Normal spaces

In the next section we will see that in the ´etale topology we can describe the local structure of representation varieties in the neighborhood of a closed orbit in terms of the normal space to this orbit. In this section we will give a representation theoretic description of this normal space.

We recall some standard facts about tangent spaces first. Let Xbe a not necessarily reduced affine variety with coordinate ring C[X] = C[x1, . . . , xn]/I. If the origin o = (0, . . . ,0) ∈ V(I), elements ofI have no constant terms and we can write anyp∈Ias

p=

∞

X

i=1

p⁽ⁱ⁾ withp⁽ⁱ⁾ homogeneous of degreei.

Theorder ord(p) is the least integerr≥1 such thatp^(r) 6= 0. Define the following two ideals in C[x1, . . . , xn]

Il={p⁽¹⁾|p∈I} andIm={p^(r)|p∈I andord(p) =r}.

3.7. Normal spaces 147

•

• '''' '''' '''' '''' ''

Figure 3.9: The Artin-Mumford bundle

The subscriptsl(respectivelym) stand forlinear terms(respectively, terms ofminimal degree).

The tangent space to X in o, To(X) is by definition the subscheme of Cⁿ determined by Il. Observe that

Il= (a11x1+. . .+a1nxn, . . . , al1x1+. . .+alnxn)

for somel×nmatrixA= (aij)i,j of rankl. That is, we can express allxkas linear combinations of some{xi₁, . . . , xin−l}, but then clearly

C[To(X)] =C[x1, . . . , xn]/Il=C[xi1, . . . , xin−l]

In particular,To(X) is reduced and is a linear subspace of dimensionn−linCⁿthrough the point o.

Next, consider an arbitrary geometric pointxofXwith coordinates (a1, . . . , an). We can trans-latexto the originoand the translate ofXis then the scheme defined by the ideal

(f1(x1+a1, . . . , xn+an), . . . , fk(x1+a1, . . . , xn+an))

Now, the linear term of the translated polynomialfi(x1+a1, . . . , xn+an) is equal to

In particular, the dimension of this linear subspace can be computed from theJacobian matrixin xassociated with the polynomials (f1, . . . , fk)

Hence, we can also identify the tangent space toXinxwith the algebra morphismsC[X] ^φ- _C[ε]

whose composition with the projection π :C[ε] -- _C (sending ε to zero) is the evaluation in x= (a1, . . . , an). That is, letevx∈X(C) be the point corresponding to evaluation inx, then

Tx(X) ={φ∈X(C[ε])|X(π)(φ) =evx}.

The following two examples compute the tangent spaces to the (trace preserving) representation varieties.

Example 3.10 (Tangent space torep_n) LetAbe an affineC-algebra generated by{a1, . . . am} and ρ : A - Mn(C) an algebra morphism, that is, ρ ∈ repn A. We call a linear map A ^D- Mn(C) aρ-derivationif and only if for alla, a⁰∈Awe have that

D(aa⁰) =D(a).ρ(a⁰) +ρ(a).D(a⁰).

3.7. Normal spaces 149

We denote the vectorspace of allρ-derivations ofAbyDerρ(A). Observe that anyρ-derivation is determined by its image on the generatorsai, henceDerρ(A)⊂M_n^m. We claim that

Tρ(rep_nA) =Derρ(A).

Indeed, we know thatrep_n A(C[ε]) is the set of algebra morphisms A ^φ- Mn(C[ε])

By the functorial characterization of tangentspaces we have thatTρ(rep_nA) is equal to {D:A - Mn(C) linear|ρ+Dε:A - Mn(C[ε]) is an algebra map}.

Becauseρis an algebra morphism, the algebra map condition

ρ(aa⁰) +D(aa⁰)ε= (ρ(a) +D(a)ε).(ρ(a⁰) +D(a⁰)ε) is equivalent toD being aρ-derivation.

Example 3.11 (Tangent space totrep_n) LetAbe a Cayley-Hamilton algebra of degreenwith trace maptrA and trace generated by {a1, . . . , am}. Letρ∈trepnA, that is,ρ:A - Mn(C) is atrace preservingalgebra morphism. Because trep_n A(C[ε]) is the set of all trace preserving algebra morphismsA - Mn(C[ε]) (with the usual trace maptronMn(C[ε])) and the previous example one verifies that

Tρ(trep_nA) =Der^tr_ρ(A)⊂Derρ(A) the subset oftrace preservingρ-derivationsD, that is, those satisfying

D◦trA=tr◦D

A ^D- Mn(C)

A

tr_A

? _D

- Mn(C)

tr

?

Again, using this property and the fact thatAistracegenerated by{a1, . . . , am}a trace preserving ρ-derivation is determined by its image on theai so is a subspace ofM_n^m.

Thetangent cone to Xin o,T Co(X), is by definition the subscheme ofCⁿ determined by Im, that is,

C[T Co(X)] =C[x1, . . . , xn]/Im.

It is called aconebecause ifcis a point of the underlying variety ofT Co(X), then the linel=−→ocis contained in this variety becauseImis a graded ideal. Further, observe that asIl⊂Im, the tangent

cone is a closed subscheme of the tangent space atX ino. Again, ifx is an arbitrary geometric point ofXwe define thetangent cone toXinx,T Cx(X) as the tangent coneT Co(X⁰) whereX⁰ is the translated scheme ofXunder the translation takingxtoo. Both the tangent space and tangent cone containlocal informationof the schemeXin a neighborhood ofx.

Letmxbe the maximal ideal ofC[X] corresponding tox(that is, the ideal of polynomial functions vanishing inx). Then, its complementSx=C[X]−mx is a multiplicatively closed subset and the local algebra Ox(X) is the corresponding localizationC[X]S_x. It has a unique maximal idealmxwith residue fieldOx(X)/mx =C. We equip the local algebraOx =Ox(X) with the mx-adic filtration that is the increasingZ-filtration

Fx: ...⊂mⁱ⊂mⁱ⁻¹⊂. . .⊂m⊂ Ox=Ox=. . .=Ox=. . . withassociated graded algebra

gr(Ox) = . . .⊕ mⁱx

mⁱ⁺¹x

⊕mⁱ⁻¹x

mⁱx

⊕. . .⊕mx

m²x

⊕C⊕0⊕. . .⊕0⊕. . . Proposition 3.12 Ifxis a geometric point of the affine schemeX, then

1. C[Tx(X)]is isomorphic to the polynomial algebra C[^m_m^x2 x].

2. C[T Cx(X)]is isomorphic to the associated graded algebragr(Ox(X)).

Proof. After translating we may assume thatx=olies inV(I)^⊂ - _Cⁿ. That is, C[X] =C[x1, . . . , xn]/I and mx= (x1, . . . , xn)/I.

(1) : Under these identifications we have mx

m²_x 'mx/m²_x

'(x1, . . . , xn)/((x1, . . . , xn)²+I) '(x1, . . . , xn)/((x1, . . . , xn)²+Il)

and asIl is generated by linear terms it follows that the polynomial algebra on ^m_m^x₂

x is isomorphic to the quotient algebraC[x1, . . . , xn]/Il which is by definition the coordinate ring of the tangent space.

(2) : Again using the above identifications we have gr(Ox)' ⊕^∞i=0mⁱ_x/mⁱ⁺¹_x

' ⊕^∞i=0mⁱ_x/mⁱ⁺¹_x

' ⊕^∞_i=0(x1, . . . , xn)ⁱ/((x1, . . . , xn)ⁱ⁺¹+ (I∩(x1, . . . , xn)ⁱ)) ' ⊕^∞i=0(x1, . . . , xn)ⁱ/((x1, . . . , xn)ⁱ⁺¹+Im(i))

3.7. Normal spaces 151

whereIm(i) is the homogeneous part ofIm of degreei. On the other hand, thei-th homogeneous part ofC[x1, . . . , xn]/Imis equal to

(x1, . . . , xn)ⁱ (x1, . . . , xn)ⁱ⁺¹+Im(i)

we obtain the required isomorphism.

This gives a third interpretation of the tangent space as Tx(X) =Hom_C(mx

m²_x,C) =Hom_C(mx

m²x

,C).

Hence, we can also view the tangent space Tx(X) as the space of point derivations Derx(Ox) on Ox(X) (or of the point derivationsDerx(C[X]) onC[X]). That is,C-linear mapsD:Ox - _C(or D:C[X] - _C) such that for all functionsf, gwe have

D(f g) =D(f)g(x) +f(x)D(g).

If we define thelocal dimension of an affine schemeXin a geometric pointx dimx Xto be the maximal dimension of irreducible components of the reduced varietyX passing throughx, then

dimx X=dimoT Cx(X).

We say thatX is nonsingular atx (or equivalently, thatxis a nonsingular point of X) if the tangent cone toXinxcoincides with the tangent space toXinx. An immediate consequence is Proposition 3.13 IfXis nonsingular atx, then Ox(X)is a domain. That is, in a Zariski neigh-borhood ofx,Xis an irreducible variety.

Proof. IfXis nonsingular atx, then

gr(Ox)'C[T Cx(X)] =C[Tx(X)]

the latter one being a polynomial algebra whence a domain. Now, let 06=a, b ∈ Ox then there existk, lsuch thata∈m^k−m^k+1andb∈m^l−m^l+1, that isais a nonzero homogeneous element of gr(Ox) of degree−k and bone of degree −l. But then, a.b∈ m^k+l−m^k+l−1 hence certainly a.b6= 0 inOx.

Now, consider the natural mapφ:C[X] - Ox. Let{P1, . . . , Pl}be the minimal prime ideals ofC[X]. All but one of them, sayP1 =φ⁻¹(0), extend to the whole ringOx. Taking the product off functionsfi∈Pinonvanishing inxfor 2≤i≤lgives the Zariski open setX(f) containingx and whose coordinate ring is a domain, whenceX(f) is an affine irreducible variety.

When restricting to nonsingular points we reduce to irreducible affine varieties. From the Jaco-bian condition it follows that nonsingularity is a Zariski open condition onXand by the implicit function theorem,Xis a complex manifold in a neighborhood of a nonsingular point.

Let X ^φ- Y be a morphism of affine varieties corresponding to the algebra morphism C[Y] ^φ

∗

- _C[X]. Let x be a geometric point ofX and y = φ(x). As φ^∗(my) ⊂ mx, φ induces a linear map^m_m^y2

y - ^m_m^x2

x and taking the dual map gives thedifferential ofφinxwhich is a linear map

dφx:Tx(X) - Tφ(x)(Y).

AssumeXa closed subscheme ofCⁿand Ya closed subscheme ofC^m and letφbe determined by thempolynomials{f1, . . . , fm}inC[x1, . . . , xn]. Then, the Jacobian matrix inx

defines a linear map from Cⁿ to C^m and the differential dφx is the induced linear map from Tx(X)⊂Cⁿ toTφ(x)(Y)⊂C^m. LetD∈Tx(X) =Derx(C[X]) and xD the corresponding element of X(C[ε]) defined byxD(f) =f(x) +D(f)ε, thenxD◦φ^∗∈Y(C[ε]) is defined by

xD◦φ^∗(g) =g(φ(x)) + (D◦φ^∗)ε=g(φ(x)) +dφx(D)ε giving us theε-interpretation of the differential

φ(x+vε) =φ(x) +dφx(v)ε for allv∈Tx(X).

Proposition 3.14 Let X ^φ- Y be a dominant morphism between irreducible affine varieties.

There is a Zariski open dense subsetU ^⊂ - X such thatdφx is surjective for allx∈U. Proof. We may assume thatφfactorizes into

X -^ρ- Y ×C^d

-withφa finite and surjective morphism. Because the tangent space of a product is the sum of the tangent spaces of the components we have thatd(prW)z is surjective for allz∈Y ×C^d, hence it

3.7. Normal spaces 153

suffices to verify the claim for afinitemorphismφ. That is, we may assume that S =C[Y] is a finite module overR=C[X] and letL/Kbe the corresponding extension of the function fields. By theprincipal element theoremwe know thatL=K[s] for an elements∈L which is integral over Rwith minimal polynomial

F=tⁿ+gn−1tⁿ⁻¹+. . .+g1t+g0 withgi∈R

Consider the ringS⁰=R[t]/(F) then there is an elementr∈Rsuch that the localizationsSr⁰ and Sr are isomorphic. By restricting we may assume thatX=V(F) ^⊂- Y ×Cand that

X=V(F)^⊂- Y ×C

Y

pr_Y

?

φ

-Letx= (y, c)∈X then we have (again using the identification of the tangent space of a product with the sum of the tangent spaces of the components) that

Tx(X) ={(v, a)∈Ty(Y)⊕C|c∂F

∂t(x) +vgn−1cⁿ⁻¹+. . .+vg1c+vg0= 0}.

But then,dφxi surjective whenever^∂F_∂t(x)6= 0. This condition determines anon-emptyopen subset ofX as otherwise ^∂F_∂t would belong to the defining ideal ofX inC[Y ×C] (which is the principal ideal generated byF) which is impossible by a degree argument Example 3.12 (Differential of orbit map) LetXbe a closedGLn-stable subscheme of aGLn -representationV andxa geometric point ofX. Consider the orbitclosureO(x) ofxinV. Because the orbit map

µ: GLn -- GLn.x^⊂ - O(x)

is dominant we have thatC[O(x)]^⊂ - _C[GLn] and therefore a domain, soO(x) is an irreducible affine variety. We define thestabilizer subgroupStab(x) to be the fiberµ⁻¹(x), then Stab(x) is a closed subgroup of GLn. We claim that the differential of the orbit map in the identity matrix e=rr

n

dµe:gl_n - Tx(X) satisfies the following properties

Ker dµe=stab(x) and Im dµe=Tx(O(x)).

By the proposition we know that there is a dense open subsetU ofGLnsuch thatdµgis surjective for all g ∈ U. By GLn-equivariance ofµ it follows that dµg is surjective for all g ∈ GLn, in

particulardµe :gl_n - Tx(O(x)) is surjective. Further, all fibers ofµ overO(x) have the same dimension. But then it follows from thedimension formulaof proposition that

dim GLn=dim Stab(x) +dimO(x)

(which, incidentally gives us an algorithm to compute the dimensions of orbitclosures). Combining this with the above surjectivity, a dimension count proves thatKer dµe=stab(x), the Lie algebra ofStab(x).

Let Abe a C-algebra and letM and N be twoA-representations of dimensions saymand n.

AnA-representationP of dimensionm+nis said to be anextension ofN byM if there exists a short exact sequence of leftA-modules

e: 0 - M - P - N - 0

We define an equivalence relation on extensions (P, e) of N byM : (P, e)∼= (P⁰, e⁰) if and only if there is an isomorphismP ^φ- P⁰ of leftA-modules such that the diagram below is commutative

e: 0 - M - P - N - 0

e⁰: 0 - M

id_M

?

- P⁰

φ

?

- N

id_N

?

- 0 The set of equivalence classes of extensions ofN byM will be denoted byExt¹_A(N, M).

An alternative description ofExt¹A(N, M) is as follows. Letρ:A - Mmandσ:A - Mn

be the representations definingM andN. For an extension (P, e) we can identify theC-vectorspace withM⊕N and theA-module structure onP gives a algebra mapµ:A - Mm+nand we can represent the action ofaonP by left multiplication of the block-matrix

µ(a) =

»ρ(a) λ(a) 0 σ(a) –

, whereλ(a) is anm×nmatrix and hence defines a linear map

λ:A - Hom_C(N, M).

The condition thatµis an algebra morphism is equivalent to the condition λ(aa⁰) =ρ(a)λ(a⁰) +λ(a)σ(a⁰)

and we denote the set of all liner mapsλ:A - Hom_C(N, M) byZ(N, M) and call it the space of cycle . The extensions of N byM corresponding to two cycles λ and λ⁰ from Z(N, M) are equivalent if and only if we have anA-module isomorphism in block form

»idM β 0 idN

–

withβ∈Hom_C(N, M)

3.7. Normal spaces 155

between them. A-linearity of this map translates into the matrix relation

»idM β 0 idN

– .

»ρ(a) λ(a) 0 σ(a) –

=

»ρ(a) λ⁰(a) 0 σ(a) –

.

»idM β 0 idN

–

for alla∈A

or equivalently, thatλ(a)−λ⁰(a) =ρ(a)β−βσ(a) for alla∈A. We will now define the subspace ofZ(N, M) ofboundariesB(N, M)

{δ∈Hom_C(N, M)| ∃β∈Hom_C(N, M) :∀a∈A:δ(a) =ρ(a)β−βσ(a)}.

We then have the descriptionExt¹_A(N, M) =^Z(N,M)_B(N,M).

Example 3.13 (Normal space torep_n) LetAbe an affineC-algebra generated by{a1, . . . , am} andρ:A - Mn(C) an algebra morphism, that is,ρ∈repnA determines ann-dimensional A-representationM. We claim to have the following description of thenormal spaceto the orbitclosure Cρ=O(ρ) ofρ

Nρ(rep_n A)^def= Tρ(rep_n A)

Tρ(Cρ) =Ext¹A(M, M).

We have already seen that the space of cycles Z(M, M) is the space of ρ-derivations of A in Mn(C), Derρ(A), which we know to be the tangent spaceTρ(rep_n A). Moreover, we know that the differentialdµe of the orbit mapGLn

µ

- Cρ ^⊂- Mn^m

dµe: gl_n=Mn - Tρ(Cρ)

is surjective. Now, ρ = (ρ(a1), . . . , ρ(am)) ∈ Mn^m and the action of action of GLn is given by simultaneous conjugation. But then we have for anyA∈gl_n=Mnthat

(In+Aε).ρ(ai).(In−Aε) =ρ(ai) + (Aρ(ai)−ρ(ai)A)ε.

Therefore, by definition of the differential we have that

dµe(A)(a) =Aρ(a)−ρ(a)A for alla∈A.

that is,dµe(A) ∈B(M, M) and as the differential map is surjective we haveTρ(Cρ) =B(M, M) from which the claim follows.

Example 3.14 (Normal space totrep_n) LetAbe a Cayley-Hamilton algebra with trace map trA and trace generated by {a1, . . . , am}. Letρ∈trepn A, that is,ρ:A - Mn(C) is a trace preserving algebra morphism. Any cycleλ:A - Mn(C) inZ(M, M) =Derρ(A) determines an algebra morphism

ρ+λε:A - Mn(C[ε])

We know that the tangent space Tρ(trep_n A) is the subspace Der^tr_ρ(A) of trace preserving ρ-derivations, that is, those satisfying

λ(trA(a)) =tr(λ(a)) for alla∈A

Observe that for all boundaries δ ∈ B(M, M), that is, such that there is anm ∈ Mn(C) with δ(a) =ρ(a).m−m.ρ(a) are trace preserving as

δ(trA(a)) = ρ(trA(a)).m−m.ρ(trA(a)) =tr(ρ(a)).m−m.tr(ρ(a))

= 0 =tr(m.ρ(a)−ρ(a).m) =tr(δ(a)) Hence, we can define the space oftrace preserving self-extensions

Ext^trA(M, M) = Der^tr_ρ(A) B(M, M)

and obtain as before that thenormal spaceto the orbit closureCρ=O(ρ) is equal to Nρ(trep_nA)^def= Tρ(trep_nA)

Tρ(Cρ) =Ext^trA(M, M)

Im Dokument noncommutative geometry@n, volume 1 : the tools (Seite 154-164)