AUniversalAlgebraicSurveyof C − Rings

A Universal Algebraic Survey of C

− Rings

Jean Cerqueira Berni¹, Hugo Luiz Mariano²

1Department of Mathematics – Institute of Mathematics and Statistics University of S˜ao Paulo

Rua do Mat˜ao, 1010, S˜ao Paulo - SP – Brazil

2Department of Mathematics – Institute of Mathematics and Statistics University of S˜ao Paulo

Rua do Mat˜ao, 1010, S˜ao Paulo - SP – Brazil

jeancb@ime.usp.br,hugomar@ime.usp.br

Abstract. In this paper we present some basic results of the Universal Algebra of C^∞−rings which were nowhere to be found in the current literature. The outstanding book of I. Moerdijk and G. Reyes, [1], presents the basic (and ad- vanced) facts about C^∞−rings, however such a presentation has no universal algebraic “flavour”. We have been inspired to describeC^∞−rings through this viewpoint by D. Joyce in [2]. Our main goal here is to provide a comprehen- sive material with detailed proofs of many known “taken for granted” results and constructions used in the literature aboutC^∞−rings and their applications - such proofs either could not be found or were merely sketched. We present, in detail, the main constructions one can perform within this category, such as limits, products, homomorphic images, quotients, directed colimits, free objects and others, providing a “propaedeutic exposition” for the reader’s benefit.

Keywords –C^∞−rings, algebraic constructions, universal algebra.

MSC2020 –20A05, 16S99

Resumo. Neste artigo apresentamos alguns resultados b´asicos da ´Algebra Uni- versal dos an´eis C^∞ que n˜ao aparecem explicitamente em nenhuma parte da literatura especializada. O excelente livro de I. Moerdijk e G. Reyes, [1], a- presenta fatos b´asicos (e avanc¸ados) sobre an´eisC^∞, no entanto sem um “sa- bor” alg´ebrico-universal. Nossa motivac¸˜ao para descrever os an´eisC^∞sob este ponto de vista foi o trabalho de D. Joyce em [2]. Nosso principal objetivo aqui

´e fornecer um material abrangente com demonstrac¸˜oes detalhadas de diversos resultados sobre an´eisC^∞tidos como garantidos na literatura sobre esses an´eis

Received October 18 2021, Revised August 9 2022.

e aplicac¸˜oes — provas que n˜ao est˜ao sequer esboc¸adas. Apresentamos, deta- lhadamente, as principais construc¸˜oes que se pode fazer dentro dessa categoria, tais como limites, produtos, imagens homomorfas, quocientes, limites dirigidos, objetos livres e outros, fornecendo uma “exposic¸˜ao propedˆeutica” em benef´ıcio do leitor.

Palavras-chave –An´eisC^∞, construc¸˜oes alg´ebricas, ´algebra universal.

Introduction

As observed by E. Dubuc in [3], aC^∞−ring is a model of the algebraic theory which has asn−ary operations all the smooth functions fromRⁿintoR, and whose axioms are all the equations that hold between these functions. Since every polynomial is smooth, allC^∞−rings are, in particular,R−algebras. This point of view allows us to regard this theory as an extension of the concept ofR−algebra.

The theory of C^∞−rings has been originally studied in view of its applications to Singularity Theory and in order to construct topos-models for Synthetic Differential Geometry (the Dubuc Topos, for instance. See [4]), which “grew out of ideas of Lawvere in the 1960s” (cf. [2]). Recently, however, this theory has been explored by some eminent mathematicians like David I. Spivak and Dominic Joyce in order to extend Jacob Lurie’s program of Derived Algebraic Geometry to Derived Differential Geometry (cf. [2] and [5] ).

Just as any Lawvere theory, C^∞−rings can be interpreted within any topos. In this specific case, aC^∞−ring in a toposE is a finite product preserving functor from the category whose objects are the Cartesian products of R and whose morphisms are the smooth functions between them, intoE (see, for example, [6]). In this work, however, we focus on set-theoreticC^∞−rings (i.e.,C^∞−rings in the topos of sets,Set).

Some categorial and logical aspects ofC^∞−rings were given in [7] and an order- theoretic analysis (detailing some aspects in [6]) is provided in [8]. Here we present and analyze aC^∞−ring as a universal algebra whose functional symbols are the symbols for all smooth functions from cartesian powers of R to R. Such an approach emphasizes the power of aC^∞−ring in interpreting a broader language than the algebraic one, which is expressed in terms of the R−algebra structure. It also has the advantage of giving us explicitly many constructions, such as products, coproducts, directed colimits, among others, as well as simpler proofs of the main results, which can be found in [9].

We make a detailed exposition of the description of freeC^∞−rings in terms of a colimit, and we use it to account the often used description of an arbitraryC^∞−rings in terms of generators and relations.

Our idea of describing C^∞−rings from a universal-algebraic point of view was mainly inpired by the clear and elegant presentation made by Dominic Joyce in [2] - which we found very enlightening.

Overview of the paper: We begin by presenting the equational theory of C^∞−rings in terms of a first order language with a denumerable set of variables. We define the class ofC^∞−structures and the (equationally defined) subclass ofC^∞−rings.

In the Section 2 we present a detailed description of the main constructions involving C^∞−rings: C^∞−subrings (Definition 2.1), intersections (Proposition 2.2), the C^∞−subring generated by a set (Definition 2.3), the directed union of C^∞−rings (Proposition 2.4), products (Definition 2.5), C^∞−congruences (Definition 2.7) and quotients (Definition 2.10), homomorphic images (Proposition 2.16), directed colim- its (Theorem 2.19) and small projective limits (Theorem 2.20). We also present the

“Fundamental Theorem ofC^∞−Homomorphism” (Theorem 2.15) and we present a re- sult which states that the category ofC^∞−rings is a reflective subcategory of the category of allC^∞−structures (Theorem 2.23).

We dedicate Section 3 to describe the free C^∞−rings, first with a finite set of generators and then with an arbitrary set of generators. We use this construction in order to describe an adjunction between the category of allC^∞−rings andC^∞−homomorphisms, C^∞Ringand the category of sets,Set(Proposition 3.4).

In Section 4 we describe other constructions. We present a result which states that the ring-theoretic ideals of any finitely generatedC^∞−ring classify their congruences (Proposition 4.6), and we extend this result by presenting a proof for the general case (Proposition 4.15).

We give a result which states that any C^∞−ring can be expressed as a directed colimit of finitely generatedC^∞−rings (Theorem 3.5) and inSubsection 4.2we present an explicit description for the C^∞−coproduct of C^∞−rings (Definition 4.19). We end up this work presenting an ubiquitous construction in Algebra in the Subsection 4.3, namely theC^∞−ring ofC^∞−polynomials, constructed in terms of theC^∞−product. Such a construction play an important role in [10] and [11].

1. Preliminaries: The equational theory ofC^∞-rings

The theory ofC^∞−rings can be described within a first order languageL with a de- numerable set of variables (Var (L) = {x₁, x₂,· · · , x_n,· · · }) whose nonlogical symbols are the symbols ofC^∞−functions from R^m toRⁿ, with m, n ∈ N, i.e., the non-logical symbols consist only of function symbols, described as follows:

For eachn ∈ N, then−aryfunction symbolsof the setC^∞(Rⁿ,R), i.e.,F_(n) =

{f⁽ⁿ⁾|f ∈ C^∞(Rⁿ,R)}. Thus, the set of function symbols of our language is given by:

F = [

n∈N

F_(n) = [

n∈N

C^∞(Rⁿ)

Note that our set of constants isR, since it can be identified with the set of all 0−ary function symbols,i.e.,Const(L) = F₍₀₎ =C^∞(R⁰)∼=C^∞({∗})∼=R.

The terms of this language, T, are defined, in the usual way, as the smallest set which comprises the individual variables, constant symbols andn−ary function symbols followed bynterms (n∈N).

SinceL contains no relational symbols, the set of theatomic formulas, AF, is given simply by the equality between terms, that is

AF={t₁ =t₂|t₁, t₂ ∈T}

Finally, thewell formed formulas, WFFare constructed as one usually does in any first order theory.

Definition 1.1. AC^∞−structureon a setAis a pairA= (A,Φ), where:

Φ : S

n∈NC^∞(Rⁿ,R) → S

n∈NFunc (Aⁿ;A) (f :R^{n C}

→∞ R) 7→ Φ(f) := (f^A :Aⁿ→A) ,

that is, Φinterprets the symbols¹ of all smooth real functions of n variables as n−ary function symbols onA.

Definition 1.2. Let(A,Φ)and(B,Ψ)be twoC^∞−structures. A functionϕ: A →B is called amorphism of C^∞−structures (or, simply, aC^∞−morphism) if for any n ∈ N and anyf ∈ C^∞(Rⁿ,R)the following diagram commutes:

Aⁿ

Φ(f)

ϕ⁽ⁿ⁾ //Bⁿ

Ψ(f)

A ^{ϕ //}B

i.e.,Ψ(f)◦ϕ⁽ⁿ⁾ =ϕ◦Φ(f).

Theorem 1.3. Let(A,Φ),(B,Ψ),(C,Ω) be anyC^∞−structures, and let ϕ : (A,Φ) → (B,Ψ)andψ : (B,Ψ)→(C,Ω)be two morphisms ofC^∞−structures. We have:

(1) id_A: (A,Φ)→(A,Φ)is a morphism ofC^∞−structures;

(2) ψ◦ϕ: (A,Φ)→(C,Ω)is a morphism ofC^∞−structures.

1here considered simply as syntactic symbols rather than functions.

Proof. SeeTheorem 1, p. 5 of [9].

Theorem 1.4. Let (A,Φ),(B,Ψ),(C,Ω),(D,Γ) be any C^∞−structures, and let ϕ : (A,Φ) → (B,Ψ), ψ : (B,Ψ) → (C,Ω) and ν : (C,Ω) → (D,Γ) be morphisms of C^∞−structures. We have the following equations between pairs of morphisms of C^∞−structures:

ν◦(ψ◦ϕ) = (ν◦ψ)◦ϕ;

ϕ◦idA= idB◦ϕ.

Proof. SeeTheorem 2, p. 6 of [9].

Definition 1.5. We are going to denote by C^∞Str the category whose objects are the C^∞−structures and whose morphisms are the morphisms ofC^∞−structures.

As a full subcategory ofC^∞Str we have the category of C^∞−rings. We call a C^∞−strutureA= (A,Φ)aC^∞−ringif it preserves projections and all equations between smooth functions. More precisely:

Definition 1.6. LetA= (A,Φ)be aC^∞−structure. We say thatA(or, when there is no danger of confusion,A) is aC^∞−ringif the following is true:

•Given anyn, k ∈Nand any projectionp_k:Rⁿ →R, we have:

A|= (∀x₁)· · ·(∀x_n)(p_k(x₁,· · · , x_n) = x_k)

•For everyf, g1,· · ·gn ∈ C^∞(R^m,R)withm, n ∈N, and everyh ∈ C^∞(Rⁿ,R) such thatf =h◦(g₁,· · · , g_n), one has:

A|= (∀x₁)· · ·(∀x_m)(f(x₁,· · · , x_m) = h(g(x₁,· · · , x_m),· · · , g_n(x₁,· · · , x_m)))

Definition 1.7. Let(A,Φ)and(B,Ψ)be twoC^∞−rings. A functionϕ:A→Bis called amorphism ofC^∞−ringsorC^∞-homomorphismif for anyn ∈Nand anyf :R^nC

∞

→R the following diagram commutes:

Aⁿ

Φ(f)

ϕ⁽ⁿ⁾ //Bⁿ

Ψ(f)

A ^{ϕ //}B

i.e.,Ψ(f)◦ϕ⁽ⁿ⁾ =ϕ◦Φ(f).

One possible set of axioms for the theory of the C^∞-rings can be given by the following two sets of equations:

(E1) For eachn ∈Nand for everyk ≤n, denoting thek-th projection byp_k :Rⁿ→R, the equations:

Eq^n,k₍₁₎ ={(∀x₁)· · ·(∀x_n)(p_k(x₁,· · · , x_n) = x_k)}

(E2) for every k, n ∈ N and for every (n + 2)−tuple of function symbols, (f, g1,· · · , gn, h) such that f ∈ F(n), g1,· · · , gn, h ∈ F(k) and h = f ◦ (g₁,· · · , g_n), the equations:

Eq^n,k₍₂₎ ={(∀x₁)· · ·(∀x_k)(h(x₁,· · ·, x_k) =f(g₁(x₁,· · · , x_k),· · · , g_n(x₁,· · ·, x_k)))}

As we are going to see later on, the category ofC^∞−rings and its morphisms has many constructions, such as arbitrary products, coproducts, directed colimits, quotients and many others. It also “extends” the category of commutative unital rings, CRing, in the following sense:

Remark 1.8. Since the sum + : R² → R, the opposite, − : R → R, · : R² → R and the constant functions 0 : R → R and 1 : R → R are particular cases of C^∞−functions, any C^∞−ring (A,Φ) may be regarded as a commutative unital ring (A,Φ(+),Φ(·),Φ(−),Φ(0),Φ(1)), where:

Φ(+) : A×A → A

(a₁, a₂) 7→ Φ(+)(a₁, a₂) = a₁+a₂

Φ(−) : A → A

a 7→ Φ(−)(a) =−a

Φ(0) : A⁰ → A

∗ 7→ Φ(0)

Φ(1) : A⁰ → A

∗ 7→ Φ(1) whereA⁰ ={∗}, and:

Φ(·) : A×A → A

(a₁, a₂) 7→ Φ(·)(a₁, a₂) =a₁ ·a₂

Thus, we have a forgetful functor:

Ue : C^∞Ring → CRing

(AΦ)

ϕ

(B,Ψ)

7→ (A,Φ(+),Φ(·),Φ(−),Φ(0),Φ(1))

ϕ

(B,Ψ(+),Ψ(·),Ψ(−),Ψ(0),Ψ(1))

Ue :C^∞Ring→CRing

Analogously, we can define a forgetful functor from the category of C^∞−rings andC^∞−homomorphisms into the category of commutativeR−algebras with unity,

Uˆ :C^∞Ring→R−Alg

These functors are analyzed with detail in [10].

2. The main constructions in the category ofC^∞−rings

Since the theory ofC^∞−rings is equational, the classC^∞Ringis closed inC^∞Strunder many algebraic constructions, such as substructures, products, quotients, directed colimits and others. In this section we give explicit descriptions for some of these constructions.

2.1. C^∞−Subrings

We begin defining what we mean by aC^∞−subring.

Definition 2.1. Let(A,Φ)be aC^∞−ring and letB ⊆ A. Under these circumstances, we say that(B,Φ⁰)is aC^∞−subring of(A,Φ)if, and only if, for anyn∈N,f ∈ C^∞(Rⁿ,R) and any(b₁,· · · , b_n)∈Bⁿwe have:

Φ(f)(b₁,· · · , b_n)∈B

That is to say that B is closed under any C^∞−function n-ary symbol. Note that the C^∞−structure of B is virtually the same as the C^∞−structure of (A,Φ), since they in- terpret every smooth function in the same way. However Φ⁰ has a different codomain, as:

Φ⁰ : S

n∈NC^∞(Rⁿ,R) → S

n∈NFunc (Bⁿ, B) (Rⁿ →^f R) 7→ (Φ(f)Bⁿ:Bⁿ →B)

We observe thatΦ⁰is the uniqueC^∞−structure such that the inclusion map:

ι^A_B :B ,→A

is aC^∞−homomorphism.

We are going to denote the class of allC^∞−subrings of a givenC^∞−ring(A,Φ) bySub (A,Φ).

Next we prove that the intersection of any family of C^∞−subrings of a given C^∞−ring is, again, aC^∞−subring.

Proposition 2.2. Let{(A_α,Φ_α)|α ∈Λ}be a family ofC^∞−subrings of(A,Φ), so

(∀α∈Λ)(∀n∈N)(∀f ∈ C^∞(Rⁿ,R))(Φ_α(f) = Φ(f)Aαn:A_αⁿ →A_α)

We have that:

\

α∈Λ

A_α,Φ⁰

!

where:

Φ⁰ : S

n∈NC^∞(Rⁿ,R) → S

n∈NFunc T

α∈ΛA_αn

,T

α∈ΛA_α (Rⁿ →^f R) 7→ Φ(f)(^Tα∈ΛAα)ⁿ: T

α∈ΛA_αn

→T

α∈ΛA_α

is aC^∞−subring of(A,Φ).

Proof. SeeProposition 1, p. 9 of [9].

As an application of the previous result, we can define theC^∞−subring generated by a subset of the carrier of aC^∞−ring:

Definition 2.3. Let(A,Φ)be aC^∞−ring andX ⊆A. TheC^∞−subring of(A,Φ)gener- ated byX is given by:

hXi= \

X⊆Ai

(Ai,Φi)⊆(A,Φ)

(A_i,Φ_i),

where(A_i,Φ_i)⊆(A,Φ)means that(A_i,Φ_i)is aC^∞−subring of(A,Φ)together with the C^∞−structure given inProposition 2.2.

We note that, given anyC^∞−ring(A,Φ), the map of partially ordered sets given by:

σ : (℘(A),⊆) → (Sub(A),⊆)

X 7→ hXi

satisfies the axioms of a closure operation.

2.2. Directed union ofC^∞−rings

In general, given an arbitrary family(A_α,Φ_α)_α∈Λ of C^∞−subrings of a given C^∞−ring (A,Φ), its union, S

α∈ΛA_α, together with Φ ^∪α∈ΛAα, is not necessarily a C^∞−subring of (A,Φ). However, there is an important case in which the union of a family of C^∞−subrings of a C^∞−ring (A,Φ)is, again, aC^∞−ring. This case is discussed in the following:

Proposition 2.4. Let (A,Φ) be a C^∞−ring and let {(A_α,Φ_α)|α ∈ Λ}, Λ 6= ∅, be a directed family of C^∞−subrings of (A,Φ), that is, a family such that for every pair (α, β)∈Λ×Λthere is someγ ∈Λsuch that:

A_α ⊆A_γ and

Aβ ⊆Aγ

We have that:

[

α∈Λ

A_α,Φ⁰

!

where:

Φ⁰ : S

n∈NC^∞(Rⁿ,R) → S

n∈NFunc S

α∈ΛAα

n

,S

α∈ΛAα

(Rⁿ →^f R) 7→ Φ(f)(^Sα∈ΛAα)ⁿ: S

α∈ΛAα

n

→S

α∈ΛAα

is aC^∞−subring of(A,Φ).

Proof. SeeProposition 2, p. 10 of [9].

2.3. Products,C^∞−Congruences and Quotients

Next we describe the products in the category C^∞Ring, that is, products of arbitrary families ofC^∞−rings.

Definition 2.5. Let {(A_α,Φ_α)|α ∈ Λ} be a family of C^∞−rings. The product of this family is the pair:

Y

α∈Λ

Aα,Φ^(Λ)

!

whereΦis given by:

Φ^(Λ): S

n∈NC^∞(Rⁿ,R) → S

n∈NFunc Q

α∈ΛA_αn

,Q

α∈ΛA_α (f :R^nC

→∞ R) 7→ Φ^(Λ)(f) : Q

α∈ΛA_αn

→ Q

α∈ΛA_α ((x¹_α)_α∈Λ,· · · ,(xⁿ_α)_α∈Λ) 7→(Φ_α(f)(x¹_α,· · · , xⁿ_α))_α

Remark 2.6. In particular, given aC^∞−ring(A,Φ), we have the productC^∞−ring:

(A×A,Φ⁽²⁾)

where:

Φ⁽²⁾ : S

n∈NC^∞(Rⁿ,R) → S

n∈NFunc ((A×A)ⁿ, A×A) (f :R^{n C}

→∞ R) 7→ (Φ×Φ)(f) : (A×A)ⁿ→A×A

and:

Φ⁽²⁾(f) : (A×A)ⁿ → A×A

((x₁, y₁),· · · ,(x_n, y_n)) 7→ (Φ(f)(x₁,· · · , x_n),Φ(f)(y₁,· · · , y_n))

We turn now to the definition of congruence relations inC^∞−rings. As we shall see later on, the congruences of a C^∞−rings will be classified by their ring-theoretic ideals.

Definition 2.7. Let(A,Φ)be aC^∞−ring. AC^∞−congruenceis an equivalence relation R⊆A×Asuch that for everyn ∈Nandf ∈ C^∞(Rⁿ,R)we have:

(x₁, y₁),· · ·,(x_n, y_n)∈R ⇒Φ⁽²⁾(f)((x₁, y₁),· · · ,(x_n, y_n))∈R

In other words,a C^∞−congruence is an equivalence relations that preserves C^∞−function symbols.

A characterization of a C^∞−congruence can be given using the product C^∞−structure, as we see in the following:

Proposition 2.8. Let(A,Φ)be aC^∞−ring and letR ⊆A×Abe an equivalence relation.

Under these circumstances,R is aC^∞−congruence on(A,Φ)if, and only if,(R,Φ⁽²⁾⁰), where:

Φ⁽²⁾⁰ : S

n∈NC^∞(Rⁿ,R) → S

n∈NFunc (Rⁿ, R) (Rⁿ→^f R) 7→ Φ⁽²⁾(f)Rⁿ:Rⁿ→R

is aC^∞−subring of(A×A,Φ⁽²⁾), with the structure described in theRemark 2.6.

Proof. SeeProposition 3, p. 13 of [9].

Remark 2.9. Given aC^∞−ring(A,Φ)and aC^∞−congruenceR ⊆A×A, let:

(A/R) = {a|a∈A}

be the quotient set. Given anyn ∈ N,f ∈ C^∞(Rⁿ,R)and(a₁,· · · , a_n) ∈((A/R))ⁿwe define:

Φ : S

n∈NC^∞(Rⁿ,R) → S

n∈NFunc (((A/R))ⁿ,(A/R)) (f :R^{n C}

∞

→ R) 7→ Φ(f) : ((A/R))ⁿ→(A/R) where:

Φ(f) : ((A/R))ⁿ → (A/R) (a₁,· · · , a_n) 7→ Φ(f)(a₁,· · · , a_n)

Note that the interpretation above is indeed a function, that is, its value does not de- pend on any particular choice of the representing element. This means that given (a1,· · ·, an),(a⁰₁,· · · , a⁰_n)∈((A/R))ⁿsuch that(a1, a⁰₁),· · · ,(an, a⁰_n)∈R, we have:

Φ(f)(a1,· · · , an) = Φ(f)(a1,· · · , an)

and sinceRis aC^∞−congruence,

(a1, a⁰₁),· · · ,(an, a⁰_n)∈R⇒(Φ(f)(a1,· · · , an),Φ(f)(a⁰₁,· · · , a⁰_n))∈R

so:

Φ(a₁,· · · , a_n) = Φ(f)(a₁,· · · , a_n) = Φ(f)(a⁰₁,· · · , a⁰_n) = Φ(f)(a⁰₁,· · · , a⁰_n)

The above construction leads directly to the following:

Definition 2.10. Let(A,Φ)be aC^∞−ring and letR ⊆A×Abe aC^∞−congruence. The quotientC^∞−ringofAbyRis the ordered pair:

(A/R),Φ

where:

(A/R) = {a|a∈A}

and

Φ : S

n∈NC^∞(Rⁿ,R) → S

n∈NFunc (((A/R))ⁿ,(A/R)) (f :R^{n C}

∞

→ R) 7→ Φ(f) : ((A/R))ⁿ→(A/R)

whereΦ(f)is described inRemark 2.9.

The following result shows that the canonical quotient map is, again, a C^∞−homomorphism.

Proposition 2.11. Let(A,Φ)be aC^∞−ring and letR ⊆ A×Abe a C^∞−congruence.

The function:

q : (A,Φ) → (A/R),Φ

a 7→ a

is aC^∞−homomorphism.

Proof. SeeProposition 4, p. 15 of [9].

We remark that the structure given above is the uniqueC^∞−structure such that the quotient map is aC^∞−homomorphism.

Proposition 2.12. Let(A,Φ)and(B,Ψ)be twoC^∞−rings and letϕ: (A,Φ)→(B,Ψ) be aC^∞−homomorphism. The set:

ker(ϕ) = {(a, a⁰)∈A×A|ϕ(a) = ϕ(a⁰)}

is aC^∞−congruence on(A,Φ).

Proof. SeeProposition 5, p. 16 of [9].

Corollary 2.13. Let(A,Φ)and(B,Ψ) be twoC^∞−rings and let ϕ : (A,Φ) → (B,Ψ) be aC^∞−homomorphism. Then(ker(ϕ),Φ⁽²⁾⁰)is aC^∞−subring of(A×A,Φ⁽²⁾).

Proposition 2.14. For every C^∞−congruence R ⊆ A ×A in (A,Φ), there are some C^∞−ring (B,Ψ) and some C^∞−homomorphismϕ : (A,Φ) → (B,Ψ) such thatR = ker(ϕ).

Proof. It suffices to take(B,Ψ) = ^A_R,Φ

andϕ=q_R : (A,Φ)→ ^A_R,Φ .

Theorem 2.15. (Fundamental Theorem of theC^∞−Homomorphism)Let(A,Φ)be a C^∞−ring and R ⊆ A×A be a C^∞−congruence. For every C^∞−ring (B,Ψ) and for everyC^∞−homomorphismϕ: (A,Φ)→(B,Ψ)such thatR⊆ker(ϕ), that is, such that:

(a, a⁰)∈R ⇒ϕ(a) =ϕ(a⁰),

there is a uniqueC^∞−homomorphism:

ϕe: (A/R),Φ

→(B,Ψ)

such that the following diagram commutes:

(A,Φ)

q

ϕ //(B,Ψ)

(A/R),Φ

ϕe

55

that is, such that ϕe◦ q = ϕ, where Φ is the canonical C^∞−structure induced on the quotient ^A_R

Proof. SeeTheorem 3, p. 17 of [9].

The following result is straightforward:

Proposition 2.16. Let(A,Φ)and(B,Ψ)be twoC^∞−rings and letϕ: (A,Φ)→(B,Ψ) be aC^∞−homomorphism. The ordered pair:

(ϕ[A],Ψ⁰)

where:

Ψ⁰ : S

n∈NC^∞(Rⁿ,R) → S

n∈NFunc (ϕ[A]ⁿ, ϕ[A]) (Rⁿ →^f R) 7→ Ψ(f)ϕ[A]ⁿ:ϕ[A]ⁿ→ϕ[A]

is aC^∞−subring of(B,Ψ), calledthe homomorphic image ofAbyϕ.

Corollary 2.17. Let(A,Φ)and(B,Ψ) be twoC^∞−rings and let ϕ : (A,Φ) → (B,Ψ) be a C^∞−homomorphism. As we have noticed in Proposition 2.16, (ϕ[A],Ψ⁰) is a C^∞−subring of(B,Ψ).

Under these circumstances, there is a uniqueC^∞−isomorphism:

ϕe:

A ker(ϕ),Φ

→(ϕ[A],Ψ⁰)

such that the following diagram commutes:

(A,Φ)

q

ϕ //(ϕ[A],Ψ⁰)

A ker(ϕ),Φ

^ϕe

66

that is, such that ϕe◦ q = ϕ, where Φ is the canonical C^∞−structure induced on the quotient _ker(ϕ)^A

Proof. SeeCorollary 2, p. 19 of [9].

2.4. Directed Colimits ofC^∞−Rings

The following result is going to be used to construct directed colimits ofC^∞−rings.

Lemma 2.18. Let(A,Φ)be aC^∞−ring. The ordered pair:

(A× {α},Φ×id_α)

where:

Φ×id_α : S

n∈NC^∞(Rⁿ,R) → S

n∈NFunc ((A× {α})ⁿ, A× {α}) (Rⁿ →^f R) 7→ Φ(f)×id_α : (A× {α})ⁿ → A× {α}

((a₁, α),· · · ,(a_n, α)) 7→(Φ(f)(a₁,· · · , a_n), α) is aC^∞−ring and:

π₁ : A× {α} → A (a, α) 7→ a is aC^∞−isomorphism, that is:

(A,Φ)∼=π1 (A× {α},Φ×idα)

Proof. SeeLemma 1, p. 20 of [9].

The proof of the following result describes the construction of directed colimits of directed families ofC^∞−rings.

Theorem 2.19. Let (I,≤) be a directed set and let ((A_α,Φ_α), µ_αβ)α,β∈I be a directed system. There is an object(A,Φ)inC^∞Ringsuch that:

(A,Φ)∼= lim−→

α∈I

(A_α,Φ_α)

Proof. SeeTheorem 4, p. 22 of [9].

Theorem 2.20. Given any small categoryJ and any diagram:

D: J → C^∞Ring

(α→^h β) 7→ (A_α,Φ_α)^D(h)→ (A_β,Φ_β)

there is aC^∞−ring(A,Φ)such that:

(A,Φ)∼= lim←−

α∈I

D(α)

Proof. SeeTheorem 5, p. 26 of [9].

Remark 2.21. LetΣ = S

n∈NC^∞(Rⁿ,R)and letX ={x₁, x₂,· · · , x_n,· · · }be a denu- merable set of variables, soF(Σ, X)will denote the algebra of terms of this languageΣ.

A class ofordered pairswill be simply a subsetS ⊆F(Σ, X)×F(Σ, X). In our case, these pairs are given by the axioms, soS consists of the following:

•For anyn ∈R,i≤nand a (smooth) projection mapp_i :Rⁿ→Rwe have:

(p_i(x₁,· · · , x_i,· · · , x_n), x_i)∈S

•for every f, g1,· · · , gn ∈ C^∞(R^m,R)and h ∈ C^∞(Rⁿ,R) such thatf = h◦ (g₁,· · · , g_n), we have

(h(g₁(x₁,· · · , x_m),· · · , g_n(x₁,· · · , x_m)), f(x₁,· · · , x_m))∈S

Remark 2.22. The class ofC^∞−rings is a model of an equational theory, thus it is a vari- ety of algebras. However, if we were not given this information, noting that the category ofC^∞−rings is closed under products, subalgebras and homomorphic images, theHSP Birkhoff’s Theoremwould lead us to the same conclusion, that is, that the classC^∞Ring is a variety of algebras, and by the previous remark, C^∞Ring = V(S), the variety of algebras defined byS.

In particular, we have some classical results. We list some of them:

•for every setX there is a freeC^∞−ring determined byX;

•anyC^∞−ring is a homomorphic image of some freeC^∞−ring;

•aC^∞−homomorphism is monic if, and only if, it is an injective map;

•any indexed set ofC^∞−rings,{(Aα,Φα)|α∈I}, has a coproduct inC^∞Ring.

We end this section by stating a result which says that the (variety) of allC^∞−rings is a reflective subcategory ofC^∞Str.

Theorem 2.23. The inclusion functor ι : C^∞Ring ,→ C^∞Str has a left adjoint L : C^∞Str → C^∞Ring: given by M 7→ M/θ_M where θ_M is the least C^∞-congruence of M such that M/θM ∈ Obj(C^∞Ring). Moreover, the unit of the adjunction L a ι has components(q_M)_M∈Obj(C^∞_Str), whereq_M :M M/θ_M is the quotient homomorphism.

Proof. SeeTheorem 6, p. 27 of [9].

3. FreeC^∞−Rings

Our definition ofC^∞−ring yields a forgetful functor:

U : C^∞Ring → Set

(A,Φ) 7→ A

((A,Φ)→^ϕ (B,Ψ)) 7→ (A^U(ϕ)→ B)

In fact, as we are going to see later, this functor has a left adjoint, the “free C^∞−ring”, that we shall denote by L : Set → C^∞Ring. Before we do it, we need the following:

Remark 3.1. Given any m ∈ N, we note that the set C^∞(Rⁿ) may be endowed with a C^∞−structure:

Ω : S

n∈NC^∞(Rⁿ,R) → S

n∈NFunc (C^∞(R^m)ⁿ,C^∞(R^m)) (Rⁿ→^f R) 7→ Ω(f) =f ◦ −: C^∞(R^m)ⁿ → C^∞(R^m)

(h1,· · · , hn) 7→ f ◦(h1,· · · , hn)

so it is easy to see that it can be made into aC^∞−ring(C^∞(R^m),Ω). From now, when dealing with this“canonical”C^∞−structure, we shall omit the symbolΩ, writingC^∞(R^m) instead of(C^∞(R^m),Ω).

We have the following construction of finitely generated freeC^∞−rings:

Proposition 3.2. LetU :C^∞Ring→Set,(A,Φ)7→A, be the forgetful functor. The pair (_n,(C^∞(Rⁿ),Ω)), where:

_n : {1,· · · , n} → U(C^∞(Rⁿ),Ω) i 7→ π_i :Rⁿ→R

,

is the freeC^∞-ring withngenerators, which are the projections:

π_i : Rⁿ → R

(x₁,· · · , x_i,· · · , x_n) 7→ x_i Proof. SeeProposition 1.1of [1].

As a consequence of the above result, for any finite setX,(X,(C^∞(R^X),ΦX))is the freeC^∞−ring defined byX(seeCorollary 3, p. 42 of [9]). One extends the definition for freeC^∞−rings generated by infinite sets by using the colimit of the finitely generated free ones. Thus, given an (infinite) setE, one decomposes it as the union of its finite subsets:

E = [

E⁰⊆ finE

E⁰

and defines:

C^∞(R^E) = lim−→

E⁰⊆ finE

C^∞(R^E

0).

Formally we have:

Proposition 3.3. Let E be any set. The pair (E,(C^∞(R^E),ΦE)) (where E : E → U(C^∞(R^E),Φ_E)is given (uniquely) by the universal property of the colimitC^∞(R^E)) is the freeC^∞−ring determined byE.

Proof. SeeProposition 10, p. 44 of [9].

The underlying setC^∞(R^E)can be realized as a subset ofFunc (R^E,R), namely as the subset of all functions from R^E to R which depend smoothly on finitely many coordinates (cf.Section 3of [9]).

In the following proposition, we present a description of a left adjoint to the for- getful functorU :C^∞Ring→Set.

Proposition 3.4. The functions:

L₀ : Obj (Set) → Obj (C^∞Ring) X 7→ (C^∞(R^X),Φ_X)

and

L1 : Mor (Set) → Mor (C^∞Ring)

(X →^f Y) 7→ (C^∞(R^X),Φ_X)→^fe (C^∞(R^Y),Φ_Y)

where fe : L₀(X) → L₀(Y) is the unique C^∞−homomorphism given by the universal property of the free C^∞−ring _X : X → C^∞(R^X): given the function _Y ◦f : X → U(C^∞(R^Y)), there is a uniqueC^∞−homomorphism such that the following diagram com- mutes:

X ^{X //}

Y◦f ((

U(C^∞(R^X))

U(f)e

U(C^∞(R^Y))

define a functor L : Set → C^∞Ring which is left adjoint to the forgetful functor U : C^∞Ring→Set.

Proof. SeeProposition 11, p. 47 of [9].

3.1. Relations and Generators

Let(A,Φ)be aC^∞−ring and letX ⊆A. Given the inclusion mapι^A_X :X ,→A, there is a uniqueC^∞−homomorphismfι^A_X : (L(X),ΦX)→(A,Φ)such that the following diagram commutes:

X ^{ηX //}

ι^A_X ((

U(L(X))

U(ιf^A_X)

U(A,Φ)

(1)

We claim that ifhXi=A, thenfι^A_X is surjective.

Indeed,

X =ι^A_X[X] = im(ι^A_X) and sinceU(fι^A_X)◦η_X =ι^A_X, we have:

im(ι^A_X) = im(U(fι^A_X)◦η_X).

Since the diagram (1) commutes, on the other hand, im(U(fι^A_X)◦η_X)⊆im (U(fι^A_X))

thusX ⊆im (U(fι^A_X))andhXi ⊆ him (U(ιf^A_X))i=fι^A_X[A].

SinceX generatesAandhim (U(fι^A_X))i = im (fι^A_X)is aC^∞−subring of(A,Φ), it follows that:

hXi=A⊆im(ιf^A_X)⊆A, soim (ιf^A_X) =Aandι^A_X is surjective.

In particular, takingX = A yields ι^A_A = id_A, and sinceε_A = φ_A,(A,Φ)(id_A) = id^_(A,Φ), we have:

A ^{ηA //}

idA ((

U(L(A))

U(ε_A)

U(A,Φ)

soim (ε_A) = (A,Φ), andε_Ais surjective. Now, given anyC^∞−ring(A,Φ)we have the surjective morphism:

ε_A:L(U(A,Φ))(A,Φ).

We have seen that since ε_A is a C^∞−homomorphism, ker(ε_A) is a C^∞−congruence. By theFundamental Theorem of theC^∞−Isomorphismwe have:

(A,Φ)∼= L(A) ker(ε_A)

As an application, we register the following (very useful) result:

Theorem 3.5. Let (A,Φ) be any C^∞−ring. There is a directed system of C^∞−rings, ((A_i,Φ_i), α_ij), where each (A_i,Φ_i) is a finitely generated C^∞−ring and each α_ij : (A_i,Φ_i)→(A_j,Φ_j)is a monomorphism such that:

(A,Φ)∼= lim−→(A_i,Φ_i)

Proof. SeeTheorem 8, p. 54 of [9].

4. Other Constructions

In this section we describe further results and constructions involvingC^∞−rings.

4.1. Ring-Theoretic Ideals andC^∞−Congruences

Our next goal is to classify the congruences of anyC^∞−ring. We shall see that they are classified by their ring-theoretic ideals.

Since eachC^∞−ring has an underlying (commutative, unital) ring, it is easy to see that the following result holds:

Proposition 4.1. Given any C^∞−ring (A,Φ), let Cong (A,Φ) denote the set of all the C^∞−congruences in A and let I(A,Φ)denote the set of all ideals of A. The following map is well-defined:

ψ_A: Cong (A,Φ) → I(A,Φ) R 7→ {x∈A|(x,0)∈R}

In fact, we will show that, for eachC^∞−ring (A,Φ), the mapψ_A is a bijection whose inverse map is given by

ϕ_A: I(A,Φ) → Cong (A,Φ)

I 7→ {(x, y)∈A×A|x−y∈I}

The point here is to show that the map ϕ_A is well-defined (i.e., ifI ⊆ A is an ideal, thenϕA(I)is aC^∞-congruence). This is achieved through a sequence of steps²:

(I) We show that the statement holds for any free finitely generated C^∞-ring A (A∼=C^∞(Rⁿ,R));

(II) We show that the statement holds for any finitely generatedC^∞-ringB (B ∼= C^∞(Rⁿ,R)

Θ , for someΘ∈Cong(C^∞(Rⁿ,R)));

(III) Finally, we show that the statement holds for anyC^∞-ringC, by usingThe- orem 3.5(C ∼= lim−→^i∈IB_i, for some directed diagram of finitely generatedC^∞-rings).

We begin with the free finitely generated case.

An interesting result, which is a consequence of Hadamard’s Lemma, Theo- rem 4.2, is the description of the ideals of free finitely generatedC^∞−rings in terms of C^∞−congruences.

For the reader’s benefit, we state the following result:

Theorem 4.2. (Hadamard’s Lemma)For every smooth functionf ∈ C^∞(Rⁿ)there are smooth functionsg₁,· · · , g_n∈ C^∞(R²ⁿ)such that∀(x₁,· · · , x_n),(y₁,· · · , y_n)∈Rⁿ:

f(x₁,· · · , x_n)−f(y₁,· · · , y_n) =

n

X

i=1

(x_i−y_i)·g_i(x₁,· · · , x_n, y₁,· · · , y_n)

Proof. SeeTheorem 7, p. 51 of [9].

2This procedure works in many situations!

Corollary 4.3. Given a free finitely generatedC^∞−ring(A,Φ), considering the forgetful functor given inRemark 1.8, Ue : C^∞Ring → CRing, we have that if I is a subset of Athat is an ideal (in the ordinary ring-theoretic sense) in U(A,e Φ), thenIb= {(a, b) ∈ A×A|a−b∈I}is aC^∞−congruence inA.

Proof. SeeProposition 12, p. 51 of [9].

The following lemma is a well-known result of Universal Algebra applied toC^∞−rings:

Lemma 4.4. Let (A,Φ) be a C^∞−ring and let R ∈ Cong (A,Φ). Given the quotient C^∞−homomorphism:

qR: (A,Φ) → (A/R),Φ x 7→ x+R we have the bijection:

(q_R)∗ : {S∈Cong (A,Φ)|R⊆S} → Cong (A/R),Φ

S 7→ {(q_R(s), q_R(t))∈ ^A_R× ^A_R|(s, t)∈S}

whose inverse is given by:

(q_R)^∗ : Cong (A/R),Φ

→ {S ∈Cong(A,Φ)|R ⊆S}

S⁰ 7→ (q_R×q_R)^a[S⁰]

As a consequence of the above lemma, we have:

Lemma 4.5. Let (A,Φ) be a C^∞−ring and R ∈ Cong (A,Φ), and suppose that ψ_A : Cong (A,Φ) → I(A,Φ) is a bijection with an inverse, ϕ_A : I(A,Φ) → Cong (A,Φ).

Under those circumstances, the quotientC^∞−homomorphism:

q_R: (A,Φ)→ (A/R),Φ

induces a pair of inverse bijections:

(qR)+ : I (A/R),Φ

→ {I⁰ ∈I(A,Φ)|ψA(R)⊆I⁰}

J 7→ q_R[J]

and

(p_R)⁻ : {I⁰ ∈I(A,Φ)|ψ_A(R)⊆I⁰} → I (A/R),Φ J⁰ 7→ (q_R)^a[J⁰]

Proof. SeeLemma 4, p. 64 of [9].

The following result tells us that the ideals of the finitely generated C^∞−rings classify its congruences.

Proposition 4.6. Given any finitely generatedC^∞−ring(A,Φ). Then the following maps are well-defined and provide a pair of inverse bijections:

ψ_A: Cong (A,Φ) → I(A,Φ) R 7→ {x∈A|(x,0)∈R}

ϕ_A: I(A,Φ) → Cong (A,Φ)

I 7→ {(x, y)∈A×A|x−y∈I}

Proof. SeeProposition 13, p. 52 of [9].

For anyC^∞−ring(A,Φ)we have the function:

ψ_A: Cong (A,Φ) → I(A,Φ)

R 7→ {a∈A|(a,Φ(0))∈R}

and whenever (A,Φ)is a finitely generatedC^∞−ring, we have seen in Proposition 4.6 thatψ_Ahasϕ_Aas inverse - which is a consequence ofHadammard’s Lemma.

In order to show thatψ_Ais a bijection for anyC^∞−ring(A,Φ), first we decompose it as a directed colimit of its finitely generatedC^∞−subrings (cf.Theorem 3.5):

(A,Φ)∼= lim−→

(Ai,Φi)⊆_f.g.(A,Φ)

(Ai,Φ^Ai)

and then we useProposition 4.6to obtain a bijection:

ϕe: lim←−

(Ai,Φi)⊆_f.g.(A,Φ)

I(A_i,Φ_i)→ lim←−

(Ai,Φi)⊆_f.g.(A,Φ)

Cong (A_i,Φ_i)

such that for every(A_i,Φ_i)⊆_f.g.(A,Φ)the following diagram commutes:

lim←−(Ai,Φi)⊆_f.g.(A,Φ)I(A_i,Φ_i)

ϕe //lim←−(Ai,Φi)⊆_f.g.(A,Φ)Cong (A_i,Φ_i)

I(A_i,Φ_i) _ϕ

Ai //Cong(A_i,Φ_i)

where the vertical downward arrows are the canonical arrows of the projective limit.

Finally we show that there is a bijective correspondence,α, between lim←−

(Ai,Φi)⊆_f.g.(A,Φ)

I(A_i,Φ_i)

and

I lim−→

(Ai,Φi)⊆_f.g.(A,Φ)

(A_i,Φ_i)

!

and a bijective correspondence, β, between lim←−^(Ai,Φi)⊆f.g.(A,Φ)Cong (A_i,Φ_i) and Cong

lim−→^(Ai,Φi)⊆_f.g.(A,Φ)(A_i,Φ_i)

. In fact, the bijectionsα, β,ϕeandψare complete lat- tices isomorphisms.

By composing these bijections we prove that the congruences of(C^∞(R^E),Φ_E) are classified by the ring-theoretic ideals of(C^∞(R^E),Φ_E).

As a consequence of this latter result, we have several examples ofC^∞−rings.

Example 4.7. Given any C^∞−manifold M, the ring C^∞(M) is a finitely presented C^∞−ring (Theorem 2.3 of [1]). This is true because one can embed M in some Rⁿ, find anε−neighbourhoodU ⊃M and a retractionr :U →M, soC^∞(M)is a retract of C^∞(U). (cf. p. 25 of [1]).

Example 4.8. As pointed out by I. Moerdijk and G. Reyes in [1], any Weil algebra (i.e., a local ring with anR−algebra structure which, regarded as anR−vector space is finite dimensional) is aC^∞−ring. Thus, rings of (Ehresmann) jets -quaWeil algebras - are also C^∞−rings.

Example 4.9. The ring of dual numbers, R[ε] = ^R_hX^[X2i^] is a C^∞−ring, since by Borel’s Theorem(seeTheorem 1.3of [1]) we have:

R[ε]∼= C^∞(R) hx²i

Remark 4.10. Let((A_i,Φ_i)_i∈I, α_ij : (A_i,Φ_i)→ (A_j,Φ_j))be an inductive directed sys- tem ofC^∞−rings, and let(J_{`</sub>)`∈I ∈lim←−<sup>`∈I</sup>I(A<sub>`},Φ_`).

For everyi∈I, we have the maps:

αi∗ : I

lim−→^i∈I(Ai,Φi)

→ I(Ai,Φi) J 7→ α^a_i[J]

αb: I

lim−→^i∈I(A_i,Φ_i)

→ I(A_i,Φ_i) J 7→ (α^a_i[J])i∈I

and the following limit diagram:

lim←−^i∈II(A_i,Φ_i)

α^∗_i

uu

α^∗_j

**

I(A_i,Φ_i) I(A_j,Φ_j)

α^∗_ij

oo

,

where:

α_ij^∗ : I(A_j,Φ_j) → I(A_i,Φ_i) J_j 7→ α_ij^a[J_j]

We note, first, that α maps ideals of lim−→^i∈I(A_i,Φ_i) to an element of lim←−^i∈II(Ai,Φi).

In fact, given any ideal J ∈ I

lim−→^i∈I(Ai,Φi)

, since for every i ∈ I, αi is a C^∞−homomorphism, it follows that for every i ∈ I, α^∗_i(J) = α_i^a[J] is an ideal of (A_i,Φ_i), so α(J) = (α^∗_i(J))i∈I ∈ Y

i∈I

I(A_i,Φ_i). Moreover, the family (α^∗_i(J))i∈I is compatible, since:

(∀i∈I)(∀j ∈I)(ij)(α_j◦α_ij =α_i ⇒α^∗_i =α_ij^∗◦α_j^∗)

and

(∀i∈I)(∀j ∈I)(ij)(α_i^∗(J) = (α_ij^∗◦α^∗_j)(J) = α_ij^∗(α^∗_j(J)) so

α(J) = (α^∗_i(J))_i∈I ∈lim←−

i∈I

I(A_i,Φ_i).

Proposition 4.11. Let(I,)be a directed partially ordered set and {(A_i,Φ_i), α_ij : (A_i,Φ_i)→(A_j,Φ_j)}_i,j∈I

be a directed inductive system ofC^∞−rings andC^∞−homomorphisms. For everyi ∈ I,

we have the map:

α_i^∗ : I

lim−→^i∈I(A_i,Φ_i)

→ I(A_i,Φ_i) J 7→ α^a_i[J]

By the universal property of lim←−^i∈II(A_i,Φ_i), there is a unique α : I

lim−→^i∈I(A_i,Φ_i)

→ lim←−^i∈II(A_i,Φ_i) such that for everyi ∈ I the following diagram commutes:

I

lim−→^i∈I(A_i,Φ_i)

∃!α

α^∗_i

lim←−^i∈II(Ai,Φi)

πilim

←−^{i∈II(Ai,Φi) **}

I(A_i,Φ_i) that is, such thatα_i^∗ =π_i lim

←−^{i∈II(Ai,Φi)} ◦α.

We have, thus:

α: I

lim−→^i∈I(A_i,Φ_i)

→ lim←−^i∈II(A_i,Φ_i) J 7→ (α^∗_i(J))i∈I

For any(J_i)i∈I ∈lim←−^i∈II(A_i,Φ_i),S

i∈Iα_i[J_i]is an ideal oflim−→^i∈I(A_i,Φ_i)and the map:

α⁰ : lim←−^i∈II(A_i,Φ_i) → I

lim−→^i∈I(A_i,Φ_i) (J_i)_i∈I 7→ S

i∈Iα_i[J_i] is an inverse forα, soαis a bijection.

Proof. SeeProposition 14, p. 58 of [9].

The proof of the following result is similar to the proof of the above proposition:

Proposition 4.12. Let (I,) be a directed partially ordered set and {(A_i,Φ_i), α_ij : (A_i,Φ_i)→(A_j,Φ_j)}i,j∈Ibe a directed inductive system ofC^∞−rings andC^∞−

-homomorphisms. The following function is a bijection:

β : Cong

lim−→^i∈I(A_i,Φ_i)

→ lim←−^i∈ICong(A_i,Φ_i) R 7→ ((α_i×α_i)^a(R))i∈I