebra and is a central tool in arithmetic $D$ -module theory. What it is not is a ring for which the Primer's theorems hold.
The identification $R_1 ≅ R_2/(∂_i^,p)$ is for the polynomial ring in the standard variables. Over other base algebras in characteristic $p$ the ring of differential operators is defined by the Grothendieck (divided power) recipe and contains divided derivatives $∂^[m]$ that are not products of the $∂_i$ . That ring is again simple, and it is the right characteristic- $p$ analogue of $A_n$ - but it is not generated by $x_i$ and $∂_i$ , so it is a different object from both $R_1$ and $R_2$ .
The failure of Bernstein's inequality is not repaired by changing the filtration. A finite-dimensional module has bounded Hilbert function under any good filtration, so $d(V) = 0$ regardless.

The divided power algebra, stated precisely

For $k$ of characteristic $p$ , the full ring of $k$ -linear differential operators on $k[x]$ in Grothendieck's sense is generated by the operators $∂^[m]$ defined by $∂^[m](x^j) = jmx^j-m$ . One has $∂^[m]∂^[m'] = m+m'm∂^[m+m']$ , so $∂^[p]$ is genuinely new and is not $∂^p/p!$ . This larger ring is simple and acts faithfully on $k[x]$ ; it is not finitely generated as a $k$ -algebra. Coutinho does not treat it, and this collection does not either, but it is the reason one should not conclude that "differential operators in characteristic $p$ are never simple".

Failure Modes and Common Mistakes

Assuming the two definitions of $A_n$ always agree

They agree in characteristic zero, where the comparison map is an isomorphism, and this is proved in Chapter 1. In characteristic $p$ the map $R_2 → R_1$ has a large kernel. Any argument that silently switches between the operator picture and the relations picture is unsound in characteristic $p$ , and this is by far the commonest error in this area.

Concluding that $∂$ is nilpotent in $R_2$

It is not. In $R_2$ the element $∂^,p$ is central and non-zero; it is only its image in $R_1$ that vanishes. $R_2$ is a domain, so it has no non-zero nilpotents whatsoever. The two statements " $∂^,p$ is central" and " $∂^,p$ is zero" belong to different rings.

Using the trace argument without checking the characteristic

The argument " $tr[P,X] = 0$ but $tr(I) = V$ , hence $V = 0$ " is only a contradiction when $V ≠ 0$ in the field. In characteristic $p$ it merely says $p ∣ V$ . The worked example above realises this with $V = 3$ over $Z_3$ .

Transporting holonomicity to characteristic $p$

Holonomic modules are defined as those of minimal dimension $n$ , and the definition is only meaningful because Bernstein's inequality forbids anything smaller. In characteristic $p$ there are modules of dimension $0$ , so "minimal dimension" would mean $0$ and the class would be the finite-dimensional modules - a completely different category with none of the expected closure properties. Arithmetic $D$ -module theory does have a good finiteness notion, but it is built differently, via Frobenius descent and not by copying the definition.

Historical Notes

The observation that $∂^,p$ is central in characteristic $p$ goes back to Jacobson's work on restricted Lie algebras in the 1930s and 1940s, where the $p$ -th power operation on derivations is the defining structure. The realisation that this makes the Weyl algebra a finite module over a large centre, and hence an Azumaya algebra, was worked out in the 1970s; Revoy's study of Weyl algebras over general rings is an early systematic treatment.

$p$ -curvature as an invariant of differential equations is due to Katz, in work of the early 1970s connecting Grothendieck's conjecture on algebraic solutions to the reduction of connections modulo $p$ . Cartier's descent theorem, that vanishing $p$ -curvature means descent along Frobenius, dates from the same circle of ideas.

The characteristic- $p$ picture returned to prominence in the 2000s through two developments: Bezrukavnikov, Mirković and Rumynin's localisation theory for enveloping algebras in characteristic $p$ , which is built on exactly the Azumaya structure described here, and the Tsuchimoto and Belov-Kanel-Kontsevich proofs that the Dixmier and Jacobian conjectures are equivalent. Coutinho's brief Ch. 2 §3 is a pointer towards a subject that has since grown very large.

Comparison

The following table is the practical summary. Read the first column as "the theorem you may be about to quote".

Chapter 2 in characteristic zero versus characteristic $p$ .
Property	$A_n(K)$ , $char,0$	$R_2$ (relations), $char p$	$R_1$ (operators), $char p$
Canonical monomials a basis	yes	yes, by construction	no
Degree well defined and additive	yes	yes	not defined
Domain	yes	yes	no, $∂_i^,p = 0$
Units	$K^ ×$	$k^ ×$	$k^ ×$
Centre	$K$	$k$ on $2n$ Frobenius powers	$k$ on $n$ Frobenius powers
Simple	yes	no	no
Noetherian	yes	yes	yes
Finite over its centre	no	yes, rank $p^2n$	yes
Finite-dimensional modules	none non-zero	yes, dimension $p^n$	yes, dimension $p^n$
Bernstein's inequality	holds	fails	fails

The pattern is that the ring-theoretic statements proved by degree bookkeeping survive into $R_2$ , and the statements that use characteristic zero through a factorial or a trace do not. Everything downstream of simplicity - dimension, multiplicity, holonomicity, the whole of the second half of this collection - is a characteristic-zero theory.

Key Takeaways

Key takeaways

In characteristic $p$ the operator definition and the generators-and-relations definition of the Weyl algebra give non-isomorphic rings, $R_1$ and $R_2$ , with $R_1 ≅ R_2/(∂_1^,p, … ,∂_n^,p)$ .
$R_1$ has $∂_i^,p = 0$ , hence non-zero nilpotents; it is not a domain and the degree is not well defined on it.
$R_2$ is a domain with additive degree - that part of Chapter 2 needs no hypothesis on the characteristic.
$R_2$ is not simple: the $2n$ Frobenius powers $x_i^,p, ∂_i^,p$ are central, and $R_2$ is free of rank $p^2n$ over the polynomial ring they generate.
Both rings have non-zero finite-dimensional modules, for example $k[x]/(x_1^p, … ,x_n^p)$ of dimension $p^n$ . The trace obstruction of characteristic zero evaporates because $p = 0$ in $k$ .
Consequently dimension theory collapses: modules of dimension $0$ exist and Bernstein's inequality is false.
The action of the central $∂_i^,p$ on a module is its $p$ -curvature, a genuinely new invariant, and the engine behind the Grothendieck-Katz conjecture (still open) and the Dixmier-Jacobian equivalence.

FAQs

Which of

R_1

and

R_2

deserves the name "the Weyl algebra in characteristic

p

By near-universal convention, $R_2$ : the algebra defined by the Weyl relations. It is the one with a basis of canonical monomials, the one that is a domain, and the one whose Azumaya structure is used in applications. $R_1$ is best regarded as its quotient of vanishing $p$ -curvature. If a source says "the Weyl algebra over $Z_p$ " without qualification, it almost certainly means $R_2$ .

Does the failure of simplicity mean characteristic

p

has no simple ring of differential operators?

No. The full ring of differential operators on $k[x]$ in Grothendieck's sense, generated by the divided-power operators $∂^[m]$ with $∂^[m](x^j) = jmx^j-m$ , is simple and acts faithfully. It is not finitely generated as an algebra, and it strictly contains $R_1$ . So the correct statement is that the algebra generated by $x_i$ and $∂_i$ fails to be simple, not that differential operators fail to be simple.

Why exactly does the degree argument for simplicity break?

The descent replaces $D$ by $[x_i, D]$ or $[∂_i, D]$ and needs the result to be non-zero. On coefficient polynomials these brackets are partial derivatives, and the coefficient that appears is an exponent $α_i$ or $β_i$ regarded as an element of the field. When $p ∣ α_i$ that coefficient is zero, the bracket vanishes without lowering to a constant, and the descent halts. The element $x_1^p$ is exactly a fixed point of the whole process.

R_2

Noetherian?

Yes. It is a finitely generated module over the Noetherian commutative ring $Z$ , so it is Noetherian on both sides by the standard argument. It is also Noetherian for the reason used in characteristic zero, namely that its associated graded ring for the degree filtration is a polynomial ring in $2n$ variables.

How large can a finite-dimensional module be?

Any dimension divisible by $p^n$ is achievable, as direct sums and extensions of the $p^n$ -dimensional simples. What cannot happen is a non-zero module of dimension not divisible by $p$ : the trace argument still gives $p ∣ _k M$ , and the Azumaya structure sharpens this to $p^n ∣ _k M$ over an algebraically closed field.

Does anything of holonomic theory survive?

Not by direct translation. The class of modules of minimal dimension becomes the finite-dimensional modules, which are not closed under the operations that make holonomicity useful, and the Bernstein-Sato polynomial has no characteristic- $p$ analogue in the naive sense - its roots are rational numbers whose denominators interact badly with $p$ . Arithmetic $D$ -module theory builds a substitute finiteness notion from Frobenius structures instead.

A_1(Z)

relevant to this discussion?

Yes, as the bridge. $A_1(Z)$ is a domain but not simple - each prime $p$ generates a proper two-sided ideal - and reducing modulo $p$ gives exactly $R_2$ over $Z_p$ . Arguments that reduce a characteristic-zero problem modulo many primes, such as the Dixmier-Jacobian equivalence, live in $A_n(Z)$ and use this reduction systematically.

References

S. C. Coutinho, A Primer of Algebraic D-modules, London Mathematical Society Student Texts 33, Cambridge University Press, 1995 - Ch. 2 §3 and Exercise 4.11.
S. P. Smith, Differential operators on commutative algebras, in Ring Theory (Antwerp 1985), Lecture Notes in Mathematics 1197, Springer, 1986 - the Weyl algebra and rings of differential operators in positive characteristic.
P. Revoy, Algèbres de Weyl en caractéristique p, Comptes Rendus de l'Académie des Sciences Paris 276 (1973), A225-A228 - the centre and the finite-module structure.
N. M. Katz, Nilpotent connections and the monodromy theorem: applications of a result of Turrittin, Publications Mathématiques de l'IHÉS 39 (1970), 175-232 - p-curvature and the Grothendieck conjecture.
R. Bezrukavnikov, I. Mirković and D. Rumynin, Localization of modules for a semisimple Lie algebra in prime characteristic, Annals of Mathematics 167 (2008), 945-991 - the Azumaya property in action.
Y. Tsuchimoto, Endomorphisms of Weyl algebra and p-curvatures, Osaka Journal of Mathematics 42 (2005), 435-452 - the Dixmier and Jacobian conjectures via reduction mod p.
A. Belov-Kanel and M. Kontsevich, The Jacobian conjecture is stably equivalent to the Dixmier conjecture, Moscow Mathematical Journal 7 (2007), 209-218.
P. Berthelot, D-modules arithmétiques I: opérateurs différentiels de niveau fini, Annales Scientifiques de l'École Normale Supérieure 29 (1996), 185-272 - divided-power differential operators.
J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, revised edition, American Mathematical Society, 2001 - Ch. 13 for rings finite over their centres and PI degree.

AI Suggested Questions

Prove that a polynomial over a field of characteristic p killed by every partial derivative is a polynomial in the p-th powers of the variables.
Write down the p-curvature of the module $k[x]/(x^p)$ explicitly and check it is zero.
Construct a simple module over the Weyl algebra in characteristic p with non-vanishing p-curvature.
Explain the divided-power ring of differential operators on $k[x]$ in characteristic p and prove it is simple.
Sketch how reduction modulo p turns the Dixmier conjecture into the Jacobian conjecture.
What is the precise statement of the Grothendieck-Katz p-curvature conjecture, and in which cases is it known?
Why is the Weyl algebra in characteristic p an Azumaya algebra over its centre, and what is the associated Brauer class?