An algebraic approach to the ellipticity
of linear differential operators

arXiv:1803.08452v1 [] 15 Mar 2018

Sławomir Kapka∗

Abstract
We demonstrate a method of associating the principal symbol at a K-point with a
linear differential operator acting between modules over a commutative algebra, and we
use it to define the ellipticity of a linear differential operator in a purely algebraic way.
We prove that the ellipticity is preserved by a surjective homomorphism of algebras. As
an example, we show that for every real affine variety there is an elliptic linear differential
operator acting on the algebra of regular functions on this variety.

Keywords: elliptic operators, K-points, principal symbol, universal derivations
Mathematics Subject Classification (2010): 13N05, 13N15, 58J05

0

Introduction

Motivation and main results
Differential calculus over commutative algebras is a perfect example of the abstarction in
mathematics. It is a purely algebraic theory which evolved from the theory of linear differential
operators (LDOs). [Gro, sec. 16.8], [HS, sec. 2], [Vin] are the first papers, independent of each
other, which studied LDOs from commutative algebra point of view, and which came up with
the very same definition of an abstract LDO. This in itself already shows the naturality of such
approach. After all, A. M. Vinogradov, the author of [Vin], and his co-workers pushed this
idea further and eventually developed what we now call differential calculus over commutative
algebras.
The main object of study of this theory are abstract LDOs acting between arbitrary modules
over a commutative algebra. As we can see from this preview [Kra] of differential calculus over
commutative algebras, this theory mostly focuses on functorial properties of the whole spaces of
LDOs, rather then on individual LDOs. And from the perspective of differential geometry, this
algebraic theory treats LDOs as global objects neglecting the pointwise behaviour. However,
from the theory of LDOs on smooth manifolds, we know that there is a way to study the
character of an LDO at a single point. In fact, it is the principal symbol that allows us to
determine the pointwise behaviour of an LDO acting between global sections of smooth vector
bundles. It is natural to ask if this rather analytic tool generalises to a more abstract algebraic
framework. In this paper we give a positive answer to this question by demonstrating a method
for studying the "pointwise" behaviour of LDOs acting between modules over a commutative
algebra. In a nutshell, we follow ideas from [Nes, ch. 9], and with a given K-algebra A we
consider its K-spectrum, which contains K-algebra homomorphisms from A to K. Elements
Faculty of Mathematics and Computer Science, University of Lodz, ul. Banacha 22, 90-238 Łódź, Poland.
e-mail: slawomirkapka@gmail.com, ORCID: orcid.org/0000-0003-3480-7294
∗

1

of a K-spectrum, called K-points, are good algebraic analogues of points, and thus it allows us
to study abstract LDOs at K-points.
The main aim of this paper is to show that it is possible to study pointwise behaviour
of linear differential operators already at the level of differential calculus over commutative
algebras (Section 4). More precisely, we show that the notion of principal symbol at a point
and the ellipticity of an LDO can be formulated in a purely algebraic way (Theorem 4.3 and
the definition that follows). As an example, we use this to demonstrate that elliptic linear
differential operators can be considered beyond the realm of smooth manifolds. First we prove
that the ellipticity is preserved by a surjective homomorphism of algebras (Theorem 5.5), and
then we show that for every real affine variety there is an elliptic linear differential operator
acting on the algebra of regular functions on this variety (Corollary 5.7).
Sections preview
We devote Section 1 to develop a preliminary algebraic formalism necessary for the main
results from Sections 3, 4 and 5. We introduce the notion of φ-derivation, being here just a
slightly "twisted" derivation, and we study its basic functorial properties. We focus on universal
φ-derivations, which we call φ-differentials, and which are analogous to Kähler differentials from
commutative algebra.
In Section 2 we present constructions of three notably different models of the universal φderivation, and in Section 3 we demonstrate that these models appear naturally in differential
calculus over commutative algebras and in both differential and algebraic geometry. More
specifically, we show that for any K-algebra A and any K-point h, the map dh : A → Th∗ A is a
h-differential (Proposition 3.1). As an example, we demonstrate that the pair consisting of the
Zarisiki cotangent space at a point and the differential at a point, defined on an affine variety,
forms a universal φ-derivation for φ being an evaluation at a point (Proposition 3.2). Similarly,
we show that the pair consisting of the cotangent space at a point and the ordinary differential
at a point, defined on a smooth manifold, is a universal φ-derivation for φ being an evaluation
at a point (Proposition 3.3).
We begin Section 4 with a brief comparison of the algebraic approach to linear differential
operators with the classical analytic one. We focus on the fact that the former approach
generalises the latter. Next, we have a look at the principal symbol of LDOs acting between
sections of smooth vector bundles and we investigate possible ways to generalise it to LDOs
acting between arbitrary modules. Eventually, we develop auxiliary Lemmas 4.1, 4.2 and
present our main Theorem 4.3, which states that there exists an algebraic analogue of the
principal symbol at a point. As a corollary, we demonstrate how to define the ellipticity of a
linear differential operator in a purely algebraic way.
In Section 5 we study the behaviour of the principal symbol at a point of LDOs acting
on algebras of regular functions on affine varieties. Given an affine variety, we have a natural
surjective homomorphism from the algebra of polynomials to the algebra of regular functions on
this variety. Thus we study the principal symbol of LDOs induced by algebra homomorphisms
(Lemmas 5.3 and 5.4), and we prove that ellipticity is preserved by surjective homomorphisms
of algebras (Theroem 5.5). Finally, we demonstrate that for every real affine variety there is
an elliptic linear differential operator acting on the algebra of regular functions on this variety
(Corollary 5.7).
We add Appendix A to clarify the relation between φ-differentials and Kähler differentials.
We show how to modify Kähler differentials to obtain φ-differentials (Proposition A.1).

2

1

Universal φ-derivations

The notion of derivation has already been established firmly in commutative algebra and
algebraic geometry (see [Eis, ch. 16] for example). A derivation is just any linear map from a
commutative algebra to a module which satisfies Leibniz’s rule. Furthermore, in [Gre, p. 148]
we have the notion of φ-derivations which satisfy a slightly "twisted" Leibniz’s rule and which
take values in algebras. We begin by merging these two concepts to obtain φ-derivations which
take values in modules.
Assume that K is a fixed commutative and unitial ring. We constrain ourselves to commutative, unitial, associative K-algebras and unitial K-algebra homomorphisms. Let A, B be
two K-algebras and let φ : A → B be a K-algebra homomorphism. Henceforth, this algebraic
framework will be silently assumed unless we specify otherwise.
Definition. Let M be a B-module. A K-linear map D : A → M will be called a φ-derivation
with values in M (or just φ-derivation) if for every f, g ∈ A
D(f g) = φ(f )D(g) + φ(g)D(f ).

(1.1)

The set of all φ-derivations with values in M has a natural B-module structure, and we will
denote it by DerK (φ, M).
φ-derivations behave well when composed with B-linear maps. More specifically, any Blinear map composed with a φ-derivation is again a φ-derivation.
Proposition 1.1. Let M, M ′ be two B-modules and let G : M → M ′ be a B-module homomorphism. If D : A → M is a φ-derivation, then G ◦ D : A → M ′ is also a φ-derivation.
Proof. Simply, for any f, g ∈ A
G(D(f g)) = G(φ(f )D(g) + φ(g)D(f )) = φ(f )G(D(g)) + φ(g)G(D(f )).

(1.2)

As a result, with every B-module homomorphism G : M → M ′ we can associate the B-linear
map DerK (φ, G) : DerK (φ, M) → DerK (φ, M ′ ) defined for any D ∈ DerK (φ, M) by
DerK (φ, G)(D) = G ◦ D.

(1.3)

This assignment gives rise to the covariant endofunctor DerK (φ, ·) in the category of B-modules.
Proposition 1.2. If M, M ′ , M ′′ are B-modules and G : M → M ′ , G′ : M ′ → M ′′ are B-module
homomorphisms, then
DerK (φ, G′) ◦ DerK (φ, G) = DerK (φ, G′ ◦ G)

(1.4)

and DerK (φ, IdM ) = IdDerK (φ,M ) .
Proof. For any D ∈ Der(φ, M) we have that
DerK (φ, G′ ◦ G)(D) = (G′ ◦ G) ◦ D = G′ ◦ (G ◦ D)
= DerK (φ, G′)(G ◦ D)
= DerK (φ, G′)(DerK (φ, G)(D))
= (DerK (φ, G′) ◦ DerK (φ, G))(D).
The second part follows directly from the definition of DerK (φ, IdM ).
3

(1.5)

We are interested in finding a representation for the functor DerK (φ, ·). Such representations
will be called universal φ-derivations, and they will be the main object of our study in this
section.
Definition. A pair (Ωφ , dφ ) consisting of a B-module Ωφ and a φ-derivation dφ : A → Ωφ will
be called a universal φ-derivation if for any B-module M the assignment
α

M
HomB (Ωφ , M) ∋ F 7−→
F ◦ dφ ∈ DerK (φ, M)

(1.6)

is an isomorphism of B-modules. Such dφ will be called a φ-differential and such Ωφ will be
called a module of φ-differentials.
In order to explain the word universal in the above definition, we just need to read what the
bijectivity of αM means. By doing this, we get the so called universal factorisation property,
and as a result we obtain an equivalent description of universal φ-derivations.
Proposition 1.3. (Ωφ , dφ ) is a universal φ-derivation if and only if for any B-module M and
any D ∈ DerK (φ, M) there is a unique B-linear map F : Ωφ → M such that F ◦ dφ = D.
In the special case of B = A and φ = IdA , we have that (ΩIdA , dIdA ) is precisely the Kähler
differential of A, which is usually denoted by (ΩA/K , dA/K ). The relation between Kähler
differentials and φ-differentials will be investigated more deeply in Appendix A.
There is also another reason why we named dφ a φ-differential. In Section 3 we will show
that some differentials appearing in algebra and geometry are in fact φ-differentials for some φ.
We left constructions of universal φ-derivations for the next section because we want to
show which results can be obtained by using only the universal factorisation property. Besides,
introducing models of (Ωφ , dφ ) at this point would only lead to unnecessary confusion.
By using a standard argument applied to objects defined by a universal factorisation property, we obtain that there is a unique universal φ-derivation up to a B-module isomorphism.
Proposition 1.4. If (Ωφ , dφ ) and (Ω′φ , d′φ ) are two universal φ-derivations, then there is a
unique B-module isomorphism F : Ωφ → Ω′φ such that F ◦ dφ = d′φ .
Proof. From the universality of (Ωφ , dφ ) there is a B-module homomorphism F : Ωφ → Ω′φ such
that F ◦dφ = d′φ . Similarly, from the universality of (Ω′φ , d′φ ) there is a B-module homomorphism
F ′ : Ω′φ → Ωφ such that F ′ ◦ d′φ = dφ . Hence
(F ′ ◦ F ) ◦ dφ = F ′ ◦ (F ◦ dφ ) = F ′ ◦ d′φ = dφ ,

(1.7)

and obviously IdΩφ ◦dφ = dφ . Thus, from the uniqueness of the factor of dφ , we get that
F ′ ◦ F = IdΩφ . Likewise, using an analogous argument, we get that F ◦ F ′ = IdΩ′φ , and so F is
an isomorphism.
Due to Proposition 1.4, we will call such (Ωφ , dφ ) the universal derivation, and we mean
by that a fixed but arbitrary universal φ-derivation. Similarly, we will use the notions of the
φ-differential and the module of φ-differentials for dφ and Ωφ respectively. Moreover, for the
sake of convenience, we will sometimes identify dφ with (Ωφ , dφ ).
So far A, B and φ : A → B were fixed. We leave B untouched and allow A and φ to vary
in a controlled manner. Let us consider the comma category (AlgK ↓ B), where AlgK stands
for the category of K-algebras. From the definition of (AlgK ↓ B), objects in this category
are K-algebra homomorphisms φ : A → B with the fixed codomain B, and morphisms from
φ : A → B to φ′ : A′ → B are K-algebra homomorphisms H : A → A′ such that φ′ ◦ H = φ.
Objects of (AlgK ↓ B) can be visualised as roughly vertical arrows targeting B, and morphisms as horizontal arrows closing diagrams made of such objects.
4

A❅

❅❅
❅❅
❅
φ ❅❅

H

B

/ A′
⑥
⑥⑥
⑥⑥ ′
⑥
~⑥ φ
⑥

(1.8)

By assigning modules of φ-differentials Ωφ to K-algebra homomorphisms φ : A → B, we
obtain the following mapping between objects in categories:
Ω

(AlgK ↓ B) ∋ (φ : A → B) 7−→ Ωφ ∈ ModB ,

(1.9)

where ModB stands for the category of B-modules. We will show that this assignment Ω is a
functor. First, however, we have to make sure that there is a reasonable mapping from arrows
of (AlgK ↓ B) to arrows of ModB .
Proposition 1.5. Let φ : A → B, φ′ : A′ → B and H : A → A′ be K-algebra homomorphisms
such that φ′ ◦ H = φ. Then there is a unique B-linear map ΩH : Ωφ → Ωφ′ such that ΩH ◦ dφ =
dφ′ ◦ H.
Proof. Consider the K-linear map D : A → Ωφ′ defined by the formula
D(f ) = dφ′ (H(f )).

(1.10)

D is a φ-derivation. In fact, for any f, g ∈ A
D(f g) = dφ′ (H(f g)) = dφ′ (H(f )H(g))
= φ′ (H(f ))dφ′ (H(g)) + φ′ (H(g))dφ′ (H(f ))
= φ(f )D(g) + φ(g)D(f ).

(1.11)

Thus there is a unique B-linear map ΩH : Ωφ → Ωφ′ such that ΩH ◦ dφ = D, and since
D = dφ′ ◦ H, we get that ΩH ◦ dφ = dφ′ ◦ H.
Henceforth, ΩH will denote the map introduced in Proposition 1.5. We can visualise ΩH as
the unique B-linear map which makes the diagram
ΩH

Ωφ
O

Ωφ′
O

(1.12)

dφ′

dφ

A❇

/

H

❇❇
❇❇
❇
φ ❇❇

B

/ A′
⑤
⑤
⑤⑤
⑤⑤ φ′
⑤
}⑤

commutative. As a consequence, we obtain a well defined assignment between arrows of categories
Ω
(AlgK ↓ B) ∋ H 7−→ ΩH ∈ ModB .
(1.13)
The next proposition asserts us that this mapping is in fact a covariant functor.
Proposition 1.6. If H : A → A′ and H ′ : A′ → A′′ are morphisms of (AlgK ↓ B) from
φ : A → B to φ : A′ → B and from φ′ : A′ → B to φ′′ : A′′ → B respectively, then
ΩH ′ ◦ ΩH = ΩH ′ ◦H and ΩIdA = IdΩφ .

5

Proof.
ΩH ′ ◦H ◦ dφ = dφ′′ ◦ (H ′ ◦ H) = (dφ′′ ◦ H ′ ) ◦ H = (ΩH ′ ◦ dφ′ ) ◦ H
= ΩH ′ ◦ (dφ′ ◦ H) = ΩH ′ ◦ (ΩH ◦ dφ ) = (ΩH ′ ◦ ΩH ) ◦ dφ .

(1.14)

From Proposition 1.5 we have that there are unique maps closing diagrams of the type (1.12).
Thus
(1.15)
ΩH ′ ◦H ◦ dφ = (ΩH ′ ◦ ΩH ) ◦ dφ
implies that ΩH ′ ◦H = ΩH ′ ◦ ΩH . The second part of the proposition holds simply because
IdΩφ ◦dφ = dφ .
Now we consider a technical lemma which will help us in Section 2 with verifying whether
some φ-derivation is a universal one. The idea is to transform the uniqueness from the factorisation property into some sort of surjectivity of a φ-derivation.
Proposition 1.7. Let Ω be a B-module and let d ∈ DerK (φ, Ω). Assume that for any B-module
M and any D ∈ DerK (φ, M) there exists a B-linear map F : Ω → M such that F ◦ d = D.
Then the pair (Ω, d) is a universal φ-derivation if and only if SpanB (Im d) = Ω.
Proof. First assume that (Ω, d) is a universal φ-derivation. Let Ω0 denote the B-module
SpanB (Im d) and let ι : Ω0 ֒→ Ω be the natural inclusion. We put d0 as the map d with
the codomain truncated to Ω0 . Thus we have that ι ◦ d0 = d. d0 is clearly a φ-derivation, hence
there is a B-linear map F : Ω → Ω0 such that F ◦ d = d0 . As a result
(ι ◦ F ) ◦ d = ι ◦ (F ◦ d) = ι ◦ d0 = d,

(1.16)

and clearly IdΩ ◦d = d. From the uniqueness of the factor of d we obtain that ι ◦ F = IdΩ .
Hence ι is surjective, and so Ω = Ω0 .
Now assume that Ω = Ω0 . Let M be a B-module and let D ∈ DerB (φ, M). So there exist
F1 , FP
2 : Ω → M such that F1 ◦ d = D and F2 ◦ d = D. We need to show that F1 = F2 . If
ω=
bi d(fi ) ∈ Ω, then
X
 X
X
F1 (ω) = F1
bi d(fi ) =
bi F1 (d(fi )) =
bi F2 (d(fi ))
X

(1.17)
= F2
bi d(fi ) = F2 (ω).
The assumption Ω = Ω0 implies that any ω ∈ Ω can be written as

P

bi d(fi ), and so F1 = F2 .

As a direct consequence, we obtain that the module of φ-differentials is generated as a
B-module by the image of the φ-differential. This also implies that the functor Ω preserves
surjectivity.
Proposition 1.8. Assume that H : A → A′ is an arrow of (AlgK ↓ B) from φ : A → B to
φ : A′ → B. If H is surjective, then ΩH : Ωφ → Ωφ′ is surjective as well.
P
bi dφ′ (fi′ ) for some elements
Proof. If α′ ∈ Ωφ′ , then, from Proposition 1.7, we get that α′ =
bi ∈ B and fi′ ∈ A′ . The surjectivity of H implies that fi′ = H(fi ) for some fi ∈ A, and thus
dφ′ fi′ = ΩH (dφ (fi )). As a result,
X

X
X
bi ΩH (dφ (fi )) = ΩH
α′ =
bi dφ′ (fi′ ) =
bi dφ (fi ) .
(1.18)

6

We will make use of Proposition 1.8 in Section 5 when we investigate linear differential
operators acting on the ring of regular functions of an affine variety. In a nutshell, this will be
useful because for any affine variety there is a natural surjection from the algebra of polynomials
to the algebra of regular functions.
In the realm of smooth manifolds, the differential at a point takes smooth functions and
outputs cotangent vectors at this point. By definition, cotangent vectors are the dual objects
to tangent vectors, which themselves may be seen as φ-derivations for φ being an evaluation
at a point. Due to this behaviour of the ordinary differential, we will now consider this kind
of biduality in a more abstract framework. If M is a B-module, then the dual B-module will
∨
be denoted by M ∨ . This means that M ∨ = HomB (M, B). Let dM
φ : A → DerK (φ, M) be the
map defined by the formula
dM
(1.19)
φ (f )(D) = D(f ),
where f ∈ A and D ∈ DerK (φ, M).
Proposition 1.9. For any B-module M the map dM
φ is a φ-derivation.
Proof. Simply, for any f, g ∈ A and D ∈ DerK (φ, M) we have that
dM
φ (f g)(D) = D(f g) = φ(f )D(g) + φ(g)D(f )
M
= (φ(f )dM
φ (g) + φ(g)dφ (f ))(D).

(1.20)

The K-linearity of dM
φ is clear because everything here is K-linear.
Consider now the special case where M = B. In this situation we have a necessary and sufficient
∨
condition for dB
φ : A → DerK (A, B) to be a φ-differential.
Proposition 1.10. (DerK (φ, B)∨ , dB
φ ) is a universal φ-derivation if and only if the module of
φ-differentials Ωφ is reflexive as a B-module.
Proof. Let j : Ωφ → Ω∨∨
φ be the natural map to the bidual. The reflexivity of Ωφ means that
j is an isomorphism of B-modules. The universality of Ωφ implies that B-modules Ω∨φ and
DerK (φ, B) are isomorphic via the map αB defined as in (1.6). Let β denote the inverse map
of αB . This isomorphism β : DerK (φ, B) → Ω∨φ , from the definition of αB , is such that for any
f ∈ A and any D ∈ DerK (φ, B)
β(D)(dφ (f )) = D(f ).
(1.21)
∨
∨∨
∨
We have that (β ∨ ◦ j) ◦ dφ = dB
φ , where β : Ωφ → DerK (φ, B) is the dual map to β. In fact,
for any f ∈ A and D ∈ DerK (φ, B)

(β ∨ (j(dφ (f )))(D) = (j(dφ (f ))(β(D)) = β(D)(dφ (f )) = D(f ) = (dB
φ (f ))(D).

(1.22)

∨
Thus (DerK (φ, B)∨ , dB
φ ) is a universal φ-derivation if and only if β ◦ j is an isomorphism.
But since β is an isomorphism, we get that β ∨ ◦ j is an isomorphism if and only if j is an
isomorphism.

The case which will concern us the most is the one in which B is a field. If a vector space
is finite dimensional, then it is reflexive. Thus, as a corollary, we get a degenerated version of
Proposition 1.10.
Corollary 1.11. If B is a field and the module of φ-differentials Ωφ is a finite dimensional
vector space over the field B, then dB
φ is a φ-differential.

7

2

Three models of the universal φ-derivation

We have from Proposition 1.4 that all universal φ-derivations are isomorphic and from
perspective of B-modules they are the same object. However, if we wish to find a K-linear map
from Ωφ to some K-module, then we cannot rely only on the universal property of (Ωφ , dφ ). In
such case, it may happen that we have to refer to models of the universal φ-derivation. Such
issue appears with Kähler differentials in the construction of the algebraic de-Rham complex.
Compare for instance constructions of the algebraic de-Rham complex provided by [Gro, thm
16.6.2] and by [Bra, prop. 4.5.3]. The latter is much more straightforward due to the use of a
different model of the Kähler differential.
In this section, we will give detailed constructions of three notably different models of the
universal φ-derivation. For the sake of unambiguity and to omit confusions we will give a
distinct name to each model.
First model: Classical universal φ-derivation
Let us consider B ⊗K A with the following B-module structure:
a · b ⊗ f = ab ⊗ f,

(2.1)

and with the following B-algebra structure:
a ⊗ f · b ⊗ g = ab ⊗ f g.

(2.2)

Let pφ : B ⊗K A → B be a B-algebra homomorphism given by the formula
pφ (a ⊗ f ) = aφ(f ),

(2.3)

and let δφ : A → B ⊗K A be a K-linear map defined as
δφ (f ) = 1B ⊗ f − φ(f ) ⊗ 1A .

(2.4)

Let Iφ = ker pφ and let Jφ be the B-module generated by Im δφ . I.e., Jφ = SpanB (Im δφ ). We
claim that Iφ = Jφ . In fact, theP
inclusion Iφ ⊃ Jφ follows from the fact that pφ ◦ δφ = 0.
Conversely, consider an arbitrary
ai ⊗ fi ∈ Iφ . Then
X
X
X
ai ⊗ fi =
(ai ⊗ 1A )(1B ⊗ fi − φ(fi ) ⊗ 1A ) =
ai δφ (fi ).
(2.5)
This shows that Iφ ⊂ Jφ . Iφ ⊂ B ⊗K A is an ideal because it is the kernel of a B-algebra
2
homomorphism. Hence Iφ2 is a well defined B-module. We define the B-module ΩC
φ = Iφ /Iφ
C
C
C
C
and the K-linear map dφ : A → Ωφ as π ◦ δφ , where π : Iφ → Ωφ is the natural projection.
We are aiming to prove that dC
φ is a φ-differential. First, however, we need two lemmas.
The first one states that φ-derivations vanish on the identity.
Lemma 2.1. For any B-module M and any D ∈ DerK (φ, M) we have that D(1A ) = 0.
Proof. Simply,
D(1A ) = D(1A · 1A ) = φ(1A )D(1A ) + φ(1A )D(1A ) = 2φ(1A )D(1A ).

(2.6)

Since φ is a K-algebra homomorphism, we get that φ(1A ) = 1B , and so D(1A ) = 0.
The second lemma tells us that δφ behaves almost like a φ-derivation.
Lemma 2.2. For any f, g ∈ A
δφ (f g) − φ(f )δφ (g) − φ(g)δφ (f ) = δφ (f )δφ (g).
8

(2.7)

Proof. First, from the definition of δφ , we have that
δφ (f )δφ (g) = (1 ⊗ f − φ(f ) ⊗ 1)(1 ⊗ g − φ(g) ⊗ 1)
= 1 ⊗ f g − φ(f ) ⊗ g − φ(g) ⊗ f + φ(f g) ⊗ 1.

(2.8)

By adding and subtracting φ(f g) ⊗ 1 we obtain that the above equals
(1 ⊗ f g − φ(f g) ⊗ 1) − (φ(f ) ⊗ g − φ(f g) ⊗ 1) − (φ(g) ⊗ f − φ(f g) ⊗ 1),

(2.9)

δφ (f g) − φ(f )δφ (g) − φ(g)δφ (f ).

(2.10)

and this is

C
Theorem 2.3. (ΩC
φ , dφ ) is a universal φ-derivation.
C
C
Proof. We have from Lemma 2.2 that dC
φ is a φ-derivation. The module Ωφ equals SpanB (Im dφ )
C
C
because Iφ = Jφ = SpanB (Im δφ ) and dφ = π ◦ δφ . Thus, from Proposition 1.7, we just need to
find a factor of an arbitrary φ-derivation. So let M be any B-module and let D ∈ DerK (φ, M).
Consider the B-linear map γ : B ⊗K A → M given by the formula

γ(a ⊗ f ) = aD(f )

(2.11)

and truncate it to γ̄ : Iφ → M. The map γ̄ vanishes on Iφ2 . In fact, using Lemma 2.1, we first
notice that for any f ∈ A
γ̄(δφ (f )) = D(f ) − φ(f )D(1A) = D(f ).

(2.12)

Hence, using (2.7) from Lemma 2.2, we get that for any f, g ∈ A
γ̄(δφ (f )δφ (g)) = γ̄(δφ (f g) − φ(f )δφ (g) − φ(g)δφ (f ))
= D(f g) − φ(f )D(g) − φ(g)D(f ) = 0.

(2.13)

C
Thus there is a B-linear map F : ΩC
φ → M such that F ◦ dφ = D.
C
Definition. We will call (ΩC
φ , dφ ) the classical universal φ-derivation.

We call it classical because it is just a modification of the standard construction of the
Kähler differential appearing in classic sources like [Bou, p. 568-569] or [HS, p. 210-215]. This
model will be most interesting for us because, as we will see in Section 3, modules ΩC
φ for
particular homomorphisms φ appear naturally in algebra and geometry. Moreover, in Sections
4 and 5 we will extensively use the map δφ introduced in this model.
Second model: Rough universal φ-derivation
Let B (A) be the free B-module generated by the set A. Consider its B-submodule S generated by the set
{hki : k ∈ K} ∪ {hf gi − φ(f ) hgi − φ(g) hf i : f, g ∈ A} ,

(2.14)

where hf i : A → B stands for the base element of B (A) corresponding to f . This means that
(
1B , if g = f
.
(2.15)
hf i (g) =
0,
if g =
6 f
(A)
R
We define the B-module ΩR
/S and the K-linear map dR
φ as the quotient B
φ : A → Ωφ by the
formula
R
dR
(2.16)
φ (f ) = π (hf i),

where π R : B (A) → ΩR
φ is the natural projection.
9

R
Theorem 2.4. (ΩR
φ , dφ ) is a universal φ-derivation.

Proof. Let M be a B-module and let D ∈ DerK (φ, M). Consider the B-linear map γ : B (A) →
M defined on base elements by the formula
γ(hf i) = D(f ).

(2.17)

Since D is a φ-derivation, it follows that γ vanishes on S. Hence there is a unique B-linear
R
map F : ΩR
φ → M such that D = F ◦ dφ .
R
Definition. We will call (ΩR
φ , dφ ) the rough universal φ-derivation.

The above construction is very similar to the construction of the tensor product. We consider
a huge free module and divide it by a submodule generated by appropriately tailored elements.
This rough approach explains our choice of the name. This model is also a modification of some
model of the Kähler differential (see [Eis, p. 384]).
Third model: Algebraic universal φ-derivation
Let us consider B ⊗K A ⊗K A and B ⊗K A with the natural B-module structures endowed
by the left most factors. Let hφ : B ⊗K A ⊗K A → B ⊗K A be a B-linear map defined by the
formula
hφ (a ⊗ f ⊗ g) = a ⊗ f g − aφ(f ) ⊗ g − aφ(g) ⊗ f.
(2.18)
A
We define the B-module ΩA
φ as the cokernel of hφ . This means that Ωφ = (B ⊗K A)/ Im(hφ ),
A
and we let dA
φ : A → Ωφ be defined by the formula
A
dA
φ (f ) = π (1 ⊗ f ),

(2.19)

where π A : B ⊗K A → ΩA
φ is the natural projection.
A
Theorem 2.5. (ΩA
φ , dφ ) is a universal φ-derivation.

Proof. Let M be a B-module and let D ∈ DerK (φ, M). Consider the B-linear map γ : B⊗K A →
M defined by the formula
γ(a ⊗ f ) = aD(f ).
(2.20)
Then for any a ∈ B and f, g ∈ A we have that
γ(hφ (a ⊗ f ⊗ g)) = aD(f g) − aφ(f )D(g) − aφ(g)D(f ),

(2.21)

and since D is a φ-derivation, we get that γ ◦ hφ = 0. Thus there is a unique B-linear map
A
F : ΩA
φ → M such that D = F ◦ dφ .
A
Definition. We will call (ΩA
φ , dφ ) the algebraic universal φ-derivation.

The above construction is just a modification of the model of the Kähler differential discovered in [Bra, p. 74-79].

3

Examples of φ-differentials

The aim of this section is to show that some differentials appearing in algebra and geometry
are in fact φ-differentials.
First of all, as we have seen earlier, the notion of φ-differential is a straightforward generalisation of the notion of Kähler differential. More precisely, for a given K-algebra A the Kähler
10

differential of A is the same as the IdA -differential. However, it is merely a consequence of
definitions we introduced, rather then an actual example.
The case which is the most interesting for us is the one in which, using notations from
previous sections, the K-algebra B is just K itself. This is due to the philosophy from [Nes,
ch. 9], which suggests to look at differential calculus as a part of commutative algebra. So here
is a brief overview of the main concepts which will concern us the most. For more details we
refer to [Nes, ch. 9].
Given a K-algebra A, we denote the set of all K-algebra homomorphisms h : A → K by
|A|. |A| is called the K-spectrum of A and its elements are called K-points. For any h ∈ |A|
the cotangent space at h is defined as Ih /Ih2 and it is denoted by Th∗ A, where Ih = ker h.
Additionally, we allow ourselves to define the differential at h as the map dh : A → Th∗ A given
by the formula dh (f ) = [f − h(f )].
Since any K-point h is a K-algebra homomorphism, it is reasonable to talk about hderivations. Hence let us focus for a moment on how the classical universal h-derivation looks
∗
like. Since h takes values in K, we have that ΩC
h is precisely the cotangent space Th A, and
similarly the h-differential dC
h is just dh . As a result, we obtain that dh is a h-differential.
Proposition 3.1. For any K-algebra A and any h ∈ |A| the pair (Th∗ A, dh ) is a universal
h-derivation.
The above algebraic framework is general enough to work with various cases in geometry.
We will show that differentials appearing in both algebraic and differential geometry are φdifferentials for φ being evaluations at a point.
Let F be a field and assume that V is an affine variety given by an ideal I ⊂ F[X1 , . . . , Xn ].
Let F[V ] = F[X1 , . . . , Xn ]/I be the associated ring of regular functions on V . For a given point
z ∈ V let evz : F[V ] → F denote the evaluation function at z. By so, we mean that evz (f ) =
f (z), where f ∈ F[X1 , . . . , Xn ] is any representative of f ∈ F[V ]. The Zariski cotangent space
at z is defined as mz /m2z , where mz is the maximal ideal of regular functions vanishing at z and
it is denoted by Tz∗ V . Equivalently, Tz∗ V = Iz /Iz2 , where Iz = ker(evz ). Finally, the differential
at z is the map dz : F[V ] → Tz∗ V given by the formula dz (f ) = [f − evz (f )].
Since evz is clearly an F-algebra homomorphism, we get that this geometric situation is a
special case of the previous algebraic one with K = F, A = F[V ] and h = evz . Hence the
differential dz is an evz -differential.
Proposition 3.2. For any affine variety V over a field F and any point z ∈ V the pair (Tz∗ V, dz )
is a universal evz -derivation.
A similar result holds for ordinary differentials defined on smooth manifolds (by smooth we
mean C ∞ ). Let X be a smooth real Hausdorff manifold and let z ∈ X be some point. Assume
that evz : C ∞ (M) → R is the standard evaluation map at z. Tangent vectors at z are defined
as evz -derivations with values in R and the vector space of all tangent vectors at z is denoted
by Tz X. The cotangent space at z is the dual space to Tz X and it is denoted by Tz∗ X. The
differential at z is the map dz : C ∞ (M) → Tz∗ X given by the formula dz (f )(v) = v(f ).
Straight from the definition of Tz X we have that Tz X = DerR (evz , R). As a result, the
differential at z is precisely the map dRevz : C ∞ (X) → DerR (evz , R)∨ introduced in (1.19) with
A = C ∞ (X), M = B = K = R and φ = evz . So in order to prove that dz is an evz -differential,
from Corollary 1.11, it is enough to prove that the module of evz -differentials Ωevz is a finite
dimensional real vector space. This clearly holds. In fact, from the universal property of the
universal evz -derivation we have that the dual vector space to Ωevz is precisely the tangent
space Tz X, which has a finite dimension.
Proposition 3.3. For any smooth real Hausdorff manifold X and any point z ∈ X the pair
(Tz∗ X, dz ) is a universal evz -derivation.
11

Since universal φ-derivations are unique up to an isomorphism commuting with differentials,
we get from Section 2 that (Tz∗ X, dz ) has alternative descriptions. From the construction of the
classical universal evz -derivation, we have that Tz∗ X can be seen as Iz /Iz2 , where Iz = ker(evz )
with dz mapping a smooth function f to the the class [f − f (z)] in Iz /Iz2 . As a result, which
was noticed in [Nes, sec. 9.27], this geometric case can be seen as a special case of the previous
algebraic one with K = R, A = C ∞ (X) and h = evz .
From the construction of the algebraic universal evz -derivation, we get that Tz∗ X is isomorphic with C ∞ (M)/Sz , where Sz is a vector space generated by elements of the form
f g − f (z)g − g(z)f . Interestingly, it coincides with the space of differentials, introduced in
[Nar, sec. 2.1.9], which is defined as the space of smooth functions modulo stationary functions
at the point z.
In summary, even though the notion of h-differential is defined via the universal property, it
generalises differentials from distinct categories which have very concrete set theoretic models.
Additionally, being defined axiomatically and model independent has some practical applications. For example, in order to determine whether some K-linear map from a K-algebra A
factors through some h-differential, it is enough to check if it is a h-derivation. We will make
use of this observation in upcoming Section 4 to define the principal symbol at a K-point in a
purely algebraic way.

4

Ellipticity in differential calculus over commutative algebras

Let us go back to the algebraic framework from the first section. So let A, B be two Kalgebras over some fixed ring K. For any A-module P and any B-module M there is a natural
associative action of the K-algebra B ⊗K A on HomK (P, M) given by the formula
((a ⊗ f )F )(p) = aF (f p),

(4.1)

where a ∈ B, f ∈ A, F ∈ HomK (P, M) and p ∈ P . From the construction of the classical
universal φ-derivation we have that for a given K-algebra homomorphism φ : A → B there is
the K-linear map δφ : A → B ⊗K A defined by the formula
δφ (f ) = 1B ⊗ f − φ(f ) ⊗ 1A .

(4.2)

So, in particular, for F ∈ HomK (P, M), f ∈ A and p ∈ P we have that
(δφ (f )F )(p) = F (f p) − φ(f )F (p).

(4.3)

If A = B and φ = IdA , then we will denote δφ by δ.
Using δ, just like it is done in [Nes, sec. 9.67], we can define a linear differential operator
between two modules over a commutative algebra.
Definition. Let A be an arbitrary K-algebra and let P, Q be two A-modules. A K-linear map
L : P → Q is called a linear differential operator (LDO) of order 6 k if for any f0 , . . . , fk ∈ A
δ(f0 ) . . . δ(fk )L = 0.

(4.4)

Naturally, if L is of order 6 k but not of order 6 k − 1, then we will say that L is of order k.
This algebraic definition coincides with the usual definition of linear differential operators
acting between global sections of smooth vector bundles on a smooth manifold [Nes, sec. 9.66].
Similarly, in the case of the algebra of polynomials F[X1 , . . . , Xn ] over a field F of characteristics
0, we have from [Gro, sec. 16.11] that LDOs in the usual sense acting on F[X1 , . . . , Xn ] coincide
12

with the above algebraic ones with P = Q = F[X1 , . . . , Xn ]. Hence the above algebraic notion
of LDO is coherent with the standard notion of LDO, and so it does not lead to any ambiguity.
Let us go back to the smooth case. If X is a smooth manifold and L : Γ(E) → Γ(F ) is an
LDO of order k on X acting between smooth sections of some vector bundles E and F , then
we can use the principal symbol to study the character of L. The principal symbol at z ∈ X of
L may be defined intrinsically as the function σL,z : Tz∗ X ⊗R Ez → Fz given by the formula
σL,z (α ⊗ ξ) = Lz (f k s),

(4.5)

where f ∈ C ∞ (M) and s ∈ Γ(E) are such that dz f = α, f (z) = 0 and s(z) = ξ [Nar, def.
3.3.13]. If for any non-zero α ∈ Tz∗ M the R-linear map σL,z (α) : Ez → Fz is injective, then we
say that L is elliptic at z.
We wish to generalise the notions of principal symbol and ellipticity to the case of linear
differential operators acting between modules over an arbitrary commutative K-algebra A.
We know from [KLV, sec. 1.5.3] that with every LDO L : P → Q of order 6 k we can
associate an A-linear map σ(L) : (ΩA/K )⊙k ⊗A P → Q such that
σ(L)((dA/K (f1 ) ⊙ · · · ⊙ dA/K (fk )) ⊗ p) = (δ(f1 ) . . . δ(fk )L) (p),

(4.6)

where f1 , . . . , fk ∈ A, p ∈ P , and where (ΩA/K )⊙k denotes the k-th symmetric power of the
A-module ΩA/K . When A = C ∞ (M) and P, Q are modules of sections of some vector bundles,
then this map σ(L) corresponds to the global section of the principal symbol treated as a bundle
map. Since the functor of global section Γ behaves neatly on manifolds, in this geometric case
it is quite easy to recover σL,z for any z from σ(L) (just apply [Swa, thm 1]). However, in the
case of an arbitrary algebra A and A-modules P, Q a more subtle approach is required.
First we notice that the principal symbol at a point is just a map between fibres of vector
bundles. So in order to generalise this notion to our algebraic case, we need to have an algebraic
analogue of points for an algebra and fibres for a module.
We know from [Nes, sec. 7.3] that on a paracompact manifold X ordinary points corresponds
bijectively to R-points of the algebra C ∞ (X). More precisely, every R-algebra homomorphism
from C ∞ (X) to R must be of the form evz for some z ∈ X. Hence given a K-algebra A and
an LDO acting between A-modules we are going to define the principal symbol at h for any
K-point h ∈ |A|.
We know from [Swa, cor. 3] that for any smooth vector bundle E on a manifold X and
any z ∈ X the fibre Ex is isomorphic with Γ(E)/Iz Γ(E), where Iz = ker(evz ). As a result,
repeating the ideas from [Nes, cor. 11.9], we obtain a natural way to define an analogue of fibres
at K-points for an arbitrary A-module. More specifically, for any A-module P and any K-point
h ∈ |A| let Ph be the quotient K-module P/Ih P , where Ih is the kernel of h. We will denote
the natural projection P → Ph by evh , and for p ∈ P we will denote the value evh (p) by ph .
Similarly, for any K-linear map F : P → Q we will denote the composition evh ◦F : P → Qh by
Fh . In the special case of P = A we have that Ah = A/Ih . As a result, Ah can be canonically
identified with K via h. Thus, using this identification, the map evh : A → Ah is just h itself.
Lemma 4.1. Let P, Q be two A-modules, h ∈ |A| and let F : P → Q be a K-linear map. Then
for any natural number j and any f1 , . . . , fj ∈ A
evh ◦δ(f1 ) . . . δ(fj )F = δh (f1 ) . . . δh (fj )Fh .

(4.7)

Proof. We will show it inductively. First, for any f ∈ A we have that
evh ◦(δ(f )F ) = δh (f )Fh .

13

(4.8)

In fact, for any p ∈ P we have that
evh ((δ(f )F )(p)) = evh (F (f p) − f F (p)) = Fh (f p) − h(f )Fh (p)
= (δh (f )Fh )(p).

(4.9)

Now fix f0 , . . . , fj ∈ A and put Φ = δ(f1 ) . . . δ(fj )F . Then we have that
evh ◦(δ(f0 )δ(f1 ) . . . δ(fj )F ) = evh ◦(δ(f0 )Φ)
//(4.8) with f = f0 and F = Φ// = δh (f0 )Φh
// evh ◦Φ = Φh // = δh (f0 )(evh ◦Φ)
//inductive assumption (4.7)// = δh (f0 )δh (f1 ) . . . δh (fj )Fh .

(4.10)

In the realm of smooth manifolds, given an R-linear map between sections of vector bundles,
we can ask whether this map gives rise to a linear map on fibres. Such mappings are precisely
linear maps over the ring of smooth functions, which are usually called tensors. The following
lemma is an algebraic analogue of that fact, but reduced to a single fibre.
Lemma 4.2. Let P be an A-module, V be a K-module and let h ∈ |A|. If G : P → V is
K-linear and for every f ∈ A we have that δh (f )G = 0, then the map Gh : Ph → V given by
the formula
Gh (ph ) = G(p)
(4.11)
is well defined.
Proof. Let p = f0 p0 for some f0 ∈ Ih and p0 ∈ P. Then
0 = (δh (f0 )G)(p0 ) = G(f0 p0 ) − h(f0 )G(p0 ).

(4.12)

Since f0 ∈ Ih , we obtain that h(f0 ) = 0. As a result, G(f0 p0 ) = 0.
Now we are able to show the existence of the principal symbol at a K-point of an arbitrary
linear differential operator.
Theorem 4.3. Let P, Q be two A-modules and let L : P → Q be an LDO of order 6 k. Then
for any h ∈ |A| there exists the unique K-linear map
σL,h : (Ωh )⊙k ⊗K Ph → Qh

(4.13)

such that for any f ∈ Ih and any p ∈ P
σL,h ((dh (f ))⊙k ⊗ ph ) = Lh (f k p),

(4.14)

where (Ωh )⊙k stands for the k-th symmetric power of the K-module Ωh .
Proof. Fix f1 , . . . , fk ∈ A and consider the K-linear map Gf1 ,...,fk : P → Qh given by the
formula
Gf1 ,...,fk = δh (f1 ) . . . δh (fk )Lh .
(4.15)
From the assumption that L is an LDO of order 6 k and Lemma 4.1 we obtain that for any
f ∈ A the function δh (f )Gf1 ,...,fk = 0. Thus, from Lemma 4.2 applied to Gf1 ,...,fk , we get that
there exists a K-linear map Ghf1 ,...,fk : Ph → Qh such that for any p ∈ P
Ghf1 ,...,fk (ph ) = (δh (f1 ) . . . δh (fk )Lh ) (p).
14

(4.16)

Hence, by changing f1 , . . . , fk , we obtain the K-linear mapping
A⊗k ∋ f1 ⊗ · · · ⊗ fk 7→ Ghf1 ,...,fk ∈ HomK (Ph , Qh ).

(4.17)

We know from Lemma 2.2 that δh behaves almost like an h-derivation. Thus, using the assumption that L is an LDO of order 6 k, we obtain that the mapping (4.17) is an h-derivation
on every coordinate. In fact, for any f, g, f2 , . . . , fk ∈ A
Ghfg,f2 ,...,fk = δh (f g)δh (f2 ) . . . δh (fk )Lh
//(2.7)// = (h(f )δh (g) + h(g)δh (f ) + δh (f )δh (g))δh (f2 ) . . . δh (fk )Lh
//L is an LDO// =

h(f )Ghg,f2,...,fk

+

(4.18)

h(g)Ghf,f2,...,fk .

Since the K-algebra A is commutative, we get that the mapping (4.17) is symmetric. Hence,
from the universal property of (Ωh , dh ), there exists a K-linear map
σL,h : (Ωh )⊙k ⊗K Ph → Qh

(4.19)

such that for any f1 , . . . , fk ∈ A and p ∈ P
σL,h ((dh (f1 ) ⊙ · · · ⊙ dh (fk )) ⊗ ph ) = (δh (f1 ) . . . δh (fk )Lh ) (p).

(4.20)

Notice that we moved Ph into the argument using the tensor-hom adjunction. Finally, if
f1 , . . . , fk ∈ Ih , then clearly
δh (f1 ) . . . δh (fk ) = 1K ⊗ f1 · · · fk .

(4.21)

As a result, for any f ∈ Ih and p ∈ P
σL,h ((dh (f ))⊙k ⊗ ph ) = Lh (f k p).

(4.22)

The uniqueness of σL,h follows from Proposition 1.7, which implies that the K-module Ωh is
generated by the image of dh .
In the consequence of Theorem 4.3 we are able to define the principal symbol at any K-point
and the ellipticity of a linear differential operator in a purely algebraic way.
Definition. Let L : P → Q be an LDO of order k and let h ∈ |A|. We will call σL,h the
principal symbol of L at h, where σL,h is defined as in Theorem 4.3. We will say that L is of
order k at h if the map σL,h 6= 0. If L is of order k at h and for every non-zero α ∈ Ωh the
K-linear map σL,h,α : Ph → Qh given by the formula
σL,h,α (ph ) = σL,h (α⊙k ⊗ ph )

(4.23)

is injective, then we will say that L is elliptic at h. Finally, we will say that L is elliptic if it
is elliptic at h for every h ∈ |A|.

5

Elliptic LDOs on real affine varieties

In this section we will show that the algebraic approach to elliptic LDOs from the previous
section goes beyond the realm of smooth manifolds. We will show that for every real affine
variety there is an elliptic linear differential operator acting on the algebra of regular functions
on this variety.
First, we notice that the algebra of regular functions on an affine variety has all K-points
of a fixed form. More specifically, K-points of the algebra of regular functions correspond
bijectively to ordinary points of variety, and they are precisely the evaluations.
15

Proposition 5.1. Let F be a field and assume that V is an affine variety embedded in Fn .
Then every F-point of F[V ] is of the form evz for some z ∈ V .
Proof. Let V be given by an ideal I ⊂ F[X1 , . . . , Xn ] and assume that h : F[V ] → F is an
F-algebra homomorphism. Since the F-algebra F[V ] is generated by X 1 , . . . , X n , we get that
h(f ) = f (h(X 1 ), . . . , h(X n ))

(5.1)

for any f ∈ F[V ], where f is some representative of f . Thus h = evz , where z equals
(h(X 1 ), . . . , h(X n )). It is left to prove that z ∈ V . For any g ∈ I we have that
g(z) = h(g) = h(0) = 0.

(5.2)

So z ∈ V .
From Proposition 5.1 and already mentioned fact from [Nes, sec. 7.3] (see the second
paragraph preceding Lemma 4.1), which characterises R-points of C ∞ (X), we obtain some
curious observation. Even though characterisations of K-points on algebraic varieties and
smooth manifolds are practically the same, the reasons why it is so are completely different.
In the case of affine varieties, as we have seen above, the reasoning is purely algebraic. On the
other hand, in the case of smooth manifolds the characterisation follows from the existence of
a smooth function with compact level surfaces, which is rather a topological argument.
Given an affine variety, we have a natural surjective algebra homomorphism from the algebra
of polynomials to the algebra of regular functions. Hence we are going to study the behaviour
of δ and the principal symbol under the change of an algebra by a homomorphism.
Proposition 5.2. Assume that S, T are two K-algebras and α : S → T is a K-algebra homomorphism. If Φ : S → S and Ψ : T → T are K-linear maps such that α ◦ Φ = Ψ ◦ α, then for
any f ∈ S
(δ(α(f ))Ψ) ◦ α = α ◦ (δ(f )Φ).
(5.3)
Proof. Simply, for any f, g ∈ S we have that
(δ(α(f ))Ψ)(α(g)) = Ψ(α(f g)) − α(f )Ψ(α(g))
//α ◦ Φ = Ψ ◦ α// = α(Φ(f g)) − α(f Φ(g))
//δ(f ) acts on Φ// = α((δ(f )Φ)(g)).

(5.4)

Lemma 5.3. Assume that S, T are two K-algebras and α : S → T is a K-algebra homomorphism. If F : S → S and G : T → T are K-linear maps such that α ◦ F = G ◦ α, then for any
natural number j and any f1 , . . . , fj ∈ S
(δ(α(f1 )) . . . δ(α(fj ))G) ◦ α = α ◦ (δ(f1 ) . . . δ(fj )F ),

(5.5)

δh◦α (f1 ) . . . δh◦α (fj )Fh◦α = δh (α(f1 )) . . . δh (α(fj ))Gh ◦ α.

(5.6)

and for any h ∈ |T |

Proof. We will show the first part inductively. The first step of the induction is precisely
Proposition 5.2. So fix f0 , . . . , fj ∈ S and put
Φ = δ(f1 ) . . . δ(fj )F,

Ψ = δ(α(f1 )) . . . δ(α(fj ))G.
16

(5.7)

The inductive assumption states that α ◦ Φ = Ψ ◦ α, so by using again Proposition 5.2 we have
that
(δ(α(f0 ))δ(α(f1 )) . . . δ(α(fj ))G) ◦ α = (δ(α(f0 ))Ψ) ◦ α = α ◦ (δ(f0 )Φ)
= α ◦ (δ(f0 )δ(f1 ) . . . δ(fj )F ).

(5.8)

The second part of the lemma follows from the first one. From Lemma 4.1 for any f1 , . . . , fj ∈ S
and any K-point h ∈ |A| we have that
δh◦α (f1 ) . . . δh◦α (fj )Fh◦α = evh◦α ◦(δ(f1 ) . . . δ(fj )F )
// evh◦α = evh ◦α// = evh ◦(α ◦ (δ(f1 ) . . . δ(fj )F ))
//(5.5)// = evh ◦(δ(α(f1 )) . . . δ(α(fj ))G ◦ α)
//change of brackets// = (evh ◦δ(α(f1 )) . . . δ(α(fj ))G) ◦ α
//Lemma 4.1// = δh (α(f1 )) . . . δh (α(fj ))Gh ◦ α.

(5.9)

We recall that for every K-algebra A and h ∈ |A| the K-module Ah is naturally isomorphic
with K. Thus if L : A → A is an LDO of order 6 k, then the principal symbol of L at h
σL,h : (Ωh )⊙k ⊗K Ah → Ah

(5.10)

can be identified with the K-linear functional
σL,h : (Ωh )⊙k → K

(5.11)

σL,h (dh (f1 ) ⊙ · · · ⊙ dh (fk )) = (δh (f1 ) . . . δh (fk )Lh )(1A ).

(5.12)

given by the formula

We will use this identification when we consider LDOs acting on K-algebras.
Lemma 5.4. Assume that S, T are two K-algebras and π : S → T is a surjective homomorphism of K-algebras. If L : S → S is an LDO of order 6 k and L′ : T → T is a K-linear map
such that π ◦ L = L′ ◦ π, then L′ is also an LDO of order 6 k and for any h ∈ |T |
σL′ ,h ◦ (Ωπ )⊙k = σL,h◦π ,

(5.13)

where Ωπ : Ωh◦π → Ωh is the induced map from π by the functor Ω as in Proposition 1.5.
Proof. Since π ◦ L = L′ ◦ π, by using Lemma 5.3 we obtain that for any f0 , . . . , fk ∈ S
(δ(π(f0 )) . . . δ(π(fk ))L′ ) ◦ π = π ◦ (δ(f0 ) . . . δ(fk )L) = 0.

(5.14)

Hence, from the surjectivity of π, we get that δ(g0 ) . . . δ(gk )L′ = 0 for any g0 , . . . , gk ∈ T . This
means that L′ is an LDO of order 6 k. Now let h ∈ |T | and f1 , . . . , fk ∈ S. From the definition
of Ωπ we have that Ωπ (dh◦π (f )) = dh (π(f )) for every f ∈ S, and thus for any f1 , . . . , fk ∈ S
σL′ ,h ((Ωπ )⊙k )(dh◦π (f1 ) ⊙ · · · ⊙ dh◦π (fk )) = σL′ ,h (dh (π(f1 )) ⊙ · · · ⊙ dh (π(fk )))
//(5.12)// = (δh (π(f1 )) . . . δh (π(fk ))L′h )(1T )
//1T = π(1S )// = (δh (π(f1 )) . . . δh (π(fk ))L′h )(π(1S ))
//Lemma 5.3// = (δh◦π (f1 ) . . . δh◦π (fk )Lh◦π )(1S )
//(5.12)// = σL,h◦π (dh◦π (f1 ) ⊙ · · · ⊙ dh◦π (fk )).

17

(5.15)

As a result, we get that the ellipticity is preserved by a surjective homomorphism of Kalgebras in the following sense.
Theorem 5.5. Assume that S, T are two K-algebras, π : S → T is a surjective homomorphism
of K-algebras and L : S → S is an LDO of order k at h ◦ π and elliptic at h ◦ π for every
h ∈ |T |. If L′ : T → T is K-linear and such that π ◦ L = L′ ◦ π, then L′ is an elliptic LDO of
order k.
Proof. From Lemma 5.4 we have that L′ is an LDO of order 6 k and for every h ∈ |T |
σL′ ,h ◦ (Ωπ )⊙k = σL,h◦π .

(5.16)

Thus we just need to show that σL′ ,h (α⊙k ) 6= 0 for any non-zero α ∈ Ωh . Proposition 1.8 states
that the functor Ω preserves surjectivity. So we have that any α ∈ Ωh is of the form Ωπ (β) for
some β ∈ Ωh◦π . Thus we have that
σL′ ,h (α⊙k ) = σL′ ,h ((Ωπ (β))⊙k ) = σL,h◦π (β ⊙k ),

(5.17)

and this is non-zero for non-zero α because of the assumption about L.
Now, let V be a real affine variety defined by some ideal I = (f1 , . . . , fk ) ⊂ R[X1 , . . . , Xn ].
Let R[V ] = R[X1 , . . . , Xn ]/I denote the R-algebra of regular functions on V and let π :
R[X1 , . . . , Xn ] → R[V ] be the canonical projection. We know from Proposition 5.1 that Rpoints of R[X1 , . . . , Xn ] and R[V ] correspond bijectively to Rn and V respectively. Thus we
can talk about the ellipticity at ordinary points. By applying Theorem 5.5 to the homomorphism π we obtain a simple tool for constructing elliptic linear differential operators on real
affine varieties.
Corollary 5.6. If L : R[X1 , . . . , Xn ] → R[X1 , . . . , Xn ] is an LDO of order k at z and elliptic
at z for every z ∈ V and such that L(I) ⊂ I, then the induced map L′ : R[V ] → R[V ] is an
elliptic linear differential operator of order k.
Proof. Since L(I) ⊂ I, we have that L induces an R-linear map L′ : R[V ] → R[V ] such that
L′ ◦ π = π ◦ L. The rest follows directly from Theorem 5.5.
Corollary 5.7. Every real affine variety admits an elliptic linear differential operator acting
on its algebra of regular functions.
Proof. For the sake of readability we constrain ourselves to varieties on the plane. The general
case can be proved similarly. So let I ∈ R[X, Y ] be an ideal generated by some polynomials
f1 , . . . , fk . For each fi we define natural numbers si and ti as the degrees of fi in variables X
and Y respectively. Let s be the minimal even number greater than all si and analogously let
t be the minimal even number greater than all ti . We define the map ∆I : R[X, Y ] → R[X, Y ]
as
∂s
∂t
∆I =
+
.
(5.18)
∂X s ∂Y t
Clearly, ∆I is an elliptic LDO, and the numbers s and t were tailored in such a way that
∆I (I) = 0. In particular, ∆I (I) ⊂ I. So, using Corollary 5.6, we get that the induced map
∆′I : R[V ] → R[V ] is an elliptic linear differential operator.
For a concrete example, take the ideal I = (Y 3 − X 2 ) ⊂ R[X, Y ] and let V be the real affine
variety generated by I. According to the above description, we consider ∆I : R[X, Y ] → R[X, Y ]
by the formula
∂4
∂4
∆I =
+
,
(5.19)
∂X 4 ∂Y 4
18

and this LDO induces an elliptic LDO ∆′I : R[V ] → R[V ] of order 4.
The construction of ∆′I , presented in the proof of Corollary 5.7, is far from pleasing. ∆′I is
just a "restriction" of some operator of the form (5.18) defined on the ambient space. What is
even worse, it depends on the generator I of V . However, it is interesting whether there is a
way to associate an elliptic LDO to a real affine variety in some natural manner.

A

Appendix: Universal φ-derivations and Kähler differentials

One familiar with the theory of Kähler differentials may notice that most constructions
from Sections 1 and 2 are just modified versions of objects appearing in the theory of Kähler
differentials. In this appendix we will make clear why it is so.
If A is a K-algebra and P is an A-module, then the set of all derivations with values in P
is denoted by DerK (A, P ). The universal derivation of A, which is the same as the universal
IdA -derivation in the language from Section 1, is called the Kähler differential of A and it is
denoted by (ΩA/K , dA/K ).
Let A, B be two K-algebras and let φ : A → B be a K-algebra homomorphism. Because
of φ, we have the restriction of scalars by φ which endows any B-module with an A-module
structure. More specifically, if M is a B-module, then Mφ denotes the set M with an A-module
structure such that for any a ∈ A and m ∈ Mφ
am = φ(a)m.

(A.1)

As a result, φ-derivations with values in M are the same as derivations with values in Mφ . In
fact, if D : A → Mφ is a derivation, then for any f, g ∈ A
D(f g) = gD(f ) + f D(g) = φ(g)D(f ) + φ(g)D(f ).

(A.2)

The restriction of scalars by φ is a functor from the category of B-modules to the category
of A-modules. It has a left adjoint, called the extension of scalars by φ, which is just the tensor
multiplication by B (see [Bou, p. 277] for instance). This means that for every A-module P
and every B-module M there is a natural isomorphism
HomA (P, Mφ ) ∼
= HomB (B ⊗A P, M).

(A.3)

As a result, for any B-module M we have that
DerK (φ, M) ∼
= DerK (A, Mφ )
∼
= HomA (ΩA/K , Mφ )
∼
= HomB (B ⊗A ΩA/K , M).

(A.4)

So the B-module B ⊗A ΩA/K with the function d′φ : A → B ⊗A ΩA/K which maps f from A to
1B ⊗ dA/K (f ) in B ⊗A ΩA/K is a representing object for the functor DerK (φ, ·).
Proposition A.1. If A, B are K-algebras and φ : A → B is a K-algebra homomorphism, then
the pair (B ⊗A ΩA/K , d′φ ) is a universal φ-derivation.
Hence most of results from Sections 1 and 2 can be obtained by combining Proposition
A.1 and certain properties of Kähler differentials. However, such approach requires additional
categorial tools about representing objects and some specific properties of Kähler differentials
which are scattered across several sources. Thus, for the sake of clarity and readability of
Sections 3, 4 and 5, we had chosen a simpler path and done a concise analysis of φ-derivations
on its own.
19

Acknowledgements: First of all, I am most grateful to Antoni Pierzchalski for spending his
time to listen about the main ideas from this paper and for many useful comments. I am
also grateful to Sameer Kailasa for showing me the proof of Proposition 5.1 through online
discussion. Last but not least, I would like to thank my brother Radosław Kapka for careful
reading the final draft of this paper and for numerous hints regarding writing in English.

References
[Bou] N. Bourbaki, Algebra I. Elements of Mathematics (1998).
[Bra] M. Brandenburg,
Tensor
arXiv:1410.1716, (2014).
[Eis]

categorical

foundations

of

algebraic

geometry.

D. Eisenbud, Commutative Algebra, with a View Toward Algebraic Geometry. Graduate
Texts in Mathematics, 150 (1995).

[Gre] W. H. Greub, Linear Algebra. Graduate Texts in Mathematics, 23 (1975).
[Gro] A. Grothendieck, Éléments de géométrie algébrique. IV. Publications mathématiques de
l’I.H.É.S., 32, 5-361 (1967).
[HS]

R. G. Heyneman, M. E. Sweedler, Affine Hopf Algebras, I. Journal of Algebra, 13,
192-241 (1969).

[Kra] I. S. Krasil’shchik, Calculus over Commutative Algebras: A Concise User Guide. Acta
Applicandae Mathematicae 49, 235–248 (1997).
[KLV] I. S. Krasil’shchik, V. V. Lychagin, A. M. Vinogradov, Geometry of Jet Spaces and Nonlinear Partial Differential Equations. Advanced Studies in Contemporary Mathematics
(1986).
[Nar] R. Narasimhan, Analysis on real and complex manifolds. Advanced Studies in Pure
Mathematics (1973).
[Nes]

J. Nestruev, Smooth Manifolds and Observables. Graduate Texts in Mathematics, 220
(2003).

[Swa] R. G. Swan, Vector Bundles and Projective Modules. Transactions of the American
Mathematical Society, 105(2), 264–277 (1962).
[Vin]

A. M. Vinogradov, The logic algebra for the theory of linear differential operators. Soviet
Math. Dokl., 13(4), 1058-1062 (1972).

20

