Exercises. II
A- Let A be a ring and M an A-module. M is said to be free (of
finite type) if it is isomorphic to An for some n ∈ N. We say that M
is locally free if for every p ∈ Spec(A), there exists f ∈ A\p such that
Mf is a free Af -module.
(1) Let S ⊂ A be a multiplicative subset and M a locally free
A-module. Show that S −1 M is a locally free S −1 A-module.
(2) Show that if A is a local ring, any locally free A-module is free.
(3) Assume that M is finitely presented. Show that M is locally
free if and only if Mp is a free Ap -module for all prime ideals
p ⊂ A.
(4) Show that MP
is locally free if and only if there exists f1 , . . . , fr ∈
A such that ri=1 fi = 1 and Mfi are free Afi -modules.
(5) Let k be a field. Show that every locally free k[T ]-module is
free.
(6) Consider the ring A = k[U, V ]/(U 2 − V 3 ). Show that the ideal
(U − 1, V − 1) is a locally free A-module which is not free.
B- Let A = C 0 ([0, 1]) be the ring of continuous real-valued functions
on the unit segment [0, 1].
(1) For x ∈ [0, 1] let px ⊂ A be the set of continuous functions f
such that f (x) = 0. Show that px is a maximal ideal of A.
(2) Let a ⊂ A be an ideal. Let V(a) be the set of x ∈ [0, 1] such
that a ⊂ px . Show that V(a) is a closed subset of [0, 1]. (Hint:
Show that the complement of V(a) is open.)
(3) Conversely, given a closed subset Z ⊂ [0, 1], let I(Z) be the
ideal of functions vanishing on Z. Show that V(I(Z)) = Z.
(4) Let a ⊂ A be an ideal such that V(a) = ∅. Show that a = A.
(5) Let p be a prime ideal of A. Show that [0, 1]\V(p) contains at
most two connected components. (Hint: Assume that [0, 1]\V(p)
contains three open intervals ]a, b[, ]c, d[ and ]u, v[ with a < b ≤
c < d ≤ u < v and such that [b, c] ∩ V(p) and [d, u] ∩ V(p)
are both non-empty. Construct a continuous function θ1 (resp.
θ2 ) which vanishes exactly at [c, v] (resp. [a, d]). Using that
0 = θ1 .θ2 ∈ p, deduce a contradiction.) Deduce then that V(p)
contains exactly one element.
(6) Show that all maximal ideals of A are of the form px for some
x ∈ [0, 1]. Are there prime ideals in A which are not of the form
1
2
−1
px ? (Hint: Consider the function θ(x) = e−x extended on 0
by θ(0) = 0. The class of f (x) = x in A/(θ) is not a nilpotent
element. Hence, there exists a prime ideal q in A/(θ) such that
f + (θ) 6∈ q...)
C- (The algebra of smooth functions) Let A be an R-algebra. Denote
SpecR (A) ⊂ Spec(A) the set of all maximal ideals of A whose residue
field is R. We call it the real spectrum of A.
(1) Prove that
SpecR (A) = HomR−alg (A, R).
We call the Zariski topology on SpecR (A) to the topology induced
in SpecR (A) by the Zariski topology in Spec(A). We call the Gelfand
topology on SpecR (A) to the coarsest topology which makes the functions a : SpecR A → R, a 7→ [a] ∈ A/m = R continuous.
Let M ⊂ Rn be a compact smooth manifold1. Denote C ∞ (M ) the
R-algebra of smooth functions on M . Every point x ∈ M defines a
maximal ideal
px : = {f ∈ C ∞ (M ) : f (x) = 0}
(2) The map Φ : M −→ SpecR (C ∞ (M )), x 7→ px is an homeomorphism, considering the Zariski topology. In addition, both the
Gelfand and Zariski topology coincide.
(a) Prove Φ is injective (Hint: given two points x, y ∈ M
construct a smooth function f such that f (x) = 0 and
f (y) 6= 0).
T
(b) Consider m ∈ SpecR (C ∞ (M )), prove that if g∈m (g)0 6= ∅
(here (g)0 = {m ∈ SpecR (C ∞ (M )) : g ∈ m}) then there
exist a point x ∈ M such that m = px .
T
(c) Consider m ∈ SpecR (C ∞ (M )) prove that if g∈m (g)0 = ∅
then there exist g1 , · · · , gn ∈ m such that (g1 )0 ∩ · · · ∩
(gn )0 = ∅ (Hint: note that M is compact). T
(d) Consider m ∈ SpecR (C ∞ (M )), prove that g∈m (g)0 6= ∅
(Hint: If not, use (c) to construct an invertible smooth
function which belongs to m). Conclude Φ is surjective.
(e) Conclude Φ is an homeomorphism (Hint: Check all closed
subsets Y ⊂ M are of the form V (I) for an ideal I ⊂
C ∞ (M )).
1All
results of this exercise actually hold for any smooth manifold. Additional
assumptions are intended only for simplicity.