nLab
Fermat theory

Idea

A ‘Fermat theory’ is a Lawvere theory that extends the usual theory of commutative rings by permitting differentiation.

The idea of a Fermat theory seems to have been invented by Anders Kock. But as the name suggests, it has its roots in an old observation of Fermat. Namely: if $f:ℝ\to ℝ$ is a smooth function, then

$$f(x+y)=f(x)+y\tilde{f}(x,y)$$f (x+y) = f(x) + y \tilde{f}(x,y)

for a unique smooth function $\tilde{f}:{ℝ}^2\to ℝ$. Clearly

$$\tilde{f}(x,y)=\frac{f(x+y)-f(x)}{y}$$\tilde{f}(x,y) = \frac{f(x+y) - f(x)}{y}

for $y\ne 0$, but the interesting thing is that

$$\tilde{f}(x\mathrm{,0})=f\prime (x)$$\tilde{f}(x,0) = f'(x)

So, the function $\tilde{f}$ knows about the derivative of $f$!

Fermat noted this for $f$ a polynomial; Hadamard generalized it to smooth $f$. The function $\tilde{f}$ is also called a Hadamard quotient. See Hadamard lemma.

If we take $\tilde{f}(x\mathrm{,0})=f\prime (x)$ as a definition of the derivative, we can derive many of the basic rules for derivatives from the formula

$$f(x+y)=f(x)+y\tilde{f}(x,y)$$f(x+y) = f(x) + y \tilde{f}(x,y)

using just algebra — no limits! As an exercise, the reader should check these rules:

$$(f+g)\prime =f\prime +g\prime$$(f + g)' = f' + g'

$$(fg)\prime =f\prime g+fg\prime$$(f g)' = f' g + f g'

$$(f\circ g)\prime =(f\prime \circ g)g\prime$$(f \circ g)' = (f' \circ g) g'

All these ideas also work if $f$ is parametrized, that is, a smooth function from $ℝ\times {ℝ}^n$ to $ℝ$, where the $n$ extra variables serve as ‘parameters’.

Definition

The above observations suggest defining the following kind of Lawvere theory. A Fermat theory is an extension of the algebraic theory of commutative rings, such that for any $(n+1)$-ary operation $f$ there is a unique $(n+2)$-ary operation $\tilde{f}$ such that

$$f(x+y,\stackrel{\rightharpoonup }{z})=f(x,\stackrel{\rightharpoonup }{z})+y\tilde{f}(x,y,\stackrel{\rightharpoonup }{z})$$f(x + y, \vec{z}) =f(x, \vec{z}) + y \tilde{f}(x,y,\vec{z})

where $\stackrel{\rightharpoonup }{z}$ is a list of $n$ variables which act as parameters. Here we are abusing language by writing the operations $f$ and $\tilde{f}$ as if they were functions, to avoid unintuitive commutative diagrams.

Examples

$C^{\mathrm{\infty }}$-rings

There is a Lawvere theory called the theory of $C^{\mathrm{\infty }}$-rings, whose $n$-ary operations are the smooth maps $f:{ℝ}^n\to ℝ$,

$$T(n):=C^{\mathrm{\infty }}({ℝ}^n,ℝ),$$T(n) := C^\infty(\mathbb{R}^n, \mathbb{R}) \,,

with composition of operations defined in the obvious way. An algebra of this Lawvere theory is called a C∞-ring.

The theory of $C^{\mathrm{\infty }}$-rings is a Fermat theory. For any smooth manifold $M$, the algebra of smooth real-valued functions $C^{\mathrm{\infty }}(M)$ is a $C^{\mathrm{\infty }}$ ring. More generally, if $M$ is any diffeological space, Chen space or Frolicher space, we can define $C^{\mathrm{\infty }}(M)$, and this will be a $C^{\mathrm{\infty }}$-ring.

In formulas, and even more generally: for any generalized space given by a presheaf $X$ on CartSp, the corresponding $C^{\mathrm{\infty }}$-ring is the copresheaf

$$C^{\mathrm{\infty }}(X):{ℝ}^n\mapsto [{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{Set}](X,Y({ℝ}^n))$$C^\infty(X) : \mathbb{R}^n \mapsto [CartSp^{op},Set](X,Y(\mathbb{R}^n))

that sends each object ${ℝ}^n\in \mathrm{CartSp}$ to the hom-set in the functor category $[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{Set}]$ from $X$ to the presheaf represented by ${ℝ}^n$ under the Yoneda embedding. By the canonical right exactness of the hom-functor, this preserves limits and hence in particular products in CartSp.

Partial derivatives

If $T$ is a Fermat theory and $f$ is any $(n+1)$-ary operation, we may define an operation ${\partial }_{1}f$

by

$${\partial }_1(x,\stackrel{\rightharpoonup }{z})=\tilde{f}(x\mathrm{,0},\stackrel{\rightharpoonup }{z})$$\partial_1(x, \vec{z}) = \tilde{f}(x,0,\vec{z})

This acts like the partial derivative of $f$ with respect to its first argument. With a bit of more work we get a list of $n$-ary operations ${\partial }_{i}f$. So, if $T(n)$ denotes the set of $n$-ary operations in our algebraic theory $T$, we get maps

$${\partial }_i:T(n)\to T(n)$$\partial_i : T(n) \to T(n)

for $1\le i\le n$.

For any Lawvere theory, $T(n)$ is an algebra of that theory — the free $T$-algebra on $n$ generators. So, when $T$ is a Fermat theory, $T(n)$ is automatically a commutative ring. One can check that each map

$${\partial }_i:T(n)\to T(n)$$\partial_i : T(n) \to T(n)

is a derivation of this ring — this is really just the chain rule.

Modules and derivations

Let $A$ be a $T$-algebra. A module $N$ over $A$ is simply a module for the underlying ring of $A$.

But the notion of derivation $\delta :A\to N$ of such modules depends on the $T$-structure:

To motivate the concept, let first $A$ be an ordinary ring and $N$ an ordinary module. Then the three axioms of an ordinary derivation $\delta :A\to N$

1. $\delta (a+b)=\delta (a)+\delta (b)$

2. $\delta (\lambda a)=\lambda \delta (a)$

3. $\delta (a\cdot b)=a\delta (a)+b\delta (b)$

are equivalent to the condition that for any polynomial $p\in ℝ[x_1,\cdots ,x_n]$ and ring elements $a_i$ we have

$$\delta (p(a_1,\cdots ,a_n))=\sum _{i=1}^n\frac{\partial p}{\partial x_i}(a_1,\cdots ,a_n)\delta (a_i).$$\delta\left( p(a_1, \cdots, a_n) \right) = \sum_{i= 1 }^{n} \frac{\partial p}{\partial x_i} \left( a_1, \cdots, a_n \right) \delta(a_i) \,.

(It is immediate that the first three axioms imply this one. To see the converse, apply the latter to the polynomials $p_1(x,y)=x+y$, $p_2(x)=\lambda a$ and $p_3(x,y)=xy$.)

The definition of derivations for general $T$-algebras now follows the last expression, using the notion of partial derivatives from above:

Definition

For $T$ a Fermat theory, $A$ a $T$-algebra and $N$ an $A$-module, a derivation $\delta :A\to N$ is a map such that for each $f\in T(n)$ and elements $(a_i\in A)$ we have

$$\delta (f(a_1,\cdots ,a_n))=\sum _{i=1}^n\frac{\partial f}{\partial x_i}(a_1,\cdots ,a_n)\delta (a_i).$$\delta\left( f(a_1, \cdots, a_n) \right) = \sum_{i= 1 }^{n} \frac{\partial f}{\partial x_i} \left( a_1, \cdots, a_n \right) \delta(a_i) \,.

Notice that in particular such a derivation of a $T$-algebra $A$ is a derivation of the underlying ring. (This follows again by using the above three polynomials and remembering that by definition $T(n)$ at least contains all polynomials.)

Kähler differentials

The sets $T(n)$ for $n\in \mathbb{N}$ canonically have the structures of modules over $T(n)$.

Theorem

The map

$$d:=⟨{\partial }_1,\dots ,{\partial }_n⟩:T(n)\to \prod _{i=1}^{n}T(n)$$d := \langle \partial_1, \dots, \partial_n \rangle : T(n) \to \prod_{i = 1}^{n} T(n)

obtained from the partial derivatives is the universal $T$-derivation of $T(n\mathrm{,1})$.

This means that if $N$ is a module of $T(n)$ and $\delta :T(n)\to N$ is a derivation in the above sense, then $\delta$ factors uniquely through the map $⟨{\partial }_1,\dots ,{\partial }_n⟩$.

The point of this theorem is that it gives us a version of Kähler differentials for $T(n)$.

We may think of an element $(f_i)\in {\prod }_{i=1}^{n}T(n)$ as the Kähler differential 1-form $f_{1}d{x}^1+f_{2}d{x}^2+\cdots +f_{n}d{x}^n$ and of the derivation $d:=⟨{\partial }_1,\dots ,{\partial }_n⟩$ as the operation

$$d:f\mapsto \sum _i\frac{\partial f_i}{\partial x^i}d{x}^i.$$d : f \mapsto \sum_i \frac{\partial f_i}{ \partial x^i} d x^i \,.

Indeed, when the Fermat theory is that of C-infinity rings, then this notion of Kähler differentials does coincide with the ordinary notion of smooth 1-form. The same is not true, in general, if one instead forms ring-theoretic Kähler differentials.

References

The original reference is

• E. Dubuc, Anders Kock, On 1-form classifiers, Comm. Alg. 12 (1984), no. 11-12, 1471–1531 (pdf)

Parts of the above material are a summary of the following talk:

• Anders Kock, Kähler differentials for Fermat theories, Fields Workshop on Smooth Structures in Logic, Category Theory and Physics, May 1, 2009, University of Ottawa. (abstract)

For more, see:

• Alex Hoffnung, Smooth Structure in Ottawa II. (blog entry)

and the comments on this blog entry.