2 The Tate Conjecture; Semisimplicity.
In this section, we prove that Tate’s conjecture is implied by Finiteness I. Throughout the section,kis a field and DGal.kal=k/. We begin with some elementary lemmas.
LEMMA2.1. If ˛WA ! B is an isogeny of degree prime to chark, then Ker.˛/.kal/ is a finite subgroup ofA.kal/stable under the action of ; conversely, every such subgroup arises as the kernel of such an isogeny, i.e., the quotientA=N exists overk.
PROOF. Over kal, this follows from (8.10). The only additional fact needed is that, if N.kal/is stable under the action of , then the quotientA=N is defined overk. 2 LEMMA2.2. (a) For any abelian varietyAand`¤char.k/, there is an exact sequence
0!T`A `
n
!T`A!A`n.kal/!0:
(b) An isogeny˛WA!Bof degree prime to char.k/defines an exact sequence 0!T`A!T`B !C !0
with the order ofC equal to the power of`dividingdeg.˛/..
PROOF. (a) This follows easily from the definition
T`AD f.an/n1jan2A`n.kal/; `anDan 1; `a1D0g: (b) To prove this, consider the following infinite diagram:
x element divisible by all powers of`, and so
limKn df
D f.an/jan2Kn,`anDan 1,`a1D0g
is zero. Since#B`n.kal/D.`n/2g D#A`n.kal/, we must have#KnD#Cn. Therefore
#Cnis constant fornlarge. The mapCnC1 !Cnis surjective; therefore fornlarge it is bijective, and it follows thatlimCm! Cnis a bijection for all largen. On passing to the inverse limit we get an exact sequence
0!T`B !T`A!C !0
as required. 2
Let˛WB ! Abe an isogeny. Then the image of T`˛WT`B ! T`Ais -stableZ` -module of finite index in T`A. Our final elementary lemma shows that every such sub-module arises from an isogeny ˛, and even that˛ can be taken to have degree a power of
`.
LEMMA2.3. Assume`¤char.k/. For any -stable submoduleW of finite index inT`A, there an abelian variety B and an isogeny˛WB ! A of degree a power of ` such that
LetAbe an abelian variety over a fieldk, and let`be a prime¤chark. Consider the following condition (slightly weaker than Finiteness I):
(*) up to isomorphism, there are only finitely many abelian varieties B isogenous toAby an isogeny of degree a power of`.
LEMMA2.4. Suppose Asatisfies (*). For anyW V`Astable under , there is au 2 End.A/˝Q` such thatuV`ADW.
PROOF. SetT` DT`AandV`DV`A. Let
XnD.T`\W /C`nT`:
This is aZ`-submodule ofT`stable under and of finite index inT`. Therefore, there is an isogeny
fnWB.n/!A, such that fn.T`B.n//DXn:
2. THE TATE CONJECTURE; SEMISIMPLICITY. 137 According to (*), theB.n/fall into only finitely many distinct isomorphism classes, and so at least one class has infinitely manyB.n/’s: there is an infinite setI of positive integers such that all theB.i /fori 2 I are isomorphic. Leti0 be the smallest element ofI. For and hence as an element ofEnd.A/˝Q`. Moreover, it is clear from the second diagram thatui.Xi0/DXi. BecauseXi Xi0, theui fori 2 I are in thecompactsetEnd.Xi0/, and so, after possibly replacing .ui/with a subsequence, we can assume .ui/converges to a limituinEnd.Xi0/ End.V`A/. Now End0.A/` is a subspace ofEnd.V`A/, and hence is closed. Since eachui lies inEnd.A/˝Q`, so also does their limitu. For any x 2 Xi0,u.x/ Dlim ui.x/ \Xi. Conversely, ify 2 \Xi, then there exists for each i 2I, an elementxi 2Xi0 such thatui.xi/Dy. From the compactness ofXi0 again, we deduce that, after possibly replacing I with a subset, the sequence.xi/will converge to a limitx2Xi0. Nowu.x/Dlimu.xi/Dlimui.xi/Dy. Thusu.Xi0/D \Xj DT`\W,
and it follows thatu.V`A/DW. 2
Before proving the main theorem of this section, we need to review a little of the theory of noncommutative rings (CFT, Chapter IV). By a k-algebra, I will mean a ringR, not necessarily commutative, containing k in its centre and of finite dimension over k, and by an R-module I’ll mean an R-module that is of finite dimension over k. IfR has a faithful semisimple module, then every R-module is semisimple, and the k-algebraR is said to besemisimple. A simplek-algebra, i.e., ak-algebra with no two-sided ideals except for the obvious two, is semisimple and a theorem of Wedderburn says that, conversely, a semisimplek-algebra is a finite product of simplek-algebras.
Another theorem of Wedderburn says that every simple k-algebra is isomorphic to Mn.D/for somenand some divisionk-algebraD.
LetDbe a division algebra overk. The right ideals inMn.D/are the sets of the form a.J /withJ f1; 2; : : : ; nganda.J /the set of matrices whosejth columns are zero for j … J. Note thata.J /is generated by the idempotente Ddiag.a1; : : : ; an/withaj D1 forj 2J andaj D0otherwise. On combining this remark with the Wedderburn theorems, we find that every right ideal in a semisimple k-algebraR is generated by an idempotent:
aDeRfor someewithe2 De.
ThecentralizerCE.R/of subalgebraRof ak-algebraE consists of the elements of E such that ˛D˛ for all˛2R. LetRbe ak-algebra and letE DEndk.V /for some faithful semisimpleR-moduleV; the Double Centralizer Theorem says thatCE.CE.R//D R:
IfRis a semisimplek-algebra, thenR˝kk0need not be semisimple — for example, if R DkŒ˛with˛p 2k,˛…k, thenR˝kkalcontains the nilpotent element˛˝1 1˝˛.
However, this only happens in characteristicp: ifkis of characteristic0, thenRsemisimple H)R˝k k0semisimple.
LetAbe an abelian variety. ThenEnd.A/˝Qis a finite-dimensional algebra overQ (10.15), and it is isomorphic to a product of matrix algebras over division algebras (see the first subsection of9). It is, therefore, a semisimpleQ-algebra.
THEOREM2.5. LetAbe an abelian variety overk, and assume thatAAandAsatisfy we know that a is generated by an idempotent e, and clearly eV` D W. Becausee is idempotent COROLLARY2.6. Assume (*) holds for abelian varieties overk. Then the map
Hom.A; B/˝Q` !Hom.V`A; V`B/
is an isomorphism.
PROOF. Consider the diagram of finite-dimensional vector spaces overQ`:
End.V`.AB// DEnd.V`A/ Hom.V`A;V`B/ Hom.V`B;V`A/ End.V`B/
[ [ [ [ [
End.AB/˝Q` DEnd.A/˝Q` Hom.A;B/˝Q` Hom.B;A/˝Q` End.B/˝Q`:
The theorem shows that the inclusion at left is an equality, and it follows that the remaining
inclusions are also equalities. 2
COROLLARY2.7. LetR be the image ofQ`ΠinEnd.V`A/. ThenR is the centralizer ofEnd0.A/`inEnd.V`A/.
PROOF. Theorem 2.5a shows thatV`Ais a semisimpleR-module. As it is also faithful, this implies thatRis a semisimple ring. The double centralizer theorem says thatC.C.R//D R, and (2.5b) says thatC.R/ DEnd.A/˝Q`. On putting these statements together, we
find thatC.End.A/˝Q`/DR: 2