Etale topology

A closed subvarietyX ^⊂ - _C^mcan be equipped with theZariski topology or with the much finer analytic topology . A major disadvantage of the coarseness of the Zariski topology is the failure to have animplicit function theorem in algebraic geometry. Etale morphisms are introduced to bypass this problem.

We will define ´etale morphisms which determine the´etale topology . This is no longer a usual topology determined by subsets, but rather aGrothendieck topology determined bycovers. Definition 3.1 A finite morphismA ^f- B of commutativeC-algebras is said to be´etale if and only ifB=A[t1, . . . , tk]/(f1, . . . , fk) such that theJacobian matrix

2 6 6 4

∂f1

∂t₁ . . . ^∂f_∂t¹ . k

.. ...

∂f_k

∂t₁ . . . ^∂f_∂t^k

k

3 7 7 5

has a determinant which is a unit inB.

Recall that byspecAwe denote theprime ideal spectrum or theaffine scheme of a commutative C-algebra A (even when A is not affine as a C-algebra). That is, spec A is the set of all prime ideals ofAequipped with theZariski topology, that is the open subset are of the form

X(I) ={P ∈specA | I6⊂P}

for some idealI / A. If A is an affine C-algebra, the points of the corresponding affine variety correspond to themaximal ideals of A and the induced Zariski topology coincides with the one introduced before. In this chapter, however, not allC-algebras will be affine.

Example 3.1 Consider the morphismC[x, x⁻¹]^⊂- _C[x, x⁻¹][√ⁿ

x] and the induced map on the affine schemes

specC[x, x⁻¹][√ⁿ

x] ^ψ- specC[x, x⁻¹] =C− {0}.

Clearly, every pointλ∈C− {0}has exactlynpreimagesλi=ζⁱ√ⁿ

λ. Moreover, in a neighborhood ofλi, the mapψis a diffeomorphism. Still, we do not have an inverse map in algebraic geometry as √ⁿ

xis not a polynomial map. However,C[x, x⁻¹][√ⁿ

x] is an ´etale extension ofC[x, x⁻¹]. In this way ´etale morphisms can be seen as an algebraic substitute for the failure of an inverse function theorem in algebraic geometry.

Proposition 3.1 Etale morphisms satisfy ’sorite’, that is, they satisfy the commutative diagrams of figure 3.1. In these diagrams,etdenotes an ´etale morphism,f.f.denotes a faithfully flat morphism and the dashed arrow is the ´etale morphism implied by ’sorite’.

With these properties we can define a Grothendieck topology on the collection of all ´etale morphisms.

Definition 3.2 The´etale site ofA, which we will denote byAet is the category with

• objects : the ´etale extensionsA ^f- B ofA

• morphisms : compatibleA-algebra morphisms A

B1

φ

f¹

B2 f2

-By proposition 3.1 all morphisms inAetare ´etale. We can turnAetinto a Grothendieck topology by defining

3.1. Etale topology 113

Figure 3.1: Sorite for ´etale morphisms

• cover : a collectionC={B ^f-ⁱ Bi}inAetsuch that specB=∪iIm(specBi

f

- spec B ) Definition 3.3 An´etale presheaf of groups onAet is a functor

G:Aet - groups In analogy with usual (pre)sheaf notation we denote for each

• object B∈Aettheglobal sections Γ(B,G) =G(B)

• morphism B ^φ- C in Aet the restriction map Res^B_C = G(φ) : G(B) - _G(C) and g|C=G(φ)(g).

An ´etale presheafGis an´etale sheaf provided for everyB∈Aet and every cover{B - Bi}we have exactness of theequalizer diagram

0 - _G(B)

Example 3.2 (Constant sheaf ) IfGis a group, then

G:Aet - groups B7→G^⊕π0(B) is a sheaf whereπ0(B) is the number of connected components ofspecB.

Example 3.3 (Multiplicative groupGm ) The functor Gm:Aet - groups B7→B^∗ is a sheaf onAet.

A sequence of sheaves of Abelian groups onAetis said to be exact G^{0 f}- _G ^g- _G”

if for everyB∈Aetands∈G(B) such thatg(s) = 0∈G”(B) there is a cover{B - Bi}inAet

and sectionsti∈G⁰(Bi) such thatf(ti) =s|Bi.

Example 3.4 (Roots of unityµn) We have a sheaf morphism G^m

(−)ⁿ

- _Gm

and we denote the kernel withµn. AsAis aC-algebra we can identifyµnwith the constant sheaf Zn=Z/nZvia the isomorphismζⁱ7→iafter choosing a primitiven-th root of unityζ∈C. Lemma 3.1 The Kummer sequence of sheaves of Abelian groups

0 - µn - _Gm (−)ⁿ

- _Gm - 0 is exact onAet(but not necessarily onspecA with the Zariski topology).

Proof. We only need to verify surjectivity. LetB ∈ Aet and b ∈ G^m(B) = B^∗. Consider the

´etale extensionB⁰=B[t]/(tⁿ−b) ofB, thenbhas ann-th root over inG^m(B⁰). Observe that this

n-th root does not have to belong toG^m(B).

Ifpis a prime ideal ofAwe will denote withkpthe algebraic closure of the field of fractions of A/p. An´etale neighborhood ofpis an ´etale extensionB ∈Aetsuch that the diagram below is commutative

A ^nat- kp

B

et

?

-3.1. Etale topology 115

The analogue of the localizationAp for the ´etale topology is thestrict Henselization A^shp = lim

- ^B where the limit is taken over all ´etale neighborhoods ofp.

Recall that a local algebraLwith maximal idealmand residue mapπ:L -- L/m=kis said to beHenselian if the following condition holds. Letf ∈ L[t] be a monic polynomial such that π(f) factors asg0.h0 ink[t], thenf factors asg.hwithπ(g) =g0 andπ(h) =h0. IfLis Henselian then tensoring withk induces an equivalence of categories between the ´etale A-algebras and the

´

etale k-algebras.

An Henselian local algebra is said to be strict Henselian if and only if its residue field is algebraically closed. Thus, a strict Henselian ring has no proper finite ´etale extensions and can be viewed as a local algebra for the ´etale topology.

Example 3.5 (The algebraic functionsC{x1, . . . , xd}) Consider the local algebra of C[x1, . . . , xd] in the maximal ideal (x1, . . . , xd), then the Henselization and strict Henseliza-tion are both equal to

C{x1, . . . , xd}

the ring of algebraic functions . That is, the subalgebra of C[[x1, . . . , xd]] of formal power-series consisting of those power-series φ(x1, . . . , xd) which are algebraically dependent on the coordi-nate functions xi over C. In other words, thoseφ for which there exists a non-zero polynomial f(xi, y)∈C[x1, . . . , xd, y] withf(x1, . . . , xd, φ(x1, . . . , xd)) = 0.

These algebraic functions may be defined implicitly by polynomial equations. Consider a system of equations

fi(x1, . . . , xd;y1, . . . , ym) = 0 forfi∈C[xi, yj] and 1≤i≤m Suppose there is a solution inCwith

xi= 0 andyj=y^o_j such that the Jacobian matrix is non-zero

det(∂fi

∂yj

(0, . . . ,0;y^o₁, . . . , y_m⁰))6= 0

Then, the system can be solved uniquely for power seriesyj(x1, . . . , xd) with yj(0, . . . ,0) =y^oj by solving inductively for the coefficients of the series. One can show that such implicitly defined series yj(x1, . . . , xd) are algebraic functions and that, conversely, any algebraic function can be obtained in this way.

IfGis a sheaf onAetandpis a prime ideal ofA, we define thestalk ofGinpto be G^p= lim

- ^G(B)

where the limit is taken over all ´etale neighborhoods ofp. One can verify mono- epi- or isomor-phisms of sheaves by checking it in all the stalks.

If Ais an affine algebra defined over an algebraically closed field, then it suffices to verify it in the maximal ideals ofA.

Before we define cohomology of sheaves on Aetlet us recall the definition of derived functors. LetAbe anAbelian category. An objectI ofAis said to be injective if the functor

A - abelian M 7→HomA(M, I)

is exact. We say thatAhas enough injectives if, for every objectM inA, there is a monomorphism M ^⊂ - I into an injective object.

IfAhas enough injectives andf:A - Bis a left exact functor fromAinto a second Abelian categoryB, then there is an essentially unique sequence of functors

Rⁱf:A - B i≥0

called theright derived functorsoff satisfying the following properties

• R⁰ f=f

• RⁱI= 0 forI injective andi >0

• For every short exact sequence inA

0 - M⁰ - M - M” - 0

there are connecting morphismsδⁱ:Rⁱ f(M”) - Rⁱ⁺¹f(M⁰) fori≥0 such that we have a long exact sequence

. . . - Rⁱf(M) - Rⁱ f(M”) ^δ

i

- Rⁱ⁺¹ f(M⁰) - Rⁱ⁺¹f(M) - . . .

• For any morphismM - N there are morphismsRⁱ f(M) - Rⁱ f(N) fori≥0 In order to compute the objectsRⁱf(M) define an objectNinAto bef-acyclic ifRⁱf(M) = 0 for alli >0. If we have anacyclic resolutionofM

0 - M - N0 - N1 - N2 - . . .

byf-acyclic objectNi, then the objects Rⁱ f(M) are canonically isomorphic to the cohomology objects of the complex

0 - f(N0) - f(N1) - f(N2) - . . .

One can show that all injectives aref-acyclic and hence that derived objects ofMcan be computed from aninjective resolutionofM.

3.1. Etale topology 117

Now, letS^ab(Aet) be the category of all sheaves of Abelian groups on Aet. This is an Abelian category having enough injectives whence we can form right derived functors of left exact functors.

In particular, consider the global section functor

Γ :S^ab(Aet) - abelian G7→G(A)

which is left exact. The right derived functors of Γ will be called the´etale cohomology functors and we denote

Rⁱ Γ(G) =Hetⁱ (A,G)

In particular, if we have an exact sequence of sheaves of Abelian groups 0 - _G⁰ - _G - _G” - 0, then we have a long exact cohomology sequence

. . . - H_etⁱ(A,G) - H_etⁱ(A,G”) - H_etⁱ⁺¹(A,G⁰) - . . .

IfG is a sheaf of non-Abelian groups (written multiplicatively), we cannot define cohomology groups. Still, one can define apointed setH_et¹(A,G) as follows. Take an ´etale coverC={A - Ai} ofAand define a 1-cocycle forC with values inGto be a family

gij∈G(Aij) withAij=Ai⊗AAj

satisfying the cocycle condition

(gij|Aijk)(gjk|Aijk) = (gik|Aijk) whereAijk=Ai⊗AAj⊗AAk.

Two cocyclesg andg⁰ forC are said to be cohomologous if there is a familyhi ∈G(Ai) such that for alli, j∈I we have

gij⁰ = (hi|Aij)gij(hj|Aij)⁻¹

This is an equivalence relation and the set of cohomology classes is written asH_et¹(C,G). It is a pointed set having as its distinguished element the cohomology class of gij = 1∈ G(Aij) for all i, j∈I.

We then define the non-Abelian firstcohomology pointed setas Het¹(A,G) = lim

- ^H

1 et(C,G)

where the limit is taken over all ´etale coverings ofA. It coincides with the previous definition in caseGis Abelian.

A sequence 1 - _G⁰ - _G - _G” - 1 of sheaves of groups onAetis said to be exact if for everyB∈Aetwe have

• G⁰(B) =Ker G(B) - _G”(B)

• For everyg”∈G”(B) there is a cover{B - Bi}inAetand sectionsgi∈G(Bi) such that gimaps tog”|B.

Proposition 3.2 For an exact sequence of groups onAet

1 - _G⁰ - _G - _G” - 1

there is associated an exact sequence of pointed sets

1 - _G⁰(A) - _G(A) - _G”(A) ^δ- Het¹(A,G⁰) -- Het¹(A,G) - Het¹(A,G”) ...- Het²(A,G⁰)

where the last map exists whenG⁰ is contained in the center ofG(and therefore is Abelian whence H² is defined).

Proof. The connecting mapδ is defined as follows. Letg” ∈G”(A) and letC ={A - Ai} be an ´etale covering of A such that there are gi ∈ G(Ai) that map tog | Ai under the map G(Ai) - _G”(Ai). Then,δ(g) is the class determined by the one cocycle

gij= (gi|Aij)⁻¹(gj|Aij)

with values inG⁰. The last map can be defined in a similar manner, the other maps are natural

and one verifies exactness.

The main applications of this non-Abelian cohomology to non-commutative algebra is as follows.

Let Λ be a not necessarily commutativeA-algebra andM anA-module. Consider the sheaves of groupsAut(Λ) resp. Aut(M) onAetassociated to the presheaves

B7→AutB−alg(Λ⊗AB) resp.B7→AutB−mod(M⊗AB)

for allB ∈Aet. A twisted formof Λ (resp. M) is anA-algebra Λ⁰ (resp. an A-moduleM⁰) such that there is an ´etale coverC={A - Ai}ofAsuch that there are isomorphisms

( Λ⊗AAi

φ_i

- Λ⁰⊗AAi

M⊗AAi ψ_i

- M⁰⊗AAi

of Ai-algebras (resp. Ai-modules). The set of A-algebra isomorphism classes (resp. A-module isomorphism classes) of twisted forms of Λ (resp.M) is denoted byT wA(Λ) (resp. T wA(M)). To a twisted form Λ⁰ one associates a cocycle onC

αΛ⁰ =αij=φ⁻¹_i ◦φj

with values in Aut(Λ). Moreover, one verifies that two twisted forms are isomorphic asA-algebras if their cocycles are cohomologous. That is, there are embeddings

(

T wA(Λ)^⊂ - Het¹(A,Aut(Λ)) T wA(M)^⊂ - Het¹(A,Aut(M))

3.1. Etale topology 119

In favorable situations one can even show bijectivity. In particular, this is the case if the automor-phisms group is a smooth affine algebraic group-scheme.

Example 3.6 (Azumaya algebras) Consider Λ =Mn(A), then the automorphism group isPGLn

and twisted forms of Λ are classified by elements of the cohomology group Het¹(A,PGLn)

These twisted forms are precisely theAzumaya algebrasof rank n² with center A. WhenAis an affine commutativeC-algebra and Λ is anA-algebra with centerA, then Λ is anAzumaya algebra of rankn² if and only if

Λ

ΛmΛ'Mn(C) for every maximal idealmofA.

Azumaya algebras arise in representation theory as follows. LetAbe this time anoncommutative affineC-algebra and assume that the following two conditions are satisfied

• Ahas a simple representation of dimensionn,

• rep_nAis an irreducible variety.

ThenH

nA=C[rep_nA]^GLn is a domain (whenceissnAis irreducible) and we have an onto trace preserving algebra map corresponding to the simple representation

Z

n

A=Mn(C[rep_nA])^GLn -^φ- Mn(C) Lift the standard basiseijofMn(C) to elementsaij∈R

nAand consider the determinantdof the n²×n²matrix (tr(aijakl))ij,klwith values inH

nA. Thend6= 0 and consider the Zariski open affine subset ofissn A

X(d) ={ Z

n

A ^ψ- Mn(C) | ψsemisimple anddet(tr(ψ(aij)ψ(akl)))6= 0}

If ψ ∈ X(d), then ψ :R

nA - Mn(C) is onto as the ψ(aij) form a basis of Mn(C) whence ψ determines a simplen-dimensional representation.

Proposition 3.3 With notations as above, 1. The localization of R

nA at the central multiplicative set {1, d, d², . . .}is an affine Azumaya algebra with center C[X(d)] which is the localization ofH

nAat this multiplicative set.

2. The restriction of the quotient map rep_n A -^π- issn A to the open set π⁻¹(X(d)) is a principalP GLn-fibration and determines an element in

Het¹(C[X(d)],PGLn) giving the class of the Azumaya algebra.

Proof. (1) : If m = Ker ψ is the maximal ideal of C[X(d)] corresponding to the semisimple representationψ:R

nA - Mn(C), then we have seen that the quotient R

nA R

nAmR

nA 'Mn(C) whenceR

nA⊗^H

nAC[X(d)] is an Azumaya algebra. (2) will follow from the theory of Knop-Luna

slices and will be proved in chapter 5.

An Azumaya algebra over a field is a central simple algebra. Under the above conditions we have that

Z

n

A⊗^H

nA C(issnA)

is a central simple algebra over the functionfield ofissn A and hence determines a class in its Brauer group, which is an important birational invariant. In the following section we recall the cohomological description of Brauer groups of fields.

Im Dokument noncommutative geometry@n, volume 1 : the tools (Seite 119-128)