We have shown that the deformation functor is continuous (cf. Lemma 3.2.4). Suppose that the condition (H2) holds. We can endow the tangent spacetD of D, defined by D(k[ε]), with the structure of ak-vector space as follows: Consider the local ring homomorphism which we will simply label “+”:
k[ε] ×kk[ε] −→+ k[ε]
(a+bε, a+b*ε) #→ a+ (b+b*)ε.
Let us apply D to it and use the condition (H2). We then obtain a map
+: tD ×tD →tD
called theaddition. It is easy to see that this is a law of an abelian group with zero element given byρ. Similarly, for λ∈ k, we see the local ring homomorphism
k[ε] → k[ε] a+bε #→ a+bλε.
§3.5 The Zariski tangent space and its cohomological interpretation ·25·
Applying D, we then get a map called multiplication by λ. These laws turn tD into ak-vector space.
Suppose thatA is a complete local W-algebra with residue field k which is given as a projective limitlim
←−i∈IAiof a collection of discrete artinian quotients, wherei runs through some directed index setI. We let m and mibe the maximal ideals ofA and Airespectively.
Proposition 3.5.1. The following two statements are equivalent:
(i) A is noetherian;
(ii) dimk(mi/m2i)is a bounded function of i.
Proof. Suppose that A is noetherian. Then m is generated, as an A-ideal, by a finite number d of elements of m. Since m surjects to mi, we havedimk(mi/m2i) ≤ d for each i, so (i) implies (ii).
Now assume that(ii) holds. We first claim that ma = lim
←−i∈I
ma
i for alla ≥0. The assertion is trivial fora =0, and we will proceed by induction on a. Assume the statement holds for a and consider the sequence of projective systems
0 →mia+1 →mia →mia/mia+1 →0.
Assumption(ii) implies that mai/mai+1also has bounded dimension, so the system on the right stabilizes. This implies that its limit is a finite dimensionalk-vector space N. By the induction hypothesis we have a short exact sequence
0→lim the sequence (♥) above, this gives the induction step.
We now know thatA is m-adically complete, and that m is a finitely generated A-ideal. The graded ringG(A) = 3a≥0ma/ma+1is a finitely generatedk-algebra, which is noetherian by
§3.5 The Zariski tangent space and its cohomological interpretation ·26·
Hilbert’s Basis Theorem. By [1], Corollary 10.25, this implies that A is noetherian. This show
(i). !
Definition 3.5.2. Let A be a complete noetherian local W-algebra, and let mAbe its maximal ideal.
(1) TheZariski cotangent space of A is defined to be
t∗A=mA/(m2A, mW),
where(m2A, mW) = m2A+ (image of mW)A. Note that t∗Ais a module overW/mW /k, that is, it is ak-vector space.
(2) TheZariski tangent space of A is the dual space of the cotangent space:
tA=Homk(t∗A, k).
Remark 3.5.3. Since A is noetherian, t∗Ais finite dimensional overk, so that there is no problem with the duality here.
Corollary 3.5.4. If the deformation functor D is represented by a complete local W-algebra R, then the ring R is noetherian if and only if t∗Ris finite dimensional.
Proof. Write R as a projective limit of its discrete artinian quotients Ri. Let mibe the maximal ideal of Ri. Recall thatW is noetherian, so that the k-dimension d of mW/m2W is finite. It is clear that dimk(mi/(m2i +mWRi)) anddimk(mi/m2i) differ by at mostd. Taking limit into account,t∗Ris finite dimensional overk if and only if the dimension of mi/m2i is bounded, which
by Proposition 3.5.1 is equivalent toR being noetherian. !
The following lemma gives us a functorial interpretation of the Zariski tangent space:
Lemma 3.5.5. If the deformation functor D is represented by an object R in CNLW, then there is a canonical isomorphism of k-vector spaces
tR =Homk(t∗R, k) / Homlocalg(R, k[ε]) =tD.
§3.5 The Zariski tangent space and its cohomological interpretation ·27·
Proof. Let A = k[ε] and let ϕ : R → A. Write ϕ(r) = ϕ0(r) + ϕ1(r)ε. We have, from ϕ(ab) = ϕ(a)ϕ(b), thatϕ0(ab) = ϕ0(a)ϕ0(b)and
ϕ1(ab) = ϕ0(a)ϕ1(b) +ϕ1(a)ϕ0(b).
Thus,ker(ϕ0) = mA = kε. Since ϕ is W-linear, ϕ0(r) = r = r (mod mR), and thus ϕ kills m2
R and takes mR W-linearly into kε. For r ∈ W, r = rϕ(1) = ϕ(r) = r+ϕ1(r)ε. Hence, ϕ1 killsW. Note that any element of a ∈ R can be written as a =r+x with r ∈ W and x ∈ mR. Thus, ϕ is completely determined by the restriction of ϕ1to mR, which factors throught∗R. We can then write ϕ : r+x #→ r+ϕ1(x)ε and regard ϕ1as ak-linear map from t∗R intok. Thus ϕ#→ ϕ1induces a linear mapL : Homlocalg(R, k[ε]) →Homk(t∗R, k).
Note thatR/(m2R, mW) = k⊕t∗R. For any ψ ∈ Homk(t∗R, k), we extendψ to R/m2R by declaring its value onk to be zero. Then we define ϕ : R → A by ϕ(r) =r+ψ(r)ε. Since ε2 = 0, ϕ is a W-algebra homomorphism. In particular, L(ϕ) = ψ; hence, L is surjective. Since any algebra homomorphism killing(m2R, mW)is determined by its values ont∗R,L is injective. ! Letρ : G → GL2(k) be a representation fromG into GL2(k)and letM2(k)be the set of all2×2-matrices with entries in k. We let G acts on M2(k)by the composed map
G −→ρ GL2(k) −→Ad GL(M2(k)).
Thek-vector space M2(k) with the action ofG is usually called the adjoint representation of ρ, and is denoted byAd(ρ).
Proposition 3.5.6. Suppose that the deformation functor D =Dρis represented by an object R in CNLW. Then there is a canonical isomorphism of k-vector space
tD /
−→ H1(G, Ad(ρ)).
Proof. Let ρ ∈ tD =D(k[ε])be a deformation ofρ to k[ε]. Since the maximal ideal(ε)ofk[ε]
§3.5 The Zariski tangent space and its cohomological interpretation ·28·
is principal and of square0, the map
M2(k) −→/ ker(GL2(k[ε]) →GL2(k))
X #→ 1+Xε
is an isomorphism of groups. Thus, we can liftρ to k[ε]and can compareρ and ρ. This define an elementX(σ) ∈M2(k)by
ρ(σ) =ρ(σ) +X(σ)ρ(σ)ε.
Moreover,σ #→ X(σ)is a1-cocycle for the adjoint action:
ρ(στ) = ρ(στ) +X(στ)ρ(στ)ε ρ(σ) = ρ(σ) +X(σ)ρ(σ)ε ρ(τ) = ρ(τ) +X(τ)ρ(τ)ε
ρ(σ)ρ(τ) = [ρ(σ) +X(σ)ρ(σ)ε] · [ρ(σ)−1ρ(στ) +X(τ)ρ(σ)−1ρ(στ)ε]
= ρ(στ) + [X(σ) +Ad ρ(σ)X(τ)]ρ(στ)ε.
Conversely, given an1-cocycle X : G →M2(k),
ρ(σ) = ρ(σ) +X(σ)ρ(σ)ε
defines a deformation ofρ over k[ε], hence a classρ∈ D(k[ε]).
Furthermore, ifX is a coboundary, we have X(σ) = (Ad ρ(σ) −1)Y for some Y ∈ M2(k). We have the following computation
ρ(σ) = ρ(σ) +X(σ)ρ(σ)ε = ρ(σ) + (Ad ρ(σ)Y−Y)ρ(σ)ε
= (1−Yε) · [ρ(σ) +ρ(σ)(Ad ρ(σ)Y)ε]
= (1−Yε)ρ(σ)(1+Yε).
Hence, We conclude thatX is a coboundary if and only if ρ is conjugate to ρ for some element inker(GL2(k[ε]) → GL2(k)). To complete the proof, we note that the zero element oftD isρ
§3.5 The Zariski tangent space and its cohomological interpretation ·29·
and it is sent to0 in H1(G, Ad(ρ)). !
Corollary 3.5.7. Suppose that G satisfies the p-finiteness condition Φp. If the deformation func-tor D =Dρis represented by an object R in CNLW, then tD is a finite dimensional k-vector space.
Proof. Let G0 =ker(ρ). This is an open subgroup ofG and the action of G0onAd(ρ)is trivial.
Note that
H0(G/G0, H1(G0, Ad(ρ))) =Hom(G0, M2(k)) = Hom(G0, k) ⊗kM2(k)
= Hom(Fr(G0), k) ⊗kM2(k)
whereFr(G)is the pro-p-Frattini quotient of G0. The inflation-restriction sequence yields the left exact sequence
0−→ H1(G/G0, H0(G0, Ad(ρ))) −→ H1(G, Ad(ρ)) −→H0(G/G0, H1(G0, Ad(ρ))).
The term on the left is finite sinceG/G0andH0(G0, Ad(ρ))are finite. The term on the right is finite because of the p-finiteness condition ΦpforG. Hence, this lemma is proved. !
§3.5 The Zariski tangent space and its cohomological interpretation ·30·
Chapter 4
The Existence of the Universal Deformation
Diese beklagen, daß man heute zu viel abstrakte Math-ematik lernen muß, bevor man sinnvoll arbeiten kann.
Diese Entwicklung ist zwar zu bedauern, doch darf man nicht ¨ubersehen, daßsie uns andererseits m¨achtige Hilfsmit-tel in die Hand gibt, und es erlaubt, komplizierte Sachver-halte einfach und klar darzustellen. Wer diese Metho-den ablehnt, wird bei seinen Forschungen meist an der Oberfl¨ache bleiben m¨ussen.
Gred Faltings
Let G beGL1orGL2. We will give rigidity conditions of the representationρ : G → G(k) and verify those conditions of Schlessinger’s criteria for our fixed deformation functor D ofρ in this chapter. In the process, we shall see where these assumptions are needed.