QUOTIENTS OF TANNAKIAN CATEGORIES
J.S. MILNE
ABSTRACT. We classify the “quotients” of a tannakian category in which the objects of a tan-nakian subcategory become trivial, and we examine the properties of such quotient categories.
Introduction
Given a tannakian category T and a tannakian subcategory S, we ask whether there exists a quotient ofTbyS, by which we mean an exact tensor functorq: T→QfromTto a tannakian categoryQsuch that
a. the objects ofTthat become trivial in Q(i.e., isomorphic to a direct sum of copies of11 inQ) are precisely those inS, and
b. every object ofQis a subquotient of an object in the image ofq.
WhenTis the categoryRep(G)of finite-dimensional representations of an affine group scheme Gthe answer is obvious: there exists a unique normal subgroupH ofGsuch that the objects of S are the representations on which H acts trivially, and there exists a canonical functor q satisfying (a) and (b), namely, the restriction functor Rep(G) → Rep(H) corresponding to the inclusion H ֒→ G. By contrast, in the general case, there need not exist a quotient, and when there does there will usually not be a canonical one. In fact, we prove that there exists aq satisfying (a) and (b) if and only ifSis neutral, in which case theqare classified by thek-valued fibre functors onS. Herek = End(11)def is assumed to be a field.
From a slightly different perspective, one can ask the following question: given a subgroup Hof the fundamental groupπ(T)ofT, does there exist an exact tensor functorq:T→Qsuch that the resulting homomorphism π(Q) → q(π(T)) maps π(Q) isomorphically onto q(H)? Again, there exists such aqif and only if the subcategoryTH ofT, whose objects are those on whichHacts trivially, is neutral, in which case the functorsq correspond to thek-valued fibre functors onTH.
The two questions are related by the “tannakian correspondence” between tannakian sub-categories ofTand subgroups ofπ(T)(see 1.7).
In addition to proving the above results, we determine the fibre functors, polarizations, and fundamental groups of the quotient categoriesQ.
Received by the editors 2007-07-07 and, in revised form, 2007-12-02. Transmitted by Ieke Moerdijk. Published on 2007-12-14.
2000 Mathematics Subject Classification: 18D10.
Key words and phrases: gerbes, motives, tannakian categories. c
J.S. Milne, 2007. Permission to copy for private use granted.
The original motivation for these investigations came from the theory of motives (see Milne 2002, 2007).
Notation: The notationX ≈ Y means thatX andY are isomorphic, andX ≃ Y means that
XandY are canonically isomorphic (or that there is a given or unique isomorphism).
1. Preliminaries
For tannakian categories, we use the terminology of Deligne and Milne 1982. In particular, we write11for any identity object of a tannakian category — recall that it is uniquely determined up to a unique isomorphism. We fix a field k and consider only tannakian categories with k = End(11)and only functors of tannakian categories that arek-linear.
GERBES
1.1. We refer to Giraud 1971, Chapitre IV, for the theory of gerbes. All gerbes will be for the flat (i.e., fpqc) topology on the categoryAffk of affine schemes overk. The band (= lien) of a gerbeG is denotedBd(G). A commutative band can be identified with a sheaf of groups.
1.2. Letα: G1 → G2be a morphism of gerbes overAffk, and letω0be an object ofG2,k. Define
1.3. Recall (Saavedra Rivano 1972, III 2.2.2) that a gerbe is said to be tannakian if its band is locally defined by an affine group scheme. It is clear from the exact sequence (1) that ifG1 and
1.4. The fibre functors on a tannakian categoryTform a gerbe FIB(T)overAffk(Deligne 1990, 1.13). Each objectX ofTdefines a representationω 7→ ω(X)of FIB(T), and in this way we
get an equivalenceT → Rep(FIB(T))of tannakian categories (Deligne 1989, 5.11; Saavedra Rivano 1972, III 3.2.3, p200). Every gerbe whose band is tannakian arises in this way from a tannakian category (Saavedra Rivano 1972, III 2.2.3).
FUNDAMENTAL GROUPS
1.5. We refer to Deligne 1989, §§5,6, for the theory of algebraic geometry in a tannakian cate-goryTand, in particular, for the fundamental groupπ(T)of T. It is the affine group scheme1 inT such thatω(π(T)) ≃ Aut⊗(ω)functorially in the fibre functorω on T. The groupπ(T) acts on each object X of T, and ω transforms this action into the natural action of Aut⊗(ω) onω(X). The various realizationsω(π(T))ofπ(T)determine the band ofT(i.e., the band of FIB(T)).
1.6. An exact tensor functor F: T1 → T2 of tannakian categories defines a homomorphism π(F) : π(T2)→F(π(T1))(Deligne 1989, 6.4). Moreover:
a. F induces an equivalence of T1 with a category whose objects are the objects of T2
endowed with an action ofF(π(T1))compatible with that ofπ(T2)(Deligne 1989, 6.5);
b. π(F)is flat and surjective if and only ifF is fully faithful and every subobject ofF(X), forXinT1, is isomorphic to the image of a subobject ofX(cf. Deligne and Milne 1982, 2.21);
c. π(F)is a closed immersion if and only if every object ofT2 is a subquotient of an object in the image ofq(ibid.).
1.7. For a subgroup2 H ⊂ π(T), we let TH denote the full subcategory of T whose objects are those on which H acts trivially. It is a tannakian subcategory of T (i.e., it is a strictly full subcategory closed under the formation of subquotients, direct sums, tensor products, and duals) and it follows from Deligne 1990, 8.17, that every tannakian subcategory arises in this way from a unique subgroup ofπ(T)(see Bertolin 2003, 1.6)3. The objects ofTπ(T)are exactly
the trivial objects ofT, and there exists a unique (up to a unique isomorphism) fibre functor
γT:Tπ(T)→Veck,
namely,γT
(X) = Hom(11, X).
1“T-sch´ema en groupes affines” in Deligne’s terminology.
2Note that every subgroupHofπ(T)is normal. For example, the fundamental groupπof the categoryRep(G) of representations of the affine group schemeG= Spec(A)isAregarded as an object ofInd(Rep(G)). The action
ofGonAis that defined by inner automorphisms. A subgroup ofπis a quotientA→ BofA(as a bi-algebra)
such that the action ofGonAdefines an action ofGonB. Such quotients correspond to normal subgroups ofG.
3Th´eor`eme 8.17 of Deligne 1990 applies without restriction on the fieldkbecause we are assuming thatTis tannakian (see ibid. 6.9, 8.1). Equally, the blanket hypothesis thatkis of characteristic zero in Bertolin 2003 is not
1.8. For a subgroup H ofπ(T)and an objectX ofT, we letXH denote the largest subobject ofX on which the action ofHis trivial. ThusX =XH if and only ifX is inTH.
1.9. When H is contained in the centre of π(T), then it is an affine group scheme in Tπ(T),
and soγT
identifies it with an affine group scheme overkin the usual sense. For example, γT
identifies the centre ofπ(T)withAut⊗(idT)(cf. Saavedra Rivano 1972, II 3.3.3.2, p150).
MORPHISMS OF TANNAKIAN CATEGORIES
1.10. For a group G, a right G-object X, and a left G-object Y, X ∧G Y denotes the con-tracted product ofXandY, i.e., the quotient ofX×Y by the diagonal action ofG, (x, y)g = (xg, g−1y). WhenG → H is a homomorphism of groups,X ∧GH is theH-object obtained fromX by extension of the structure group. In this last case, ifXis aG-torsor, thenX ∧GH is also anH-torsor. See Giraud 1971, III 1.3, 1.4.
1.11. LetTbe a tannakian category over k, and assume that the fundamental groupπ ofTis commutative. A torsorP underπ in T defines a tensor equivalence T → T, X 7→ P ∧π X, bound by the identity map on Bd(T), and every such equivalence arises in this way from a torsor under π (cf. Saavedra Rivano 1972, III 2.3). For any k-algebraR and R-valued fibre functorωonT,ω(P)is anR-torsor underω(π)andω(P ∧πX)≃ω(P)∧ω(π)ω(X).
2. Quotients
For any exact tensor functor q: T → T′, the full subcategoryTq ofT whose objects become trivial inT′ is a tannakian subcategory ofT(obviously).
We say that an exact tensor functor q: T → Qof tannakian categories is a quotient func-tor if every object of Q is a subquotient of an object in the image of q; equivalently, if the homomorphism π(q) : π(Q) → q(πT) is a closed immersion (see 1.6c). If, in addition, the homomorphismπ(q)is normal (i.e., its image is a normal subgroup ofq(T)), then we say that
qisnormal.
2.1. EXAMPLE. Consider the exact tensor functorωf: Rep(G)→Rep(H)defined by a homo-morphismf: H → Gof affine group schemes. The objects of Rep(G)ωf
are those on which H (equivalently, the intersection of the normal subgroups ofGcontainingf(H)) acts trivially. The functorωf is a quotient functor if and only iff is a closed immersion, in which case it is normal if and only iff(H)is normal inG.
2.2. PROPOSITION. An exact tensor functor q: T → Q of tannakian categories is a normal
quotient functor if and only if there exists a subgroup H of π(T) such that π(q) induces an
PROOF. ⇐=: Because q is exact, q(H) → q(πT) is a closed immersion. Thereforeπ(q)is a
closed immersion, and its image is the normal subgroupq(H)ofq(πT).
=⇒: Because q is a quotient functor, π(q) is a closed immersion. Let H be the kernel of the homomorphismπ(T) → π(Tq) defined by the inclusion Tq ֒→ T. The image ofπ(q) is contained in q(H), and equals it if and only if q is normal. To see this, let G = qπ(T), and identify T with the category of objects ofQ with an action ofG compatible with that of π(Q)⊂G. Thenqbecomes the forgetful functor, andTq =Tπ(Q). Thus,q(H)is the subgroup ofGacting trivially on those objects on whichπ(Q)acts trivially. It follows thatπ(Q)⊂q(H), with equality if and only ifπ(Q)is normal inG.
In the situation of the proposition, we sometimes call Qa quotient of T by H (cf. Milne 2002, 1.3).
Letq: T →Qbe an exact tensor functor of tannakian categories. By definition,qmapsTq
intoQπ(Q), and so we acquire ak-valued fibre functorωq def
a. ForX, Y inT, there is a canonical functorial isomorphism
HomQ(qX, qY)≃ωq(Hom(X, Y)H).
b. The mapω7→(ω◦q, a(ω))defines an equivalence of gerbes
r(q) : FIB(Q)→(ωq↓FIB(T)).
PROOF. (a) From the various definitions and Deligne and Milne 1982,
(see 1.2). On the other hand, we saw in the proof of (2.2) that H = Ker(π(T) → π(TH)). On comparing these statements, we seee that the morphism r(q) of gerbes is bound by an isomorphism of bands, which implies that it is an equivalence of gerbs (Giraud 1971, IV 2.2.6).
2.4. PROPOSITION. Let(Q, q)be a normal quotient ofT. An exact tensor functorq′: T →T′
factors throughqif and onlyTq′
⊃Tq andωq ≈ωq′
|Tq.
PROOF. The conditions are obviously necessary. For the sufficiency, choose an isomorphism b: ωq → ωq′
|Tq. A fibre functor ω on T′ then defines a fibre functor ω ◦ q′ on T and an
isomorphisma(ω)|Tq◦b: ωq →(ω◦q′)|Tq. In this way we get a homomorphism
FIB(T′)→(ωq↓FIB(T))≃FIB(Q)
and we can apply (1.4) to get a functorQ→T′with the correct properties.
2.5. THEOREM.LetTbe a tannakian category overk, and letω0be ak-valued fibre functor on
TH for some subgroupH ⊂π(T). There exists a quotient(Q, q)ofTbyHsuch thatωq ≃ω
0.
PROOF. The gerbe(ω0↓FIB(T))is tannakian (see 1.3). From the morphism of gerbes
(ω, a)7→ω: (ω0↓FIB(T))→ FIB(T),
we obtain a morphism of tannakian categories
Rep(FIB(T))→Rep(ω0↓FIB(T))
(see 1.4). We defineQto beRep(ω0↓FIB(T))and we defineqto be the composite of the above
morphism with the equivalence (see 1.4)
T→Rep(FIB(T)).
Since a gerbe and its tannakian category of representations have the same band, an argument as in the proof of Proposition 2.3 shows thatπ(q)mapsπ(Q)isomorphically ontoq(H). A direct calculation shows thatωqis canonically isomorphic toω
0.
We sometimes write T/ωfor the quotient of Tdefined by ak-valued fibre functorω on a subcategory ofT.
2.6. EXAMPLE. Let(T, w,T)be a Tate triple, and letSbe the full subcategory ofTof objects
isomorphic to a direct sum of integer tensor powers of the Tate objectT. Defineω0 to be the
fibre functor onS,
X 7→lim−→ n
Hom( M
−n≤r≤n
11(r), X).
2.7. REMARK. Letq: T →Qbe a normal quotient functor. ThenTcan be recovered fromQ,
the homomorphism π(Q) → q(π(T)), and the actions of q(π(T))on the objects of Q (apply 1.6a).
2.8. REMARK. A fixed k-valued fibre functor on a tannakian categoryTdetermines a Galois correspondence between the subsets ofob(T)and the equivalence classes of quotient functors
T→Q.
2.9. EXERCISE. Use (1.10, 1.11) to express the correspondence between fibre functors on tan-nakian subcategories ofTand normal quotients ofTin the language of2-categories.
2.10. ASIDE. LetGbe the fundamental group π(T)of a tannakian categoryT, and letH be a subgroup ofG. We use the same letter to denote an affine group scheme inT and the band it defines. Then, under certain hypotheses, for example, if all the groups are commutative, there will be an exact sequence
· · · →H1(k, G)→H1(k, G/H)→H2(k, H)→H2(k, G)→H2(k, G/H).
The categoryTdefines a classc(T)inH2(k, G), namely, theG-equivalence class of the gerbe
of fibre functors onT, and the image ofc(T)inH2(k, G/H)is the class ofTH. Any quotient ofTbyHdefines a class inH2(k, H)mapping toc(T)inH2(k, G). Thus, the exact sequence
suggests that a quotient ofTbyHwill exist if and only if the cohomology class ofTH is neutral, i.e., if and only ifTH is neutral as a tannakian category, in which case the quotients are classified by the elements ofH1(k, G/H)(moduloH1(k, G)). WhenTis neutral and we fix ak-valued
fibre functor on it, then the elements ofH1(k, G/H)classify thek-valued fibre functors onTH. Thus, the cohomology theory suggests the above results, and in the next subsection we prove that a little more of this heuristic picture is correct.
THE COHOMOLOGY CLASS OF THE QUOTIENT
For an affine group schemeGover a fieldk,Hr(k, G)denotes the cohomology group computed with respect to the flat topology. WhenGis not commutative, this is defined only forr = 0,1,2 (Giraud 1971).
2.11. PROPOSITION. Let (Q, q) be a quotient of T by a subgroup H of the centre of π(T).
Suppose thatTis neutral, with k-valued fibre functor ω. LetG = Aut⊗(ω), and let℘(ωq)be
theG/ω(H)-torsorHom(ω|TH, ωq). Under the connecting homomorphism
H1(k, G/H)→H2(k, H)
the class of℘(ωq)inH1(k, G/H)maps to the class ofQinH2(k, H).
PROOF. Note thatH = Bd(Q), and so the statement makes sense. According to Giraud 1971,
SEMISIMPLE NORMAL QUOTIENTS
Everything can be made more explicit when the categories are semisimple. Throughout this subsection,khas characteristic zero.
2.12. PROPOSITION. Every normal quotient of a semisimple tannakian category is semisimple.
PROOF. A tannakian category is semisimple if and only if the identity component of its
funda-mental group is pro-reductive (cf. Deligne and Milne 1982, 2.28), and a normal subgroup of a reductive group is reductive (because its unipotent radical is a characteristic subgroup).
LetTbe a semisimple tannakian category overk, and letω0be ak-valued fibre functor on a
tannakian subcategorySofT.We can construct an explicit quotientT/ω0 as follows. First, let
(T/ω0)′ be the category with one objectX¯ for each objectX ofT, and with
Hom(T/ω0)′( ¯X,Y¯) =ω0(Hom( ¯X,Y¯)H)
whereH is the subgroup of π(T)definingS. There is a unique structure of ak-linear tensor category on(T/ω0)′for whichq: T→(T/ω0)′ is a tensor functor. With this structure,(T/ω0)′
defines a fibre functorωaonT/ω0 whose action on objects is determined by
ωa( ¯X) =ω(X)
and whose action on morphisms is determined by
Hom( ¯X,Y¯) ωa Hom(ωa( ¯X), ωa( ¯Y))
whose action on objects is determined by
¯
and whose action on morphisms is determined by commutative. ThenHom(ωq1, ωq2)isπ/H-torsor, and we assume that it lifts to aπ-torsorP in
T, soP ∧π (π/H) = Hom(ωq1 , ωq2
). Then
T−−−−−−→X7→P∧πX T−q→2 Q2
realizesQ2as a quotient ofTbyH, and the corresponding fibre functor onTH isP∧πωq2 ≃ωq1.
Therefore, there exists a commutative diagram of exact tensor functors
T −−−−−−→X7→P∧πX T
which depends on the choice ofP liftingHom(ωq1, ωq2)in an obvious way.
3. Polarizations
We refer to Deligne and Milne 1982, 5.12, for the notion of a (graded) polarization on a Tate triple overR. We writeVfor the category ofZ-graded complex vector spaces endowed with a
semilinear automorphismasuch that a2v = (−1)nv ifv ∈ Vn. It has a natural structure of a Tate triple (ibid. 5.3). The canonical polarization onVis denotedΠV.
A morphism F: (T1, w1,T1) → (T2, w2,T2) of Tate triples is an exact tensor functor
F: T1 → T2 preserving the gradations together with an isomorphism F(T1) ≃ T2. We say
that such a morphism iscompatiblewith graded polarizationsΠ1 andΠ2onT1andT2(denoted
F: Π1 7→Π2) if
ψ ∈Π1(X)⇒F ψ ∈Π2(F X),
in which case, for anyX homogeneous of weightn,Π1(X)consists of the sesquilinear forms
it is a fibre functor onTw(Gm).
A CRITERION FOR THE EXISTENCE OF A POLARIZATION
3.1. PROPOSITION. LetT= (T, w,T)be an algebraic Tate triple overRsuch thatw(−1)6= 1,
and letξ: T → V be a morphism of Tate triples. There exists a graded polarization Π onT
(necessarily unique) such that ξ: Π 7→ ΠV
if and only if the real algebraic group Aut⊗(γV ◦
ξ|Tw(Gm))is anisotropic.
PROOF. LetG = Aut⊗(γV ◦ξ|Tw(Gm)). AssumeΠ exists. The restriction of Π toTw(Gm) is
a symmetric polarization, which the fibre functorγV
◦ξ maps to the canonical polarization on
VecR. This implies thatGis anisotropic (Deligne 1972, 2.6).
For the converse, let X be an object of weight nin T(C). A sesquilinear formψ: ξ(X)⊗
ξ(X) → 11(−n)arises from a sesquilinear form onX if and only if it is fixed by G. Because Gis anisotropic, there exists aψ ∈ ΠV(ξ(X))fixed byG(ibid., 2.6), and we defineΠ(X)to
consist of all sesquilinear formsφonXsuch thatξ(φ) ∈ΠV
(ξ(X)). It is now straightforward to check thatX 7→Π(X)is a polarization onT.
3.2. COROLLARY. LetF: (T1, w1,T1) → (T2, w2,T2)be a morphism of Tate triples, and let
Π2 be a graded polarization on T2. There exists a graded polarization Π1 on T1 such that
F: Π1 7→Π2if and only if the real algebraic groupAut⊗(γV◦ξΠ2 ◦F|T
w(Gm)
1 )is anisotropic.
POLARIZATIONS ON QUOTIENTS
The next proposition gives a criterion for a polarization on a Tate triple to pass to a quotient Tate triple.
3.3. PROPOSITION. LetT= (T, w,T)be an algebraic Tate triple overRsuch thatw(−1)6= 1.
Let (Q,q)be a quotient of Tby H ⊂ π(T), and let ωq be the corresponding fibre functor on
TH. AssumeH ⊃w(Gm), so thatQinherits a Tate triple structure from that onT, and thatQ
is semisimple. Given a graded polarizationΠonT, there exists a graded polarizationΠ′ onQ
such thatq: Π7→Π′ if and only ifωq ≈ω
Π|TH.
PROOF. ⇒: LetΠ′ be such a polarization onQ, and consider the functors
T→q QξΠ→′ V, ξΠ′: Π′ 7→ΠV.
BothξΠ′ ◦qandξΠare compatible withΠandΠV and with the Tate triple structures onTand
V, and soξΠ′◦q ≈ξΠ(Deligne and Milne 1982, 5.20). On restricting everything toTw(Gm)and
composing withγV,we get an isomorphismω
Π′ ◦(q|Tw( Gm)
) ≈ ωΠ. Now restrict this to TH,
and note that
ωΠ′ ◦(q|Tw( Gm))
|TH = (ωΠ′|Qπ(Q))◦(q|TH)≃ωq
becauseωΠ′|Qπ(Q) ≃γQ.
⇐: The choice of an isomorphismωq →ω
Π|TH determines an exact tensor functor
As the quotientsT/ωqandT/ω
Πare tensor equivalent respectively toQandV, this shows that
there is an exact tensor functorξ: Q→Vsuch thatξ◦q≈ξΠ. EvidentlyAut⊗(γV◦ξ|Qw(Gm))
is isomorphic to a subgroup of Aut⊗(γV ◦ξ
Π|Tw(Gm)). Since the latter is anisotropic, so also
is the former (Deligne 1972, 2.5). Henceξdefines a graded polarizationΠ′ onQ(Proposition
3.1), and clearlyq: Π7→Π′.
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Mathematics Dept., University of Michigan, Ann Arbor, MI, USA
Website: www.jmilne.org/math/
Email:[email protected]
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