nLab
Fermat theory

Context

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Contents

Idea

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

The term Fermat theory seems to have been introduced in (Kock 09) based on (Dubuc-Kock 84). But as the name suggests, it has its roots in an old observation of Fermat. Namely: if f:f \;\colon\; \mathbb{R} \longrightarrow \mathbb{R} is a polynomial function, then

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

for a unique polynomial function f˜: 2\tilde{f} \colon \mathbb{R}^2 \to \mathbb{R}. Clearly

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

for y0y \ne 0, but the interesting thing is that

f˜(x,0)=f(x)\tilde{f}(x,0) = f'(x)

So, the function f˜\tilde{f} knows about the derivative of ff! (This can be done for polynomials over any commutative ring, although Fermat wasn't working in that generality.)

Later Jacques Hadamard generalized this observation from a polynomial function ff to a continuously differentiable function ff, where now f˜\tilde{f} is unique if required to be continuous. This is the statement of the Hadamard lemma. (For a merely differentiable function ff, require f˜\tilde{f} to be continuous in yy alone.) The function f˜\tilde{f} is thus called a Hadamard quotient. If f˜\tilde{f} is to be the same class of function as ff, then we need smooth functions, and that will be our motivating context from now on.

If we take f˜(x,0)=f(x)\tilde{f}(x,0) = f'(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)+yf˜(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)=f+g(f + g)' = f' + g'
(fg)=fg+fg(f g)' = f' g + f g'
(fg)=(fg)g(f \circ g)' = (f' \circ g) g'

These ideas continue to work if ff is a smooth function from n\mathbb{R}^n to \mathbb{R}; focussing on one variable and treating the others as parameter?s, we have partial differentiation.

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)(n+1)-ary operation ff there is a unique (n+2)(n+2)-ary operation f˜\tilde{f} such that

f(x+y,z)=f(x,z)+yf˜(x,y,z) f(x + y, \vec{z}) = f(x, \vec{z}) + y \tilde{f}(x,y,\vec{z})

where z\vec{z} is a list of nn variables which act as parameters. (Here we are abusing language by writing the operations ff and f˜\tilde{f} as if they were functions, to avoid unintuitive commutative diagrams.)

Examples

C C^\infty-rings

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

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

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

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

that sends each object nCartSp\mathbb{R}^n \in CartSp to the hom-set in the functor category [CartSp op,Set][CartSp^{op},Set] from XX to the presheaf represented by n\mathbb{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

Let TT be a Fermat theory and let ff be an (n+1)(n+1)-ary operation, then we may define an operation 1f\partial_1 f

by

1(x,z)=f˜(x,0,z)\partial_1(x, \vec{z}) = \tilde{f}(x,0,\vec{z})

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

i:T(n)T(n)\partial_i : T(n) \to T(n)

for 1in1 \le i \le n.

Now T(n)T(n) is automatically an algebra of TT (this is true for any Lawvere theory: it is the free algebra on nn generators), whence T(n)T(n) is a commutative ring. One can check that each map

i:T(n)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 TT be a Fermat theory, and let AA be a TT-algebra. A module NN over AA is simply a module for the underlying ring of AA.

But the notion of derivation δ:AN\delta : A \to N of such modules depends on the TT-structure:

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

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

  2. δ(λa)=λδ(a)\delta(\lambda a) = \lambda \delta(a)

  3. δ(ab)=aδ(a)+bδ(b)\delta(a \cdot b) = a \delta(a) + b \delta(b)

are equivalent to the condition that for any polynomial p[x 1,,x n]p \in \mathbb{R}[x_1, \cdots, x_n] and ring elements a ia_i we have

δ(p(a 1,,a n))= i=1 npx i(a 1,,a n)δ(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+yp_1(x,y) = x + y, p 2(x)=λap_2(x) = \lambda a and p 3(x,y)=xyp_3(x,y) = x y.)

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

Definition

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

δ(f(a 1,,a n))= i=1 nfx i(a 1,,a n)δ(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 TT-algebra AA is a derivation of the underlying ring. (This follows again by using the above three polynomials and remembering that by definition T(n)T(n) at least contains all polynomials.)

Kähler differentials

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

Theorem

The map

d:= 1,, n:T(n) i=1 nT(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 TT-derivation of T(n,1)T(n,1).

This means that if NN is a module of T(n)T(n) and δ:T(n)N\delta : T(n) \to N is a derivation in the above sense, then δ\delta factors uniquely through the map 1,, n\langle \partial_1, \dots, \partial_n \rangle.

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

We may think of an element (f i) i=1 nT(n)(f_i) \in \prod_{i = 1}^{n} T(n) as the Kähler differential 1-form f 1dx 1+f 2dx 2++f ndx nf_1 d x^1 + f_2 d x^2 + \cdots + f_n d x^n and of the derivation d:= 1,, nd := \langle \partial_1, \dots, \partial_n \rangle as the operation

d:f if ix idx 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

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

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

For more, see:

and the comments on this blog entry.

Refinement to supergeometry and extension to a notion of super Fermat theory is discussed in

Something similar appears in def. 1.1, 1.2 of

For more on this see at synthetic differential supergeometry.

Revised on March 13, 2014 22:23:18 by Noam Zeilberger (176.189.43.179)