Furthermore A = A1 • · · · • Ak , where Ai = ei A = R · ei • Ni , and Ni is a

nilpotent ideal.

Proof. First we remark that every system of nonzero idempotents e1 , . . . , er

satisfying ei ej = 0 for i = j is linearly independent over R. Indeed, if we multiply

a linear combination k1 e1 + · · · + kr er = 0 by ei we obtain ki = 0. Consider a

non minimal idempotent e = 0. Then there exists e ∈ E with e = ee =: e = 0. ¯

Then both e and e ’ e are nonzero idempotents and e(e ’ e) = 0. To deduce the

¯ ¯ ¯ ¯

required decomposition of 1 we proceed by recurrence. Assume that we have a

decomposition 1 = e1 + · · · + er into nonzero idempotents satisfying ei ej = 0

for i = j. If ei is not minimal, we decompose it as ei = ei + (ei ’ ei ) as above.

¯ ¯

The new decomposition of 1 into r + 1 idempotents is of the same type as the

original one. Since A is ¬nite dimensional this proceedure stabilizes. This yields

1 = e1 + · · · + ek with minimal idempotents. Multiplying this relation by a

minimal idempotent e, we ¬nd that e appears exactly once in the right hand

side. Then we may decompose A as A = A1 • · · · • Ak , where Ai := ei A.

Now each Ai has only one nonzero idempotent, namely ei , and it su¬ces to

investigate each Ai separately. To simplify the notation we suppose that A = Ai ,

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298 Chapter VIII. Product preserving functors

so that now 1 is the only nonzero idempotent of A. Let N := {n ∈ A : nk =

0 for some k} be the ideal of all nilpotent elements in A.

We claim that any x ∈ A \ N is invertible. If not then xA ‚ A is a proper

ideal, and since A is ¬nite dimensional the decreasing sequence

A ⊃ xA ⊃ x2 A ⊃ · · ·

of ideals must become stationary. If xk A = 0 then x ∈ N , thus there is a k such

that xk+ A = xk A = 0 for all > 0. Then x2k A = xk A and there is some y ∈ A

with xk = x2k y. So we have (xk y)2 = xk y = 0, and since 1 is the only nontrivial

idempotent of A we have xk y = 1. So xk’1 y is an inverse of x as required.

So the quotient algebra A/N is a ¬nite dimensional ¬eld, so A/N equals R

√

or C. If A/N = C, let x ∈ A be such that x + N = ’1 ∈ C = A/N . Then

1 + x2 + N = N = 0 in C, so 1 + x2 is nilpotent and A cannot be formally real.

Thus A/N = R and A = R · 1 • N as required.

35.2. De¬nition. A Weil algebra A is a real commutative algebra with unit

which is of the form A = R · 1 • N , where N is a ¬nite dimensional ideal of

nilpotent elements.

So by lemma 35.1 a formally real and ¬nite dimensional unital commutative

algebra is the direct sum of ¬nitely many Weil algebras.

35.3. Some algebraic preliminaries. Let A be a commutative algebra with

unit and let M be a module over A. The semidirect product A[M ] of A and M

or the idealisator of M is the algebra (A — M, +, ·), where (a1 , m1 ) · (a2 , m2 ) =

(a1 a2 , a1 m2 + a2 m1 ). Then M is a (nilpotent) ideal of A[M ].

Let M m—n = {(tij ) : tij ∈ M, 1 ¤ i ¤ m, 1 ¤ j ¤ n} be the space of all

(m — n)-matrices with entries in the module M . If S ∈ Ar—m and T ∈ M m—n

then the product of matrices ST ∈ M r—n is de¬ned by the usual formula.

For a matrix U = (uij ) ∈ An—n the determinant is given by the usual formula

n

det(U ) = σ∈Sn sign σ i=1 ui,σ(i) . It is n-linear and alternating in the columns

of U .

Lemma. If m = (mi ) ∈ M n—1 is a column vector of elements in the A-module

M and if U = (uij ) ∈ An—n is a matrix with U m = 0 ∈ M n—1 then we have

det(U )mi = 0 for each i.

Proof. We may compute in the idealisator A[M ], or assume without loss of gen-

erality that all mi ∈ A. Let u—j denote the j-th column of U . Then uij mj = 0

for all i means that m1 u—1 = ’ j>1 mj u—j , thus

det(U )m1 = det(m1 u—1 , u—2 , . . . , u—n )

= det(’ mj u—j , u—2 , . . . , u—n ) = 0

j>1

Lemma. Let I be an ideal in an algebra A and let M be a ¬nitely generated

A-module. If IM = M then there is an element a ∈ I with (1 ’ a)M = 0.

n

Proof. Let M = i=1 Ami for generators mi ∈ M . Since IM = M we have

n

mi = j=1 tij mj for some T = (tij ) ∈ I n—n . This means (1n ’ T )m = 0 for

m = (mj ) ∈ M n—1 . By the ¬rst lemma we get det(1n ’ T )mj = 0 for all j. But

det(1n ’ T ) = 1 ’ a for some a ∈ I.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

35. Weil algebras and Weil functors 299

Lemma of Nakayama. Let (A, I) be a local algebra (i.e. an algebra with a

unique maximal ideal I) and let M be an A-module. Let N1 , N2 ‚ M be

submodules with N1 ¬nitely generated. If N1 ⊆ N2 +IN1 then we have N1 ⊆ N2 .

In particular IN1 = N1 implies N1 = 0.

Proof. Let IN1 = N1 . By the lemma above there is some a ∈ I with (1’a)N1 =

0. Since I is a maximal ideal (so A/I is a ¬eld), 1 ’ a is invertible. Thus

N1 = 0. If N1 ⊆ N2 + IN1 we have I((N1 + N2 )/N2 ) = (N1 + N2 )/N2 thus

(N1 + N2 )/N2 = 0 or N1 ⊆ N2 .

35.4. Lemma. Any ideal I of ¬nite codimension in the algebra of germs

∞

En := C0 (Rn , R) contains some power Mk of the maximal ideal Mn of germs

n

vanishing at 0.

Proof. Consider the chain of ideals En ⊇ I + Mn ⊇ I + M2 ⊇ · · · . Since I has

n

¬nite codimension we have I +Mk = I +Mk+1 for some k. So Mk ⊆ I +Mn Mk

n n n n

which implies Mk ⊆ I by the lemma of Nakayama 35.3 since Mk is ¬nitely

n n

generated by the monomials of order k in n variables.

35.5. Theorem. Let A be a unital real commutative algebra. Then the fol-

lowing assertions are equivalent.

(1) A is a Weil algebra.

∞

(2) A is a ¬nite dimensional quotient of an algebra of germs En = C0 (Rn , R)

for some n.

(3) A is a ¬nite dimensional quotient of an algebra R[X1 , . . . , Xn ] of poly-

nomials.

(4) A is a ¬nite dimensional quotient of an algebra R[[X1 , . . . , Xn ]] of formal

power series.

(5) A is a quotient of an algebra J0 (Rn , R) of jets.

k

Proof. Let A = R · 1 • N , where N is the maximal ideal of nilpotent ele-

ments, which is generated by ¬nitely many elements, say X1 , . . . , Xn . Since

R[X1 , . . . , Xn ] is the free real unital commutative algebra generated by these

elements, A is a quotient of this polynomial algebra. There is some k such that

xk+1 = 0 for all x ∈ N , so A is even a quotient of the jet algebra J0 (Rn , R).

k

Since the jet algebra is itself a quotient of the algebra of germs and the algebra

of formal power series, the same is true for A. That all these ¬nite dimensional

quotients are Weil algebras is clear, since they all are formally real and have

only one nonzero idempotent.

If A is a quotient of the jet algebra J0 (Rn , R), we say that the order of A is

r

at most r.

35.6. The width of a Weil algebra. Consider the square N 2 of the nilpotent

ideal N of a Weil algebra A. The dimension of the real vector space N/N 2 is

called the width of A.

Let M ‚ R[x1 , . . . , xn ] denote the ideal of all polynomials without con-

stant term and let I ‚ R[x1 , . . . , xn ] be an ideal of ¬nite codimension which

is contained in M2 . Then the width of the factor algebra A = R[x1 , . . . , xn ]/I

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300 Chapter VIII. Product preserving functors

is n. Indeed the nilpotent ideal of A is M/I and (M/I)2 = M2 /I, hence

(M/I)/(M/I)2 ∼ M/M2 is of dimension n.

=

35.7. Proposition. If M is a smooth manifold and I is an ideal of ¬nite

codimension in the algebra C ∞ (M, R), then C ∞ (M, R)/I is a direct sum of

¬nitely many Weil algebras.

If A is a ¬nite dimensional commutative real algebra with unit, then we have

Hom(C ∞ (M, R), A) = Hom(C ∞ (M, R), W (A)), where W (A) is the subalgebra

of A generated by all idempotent and nilpotent elements of A (the so-called Weil

part of A). In particular W (A) is formally real.

Proof. The algebra C ∞ (M, R) is formally real, so the ¬rst assertion follows from

lemma 35.1. If • : C ∞ (M, R) ’ A is an algebra homomorphism, then the kernel

of • is an ideal of ¬nite codimension in C ∞ (M, R), so the image of • is a direct

sum of Weil algebras and is thus generated by its idempotent and nilpotent

elements.

35.8. Lemma. Let M be a smooth manifold and let • : C ∞ (M, R) ’ A be an

algebra homomorphism into a Weil algebra A.

Then there is a point x ∈ M and some k ≥ 0 such that ker • contains the

ideal of all functions which vanish at x up to order k.

Proof. Since •(1) = 1 the kernel of • is a nontrivial ideal in C ∞ (M, R) of ¬nite

codimension.

If Λ is a closed subset of M we let C ∞ (Λ, R) denote the algebra of all real

valued functions on Λ which are restrictions of smooth functions on M . For a

smooth function f let Zf := f ’1 (0) be its zero set. For a subset S ‚ C ∞ (Λ, R)

we put ZS := {Zf : f ∈ S}.

Claim 1. Let I be an ideal of ¬nite codimension in C ∞ (Λ, R). Then ZI is a

¬nite subset of Λ and ZI = … if and only if I = C ∞ (Λ, R).

ZI is ¬nite since C ∞ (Λ, R)/I is ¬nite dimensional. Zf = … implies that f is

invertible. So if I = C ∞ (Λ, R) then {Zf : f ∈ I} is a ¬lter of nonempty closed

sets, since Zf © Zg = Zf 2 +g2 . Let h ∈ C ∞ (M, R) be a positive proper function,

i.e. inverse images under h of compact sets are compact. The square of the

geodesic distance with respect to a complete Riemannian metric on a connected

manifold M is such a function. Then we put f = h|Λ ∈ C ∞ (Λ, R). The sequence

f, f 2 , f 3 , . . . is linearly dependent mod I, since I has ¬nite codimension, so

n

g = i=1 »i f i ∈ I for some (»i ) = 0 in Rn . Then clearly Zg is compact. So this

¬lter of closed nonempty sets contains a compact set and has therefore nonempty

intersection ZI = f ∈I Zf .

Claim 2. If I is an ideal of ¬nite codimension in C ∞ (M, R) and if a function

f ∈ C ∞ (M, R) vanishes near ZI , then f ∈ I.

Let ZI ‚ U1 ‚ U 1 ‚ U2 where U1 and U2 are open in M such that f |U2 = 0.

The restriction mapping C ∞ (M, R) ’ C ∞ (M \ U1 , R) is a surjective algebra

homomorphism, so the image I of I is again an ideal of ¬nite codimension in

C ∞ (M \ U1 , R). But clearly ZI = …, so by claim 1 we have I = C ∞ (M \ U1 , R).

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993

35. Weil algebras and Weil functors 301

Thus there is some g ∈ I such that g|(M \ U1 ) = f |(M \ U1 ). Now choose

h ∈ C ∞ (M, R) such that h = 0 on U1 and h = 1 o¬ U2 . Then f = f h = gh ∈ I.

Claim 3. For the ideal ker • in C ∞ (M, R) the zero set ZI consists of one point

x only.

Since ker • is a nontrivial ideal of ¬nite codimension, Zker • is not empty and

¬nite by claim 1. For any function f ∈ C ∞ (M, R) which is 1 or 0 near the points

in Zker • the element •(f ) is an idempotent of the Weil algebra A. Since 1 is

the only nonzero idempotent of A, the zero set ZI consists of one point.

Now by claims 2 and 3 the ideal ker • contains the ideal of all functions which

∞

vanish near x. So • factors to the algebra Cx (M, R) of germs at x, compare

∞

35.5.(2). Now ker • ‚ Cx (M, R) is an ideal of ¬nite codimension, so by lemma

35.4 the result follows.

35.9. Corollary. The evaluation mapping ev : M ’ Hom(C ∞ (M, R), R),

given by ev(x)(f ) := f (x), is bijective.

This result is sometimes called the exercise of Milnor, see [Milnor-Stashe¬,

74, p. 11]. Another (similar) proof of it can be found in the mathematical short

story in the introduction to chapter VIII.

Proof. By lemma 35.8, for every • ∈ Hom(C ∞ (M, R), R) there is an x ∈ M

and a k ≥ 0 such that ker • contains the ideal of all functions vanishing at

x up to order k. Since the codimension of ker • is 1, we have ker • = {f ∈

C ∞ (M, R) : f (x) = 0}. Then for any f ∈ C ∞ (M, R) we have f ’ f (x)1 ∈ ker •,

so •(f ) = f (x).

35.10. Corollary. For two manifolds M1 and M2 the mapping

C ∞ (M1 , M2 ) ’ Hom(C ∞ (M2 , R), C ∞ (M1 , R))

f ’ (f — : g ’ g —¦ f )

is bijective.

Proof. Let x1 ∈ M1 and • ∈ Hom(C ∞ (M2 , R), C ∞ (M1 , R)). Then evx1 —¦ •

is in Hom(C ∞ (M2 , R), R), so by 35.9 there is a unique x2 ∈ M2 such that

evx1 —¦ • = evx2 . If we write x2 = f (x1 ), then f : M1 ’ M2 and •(g) = g —¦ f for

all g ∈ C ∞ (M2 , R). This also implies that f is smooth.

35.11. Chart description of Weil functors. Let A = R · 1 • N be a Weil

algebra. We want to associate to it a functor TA : Mf ’ Mf from the category

Mf of all ¬nite dimensional second countable manifolds into itself. We will give

several descriptions of this functor, and we begin with the most elementary and

basic construction, the idea of which goes back to [Weil, 53].

Step 1. If p(t) is a real polynomial, then for any a ∈ A the element p(a) ∈ A is

uniquely de¬ned; so we have a (polynomial) mapping TA (p) : A ’ A.

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302 Chapter VIII. Product preserving functors

Step 2. If f ∈ C ∞ (R, R) and »1 + n ∈ R · 1 • N = A, we consider the Taylor

∞ f (j) (») j

expansion j ∞ f (»)(t) = t of f at » and we put

j=0 j!

∞

f (j) (») j

TA (f )(»1 + n) := f (»)1 + n,

j!

j=1

which is ¬nite sum, since n is nilpotent. Then TA (f ) : A ’ A is smooth and we

get TA (f —¦ g) = TA (f ) —¦ TA (g) and TA (IdR ) = IdA .

Step 3. For f ∈ C ∞ (Rm , R) we want to de¬ne the value of TA (f ) at the vec-

tor (»1 1 + n1 , . . . , »m 1 + nm ) ∈ Am = A — . . . — A. Let again j ∞ f (»)(t) =

1± ± m

for t ∈ Rm . Then

±∈Nm ±! d f (»)t be the Taylor expansion of f at » ∈ R

we put

1±

d f (»)n±1 . . . n±m ,

TA (f )(»1 1 + n1 , . . . , »m 1 + nm ) := f (»)1 + m

1

±!

|±|≥1

which is again a ¬nite sum.

Step 4. For f ∈ C ∞ (Rm , Rk ) we apply the construction of step 3 to each com-

ponent fj : Rm ’ R of f to de¬ne TA (f ) : Am ’ Ak .

Since the Taylor expansion of a composition is the composition of the Taylor

expansions we have TA (f —¦ g) = TA (f ) —¦ TA (g) and TA (IdRm ) = IdAm .

If • : A ’ B is a homomorphism between two Weil algebras we have •k —¦

TA f = TB f —¦ •m for f ∈ C ∞ (Rm , Rk ).

Step 5. Let π = πA : A ’ A/N = R be the projection onto the quotient ¬eld

of the Weil algebra A. This is a surjective algebra homomorphism, so by step 4

the following diagram commutes for f ∈ C ∞ (Rm , Rk ):

wA

TA f

Am k

m

u u

k

πA π A

wR

f

m k

R

If U ‚ Rm is an open subset we put TA (U ) := (πA )’1 (U ) = U — N m , which is

m

an open subset in TA (Rm ) := Am . If f : U ’ V is a smooth mapping between

open subsets U and V of Rm and Rk , respectively, then the construction of steps

3 and 4, applied to the Taylor expansion of f at points in U , produces a smooth

mapping TA f : TA U ’ TA V , which ¬ts into the following commutative diagram:

wT

‘ TA f

U — Nm V — Nk

TA U AV

‘“ &

‘

u )&

pr1 π m pr1

u &

k

πA

A

wV

f

U

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35. Weil algebras and Weil functors 303

We have TA (f —¦ g) = TA f —¦ TA g and TA (IdU ) = IdTA U , so TA is now a covariant

functor on the category of open subsets of Rm ™s and smooth mappings between

them.

Step 6. In 1.14 we have proved that the separable connected smooth manifolds

are exactly the smooth retracts of open subsets in Rm ™s. If M is a smooth

manifold, let i : M ’ Rm be an embedding, let i(M ) ‚ U ‚ Rm be a tubular

neighborhood and let q : U ’ U be the projection of U with image i(M ). Then

q is smooth and q —¦ q = q. We de¬ne now TA (M ) to be the image of the smooth

retraction TA q : TA U ’ TA U , which by 1.13 is a smooth submanifold.

If f : M ’ M is a smooth mapping between manifolds, we de¬ne TA f :

TA M ’ TA M as

TA (i —¦f —¦q) TA q

TA M ‚ TA U ’ ’ ’ ’ TA U ’ ’ TA U ,

’ ’ ’’ ’’

which takes values in TA M .

It remains to show, that another choice of the data (i, U, q, Rm ) for the man-

ifold M leads to a di¬eomorphic submanifold TA M , and that TA f is uniquely

de¬ned up to conjugation with these di¬eomorphisms for M and M . Since this

is a purely formal manipulation with arrows we leave it to the reader and give

instead the following:

Step 6™. Direct construction of TA M for a manifold M using atlases.

Let M be a smooth manifold of dimension m, let (U± , u± ) be a smooth atlas

of M with chart changings u±β := u± —¦ u’1 : uβ (U±β ) ’ u± (U±β ). Then the

β

smooth mappings

wT

TA (u±β )

TA (uβ (U±β )) A (u± (U±β ))

m m

u u

πA πA

w u (U

u±β

uβ (U±β ) ±β )

±

form again a cocycle of chart changings and we may use them to glue the open

sets TA (u± (U± )) = u± (U± ) — N m ‚ TA (Rm ) = Am in order to obtain a smooth

manifold which we denote by TA M . By the diagram above we see that TA M

will be the total space of a ¬ber bundle T (πA , M ) = πA,M : TA M ’ M , since

the atlas (TA (U± ), TA (u± )) constructed just now is already a ¬ber bundle atlas.

Thus TA M is Hausdor¬, since two points xi can be separated in one chart if

they are in the same ¬ber, or they can be separated by inverse images under

πA,M of open sets in M separating their projections.

This construction does not depend on the choice of the atlas. For two atlases

have a common re¬nement and one may pass to this.

If f ∈ C ∞ (M, M ) for two manifolds M , M , we apply the functor TA to

the local representatives of f with respect to suitable atlases. This gives local

representatives which ¬t together to form a smooth mapping TA f : TA M ’

TA M . Clearly we again have TA (f —¦ g) = TA f —¦ TA g and TA (IdM ) = IdTA M , so

that TA : Mf ’ Mf is a covariant functor.

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304 Chapter VIII. Product preserving functors

35.12. Remark. If we apply the construction of 35.11, step 6™ to the algebra

A = 0, which we did not allow (1 = 0 ∈ A), then T0 M depends on the choice

of the atlas. If each chart is connected, then T0 M = π0 (M ), computing the

connected components of M . If each chart meets each connected component of

M , then T0 M is one point.

35.13. Theorem. Main properties of Weil functors. Let A = R · 1 • N

be a Weil algebra, where N is the maximal ideal of nilpotents. Then we have:

1. The construction of 35.11 de¬nes a covariant functor TA : Mf ’ Mf

such that (TA M, πA,M , M, N dim M ) is a smooth ¬ber bundle with standard ¬ber

N dim M . For any f ∈ C ∞ (M, M ) we have a commutative diagram

wT

TA f

TA M AM

πA,M πA,M

u u

wM.

f

M

So (TA , πA ) is a bundle functor on Mf , which gives a vector bundle on Mf if

and only if N is nilpotent of order 2.

2. The functor TA : Mf ’ Mf is multiplicative: it respects products.