GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS:
THE FUNCTOR CATEGORY Fquad
CHRISTINE VESPA
Abstract.In this paper, we define the functor category Fquadassociated to
F2-vector spaces equipped with a quadratic form. We show the existence of
a fully-faithful, exact functor ' : F ! Fquad, which preserves simple ob*
*jects,
where F is the category of functors from the category of finite dimensio*
*nal
F2-vector spaces to the category of all F2-vector spaces. We define the *
*sub-
category Fisoof Fquad, which is equivalent to the product of the categor*
*ies
of modules over the orthogonal groups; the inclusion is a fully-faithful*
* functor
~ : Fiso! Fquadwhich preserves simple objects.
Keywords: functor categories; quadratic forms over F2; Mackey functors; *
*rep-
resentations of orthogonal groups over F2.
Introduction
In recent years, one of the functor categories which has been particularly s*
*tudied
is the category F(p) of functors from the category Ef of finite dimensional Fp-*
*vector
spaces to the category E of all Fp-vector spaces, where Fp is the prime field w*
*ith p
elements. This category is connected to several areas of algebra and some examp*
*les
of these can be found in [4]. The category F(p) is closely related to the gener*
*al linear
groups. An important application of F(p) is given in [5], where the four authors
proved that this category is very useful for the study of the stable cohomology*
* of the
general linear groups with suitable coefficients. They showed that the calculat*
*ion
of certain extension groups in the category F determines some stable cohomology
groups of general linear groups. One of the motivations of the work presented h*
*ere
is to construct and study a category Fquadwhich could play a similar role for t*
*he
stable cohomology of the orthogonal groups.
In this paper, we restrict to the prime p = 2; the techniques can be applied*
* in
the odd prime case, but the case p = 2 presents features which make it particul*
*arly
interesting. Henceforth, we will suppose that p = 2 and we will denote the cate*
*gory
F(2) by F.
After some recollections on the theory of quadratic forms over F2, we give t*
*he
definition of the category Fquad. In order to have a good understanding of the
category Fquad, we seek to classify the simple objects of this category. The f*
*irst
important result of this paper is the following theorem.
Theorem. There is a functor
' : F ! Fquad
which satisfies the following properties:
(1)' is exact;
(2)' preserves tensor products;
(3)' is fully-faithful;
(4)if S is a simple object of F, '(S) is a simple object of Fquad.
____________
Date: February 2, 2007.
1
2 CHRISTINE VESPA
Remark. Using a study of the projective functors of the category Fquad, we will
show in [?] that '(F ) is a thick subcategory of Fquad.
To study a particular family of functors of Fquad, the isotropic functors, we*
* define
the subcategory Fisoof Fquadwhich is related to Fquadby the following theorem:
Theorem. There is a functor
~ : Fiso! Fquad
which satisfies the following properties:
(1)~ is exact;
(2)~ preserves tensor products;
(3)~ is fully-faithful;
(4)if S is a simple object of Fiso, ~(S) is a simple object of Fquad.
We obtain the classification of the simple objects of the category Fisofrom *
*the
following theorem.
Theorem. There is a natural equivalence of categories
Y
Fiso' F2[O(V )] - mod
V 2S
where S is a set of representatives of isometry classes of quadratic spaces (po*
*ssibly
degenerate).
Apart from the previous theorem, the results of this paper are contained in *
*the
Ph.D. thesis of the author [13], although several results are presented here fr*
*om a
more conceptual point of view.
The author wants to thank her PhD supervisor, Lionel Schwartz, as well as
Geoffrey Powell and Aur'elien Djament for their useful comments and suggestions
on a previous version of this paper.
1. Quadratic spaces over F2
We recall the definition and the classification of quadratic forms over the *
*field
F2. We refer the reader to [9] for details.
1.1. Definitions. Let V be a F2-vector space of finite dimension. A quadratic f*
*orm
over V is a function q : V ! F2 such that q(x + y) + q(x) + q(y) = B(x, y) is a
bilinear form. As a direct consequence of the definition, we have that the bili*
*near
form associated to a quadratic form is alternating.
The radical of a quadratic space (V, qV ) is the subspace of V given by
Rad(V, qV ) = {v 2 V |8w 2 V B(v, w) = 0}
where B(-, -) is the bilinear form associated to the quadratic form. A quadratic
space (V, qV ) is non-degenerate if Rad(V, qV ) = 0.
1.2. Non-degenerate quadratic forms.
1.2.1. Classification. In this paragraph, we recall the classification of non-d*
*egenerate
quadratic forms. The classification of non-singular alternating forms implies t*
*hat a
non-degenerate quadratic space over F2 has even dimension and has a symplectic
basis.
The space H0 is the non-degenerate quadratic space of dimension two with sym-
plectic basis {a0, b0}, and quadratic form determined by:
q0 : H0 ! F2
a0 7-! 0
b0 7-! 0.
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: THE FUNCTOR CATEGORY Fquad3
The space H1 is the non-degenerate quadratic space of dimension two with sym-
plectic basis {a1, b1}, and quadratic form determined by:
q1 :H1 ! F2
a1 7-! 1
b1 7-! 1.
The spaces H0 and H1 are not isometric, whereas the spaces H0?H0 and H1?H1
are isometric. The non degenerate quadratic spaces of dimension 2m, for m 1,
are classified by the following result.
Proposition 1.1. [9] Let m 1 be an integer.
(1)The quadratic spaces H?m0 and H1?H?(m-1)0are not isometric.
(2)A quadratic space of dimension 2m is isometric to either H?m0 or H1?H?(m*
*-1)0.
The two spaces H?m0 and H1?H?(m-1)0are distinguished by the Arf invariant
introduced in [1]. Observe that Arf(H?m0) = 0 and Arf(H1?H?(m-1)0) = 1.
1.2.2. The category Eq.
Definition 1.2. Let Eq be the category having as objects finite dimensional F2-
vector spaces equipped with a non degenerate quadratic form and with morphisms
linear maps which preserve the quadratic forms.
Observe that a linear map which preserves the quadratic forms preserves the
underlying bilinear form, but the converse is, in general, false. The following*
* propo-
sition summarizes straightforward but important results about the category Eq.
Proposition 1.3. (1)The morphisms of Eqare injective linear maps. Conse-
quently they are monomorphisms.
(2)The category Eqdoes not admit push-outs or pullbacks.
Example 1.4. The diagram V {0} ! V , where V 6' {0} does not admit a
push-out in Eq.
To resolve this difficulty, we define the notion of a pseudo push-out in Eq.
1.2.3. Pseudo push-out. To define the pseudo push-out in Eq, we need the follow*
*ing
remark, which uses the non-degeneracy of the quadratic form in an essential way.
Remark 1.5. For f an element of Hom Eq(V, W ), we have W ' f(V )?V 0, where
V 0is the orthogonal space to W in the space V . As the spaces V and f(V ) are
isometric, the spaces W and V ?V 0are also. We will write
f : V ! W ' V ?V 0.
Definition 1.6. For f : V ! W ' V ?V 0and g : V ! X ' V ?V 00morphisms in
Eq, the pseudo push-out of f and g, denoted by W ?VX, is the object V ?V 0?V 00*
*of
Eq.
We give, in the following proposition, the principal properties of the pseudo
push-out.
Proposition 1.7. Let f : V ! W ' V ?V 0and g : V ! X ' V ?V 00be morphisms
in Eq, the pseudo push-out of f and g satisfies the following properties.
(1)There exists a commutative diagram of the form
g
V _______//X
| |
f || |
fflffl| fflffl|
W _____//X?VW
4 CHRISTINE VESPA
(2)W ?VX ' X?VW ;
(3)if V ' V 0then W ?VX ' W ?VX0;
(4)Associativity: (X?VW )?ZY ' X?V(W ?ZY );
(5)Unit: V ?VW ' W .
Remark 1.8. By the first point of the previous proposition, the pseudo push-out
occurs in a commutative diagram
V ________//_V ?V "
| |
| |
fflffl| fflffl|
V ?V 0____//_V ?V 0?V 00
which is equivalent to the orthogonal sum of the diagram
{0}_______//V 00
| |
| |
fflffl| fflffl|
V 0_____//_V 0?V 00
with V . Hence, the pseudo push-out can be considered as a generalization of t*
*he
orthogonal sum.
1.3. Degenerate quadratic forms. The previous section implies that, in partic-
ular, all quadratic spaces of odd dimension are degenerate. We begin by conside*
*ring
the quadratic spaces of dimension one.
Notation 1.9. For ff 2 {0, 1}, let (x, ff) be the quadratic space of dimension *
*one
generated by x such that q(x) = ff.
1.3.1. Classification. We have the following classification:
Theorem 1.10. [9]
(1)Every quadratic space over F2 has an orthogonal decomposition
V ' H?Rad (V )
such that H is non-degenerate and Rad(V ) is isometric to either (x, 0)?r
or (x, 1)?r, where r is the dimension of Rad(V ).
(2)Let V ' H?Rad (V ) and V ' H0?Rad (V ) be two decompositions of V , if
Rad(V ) ' (x, 0)?r for r 0, then H ' H0.
(3)Let H and H0 be two non-degenerate quadratic forms such that dim(H) =
dim(H0) then, for all r > 0,
H?(x, 1)?r ' H0?(x, 1)?r.
Remark 1.11. The third point of the previous theorem, is implied by the isometr*
*y:
H0?(x, 1) ' H1?(x, 1).
This exhibits one of the particularities of quadratic forms over F2: the "non-d*
*egenerate
part" of a quadratic form is not unique in general, not even up to isometry.
1.3.2. The category Edegq.
Definition 1.12. Let Edegqbe the category having as objects finite dimensional *
*F2-
vector spaces equipped with a (possibly degenerate) quadratic form and with mor-
phisms, injective linear maps which preserve the quadratic forms.
Remark 1.13. The hypothesis that the morphisms are linear injective maps is
essential for later considerations.
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: THE FUNCTOR CATEGORY Fquad5
The following proposition underlines one of the important difference between *
*Eq
and Edegq.
Proposition 1.14. The category Edegqadmits pullbacks.
Nevertheless, Edegqdoes not contain all push-outs.
2.Definition of the category Fquad
We have emphasized the fact that all the morphisms of Eqare monomorphisms.
The constructions which link the category F to the stable homology of the gener*
*al
linear groups use, in an essential way, the existence of retractions in the cat*
*egory
Ef. Therefore, to consider analogous constructions in the quadratic case, we wo*
*uld
like to add formally retractions to the category Eq. For this, we define the ca*
*tegory
Tq inspired by the construction of the category of co-spans introduced by B'ena*
*bou
in [2].
2.1. The category Tq.
2.1.1. The categories of spans and co-spans. We will begin by recalling the con-
struction of B'enabou.
Remark 2.1. Our principal interest is in the category Fquad. The construction of
this category uses a generalization of the category of co-spans, which we have *
*chosen
to present rather than spans.
Definition 2.2. [2] Let D be a category equipped with push-outs, the category
coSpan(D) is defined in the following way:
(1)the objects of coSpan(D) are those of D;
(2)for A and B two objects of coSpan(D), Hom coSpan(D)(A, B) is the set of
equivalence classes of diagrams in D of the form A -f!D- g B, for the
equivalence relation which identifies the two diagrams A -f!D- g B and
A -u!D0 v-B if there exists an isomorphism ff : D ! D0 such that the
following diagram is commutative
B1
1
g||111
f fflffl|v11
A _____//PPP11DffB
PPPP BBB11
uPPPPB11BB
PP(1,,(P!!B
D0.
The morphism of Hom coSpan(D)(A, B) represented by the diagram A -f!
D- gB will be denoted by [A f-!D- gB];
(3)the composition is given by:
for two morphisms T1 = [A ! D B] and T2 = [B ! D0 C],
T2 O T1 = [A ! D qBD0 C].
By duality, B'enabou gives the following definition.
Definition 2.3. Let D be a category equipped with pullbacks. The category Sp(D)
is defined by: Sp(D) ' coSpan(Dop)op.
Example 2.4. The category Sp(Edegq) is defined, since Edegqadmits pullbacks by
proposition 1.14.
6 CHRISTINE VESPA
2.1.2. The category Tq. By proposition 1.3 neither the category of Spans nor co-
Spans are defined for Eq. However, we observe that the universality of the push*
*-out
plays no role in the definition of the category coSpan(D). So, by definition 1.*
*6 of
the pseudo push-out of Eqwe can give the following definition.
Definition 2.5. The category ^Tqis defined in the following way:
(1)the objects of ^Tqare those of Eq;
(2)for V and W two objects of T^q, Hom T^q(V, W ) is defined in the same way
as for the category coSpan(D) and we will use the same notation for the
morphisms;
(3)the composition is given by:
for T1 = [V ! X1 W ] and T2 = [W ! X2 Y ],
T2 O T1 = [V ! X1?WX2 Y ]
where X1?WX2 is the pseudo push-out.
The elementary properties of the pseudo push-out given in 1.7 show that the
composition is well-defined and associative. Thus, the above defines a category.
To define the category Tq, we consider the following relation on the morphis*
*ms
of ^Tq.
Definition 2.6. For V and W two objects of ^Tq, the relation R on Hom ^Tq(V, W )
for V and W objects of ^Tq, is defined by:
for T1 = [V ! X1 W ] and T2 = [V ! X2 W ], two elements of Hom ^Tq(V, W ),
T1RT2 if there exists a morphism ff of Eq such that the following diagram is co*
*m-
mutative
W 2
|222
| 2
fflffl|222
V _____//QQQQ22X1ffCC
QQQQ CC 22
QQQQCC22C
QQQ(,,2(!!C
X2.
We will denote by ~ the equivalence relation on Hom T^q(V, W ) generated by the
relation R.
Lemma 2.7. The composition in ^Tqinduces an application:
O : Hom T^q(V, W )= ~ x Hom ^Tq(U, V )= ~ ! Hom T^q(U, W )= ~ .
Proof.By the properties of the pseudo push-out given in Proposition 1.7 we veri*
*fy
that:
(1)if T1RT2, then (T3 O T1)R(T3 O T2);
(2)if T3RT4, then (T1 O T3)R(T1 O T4).
Thanks to the previous lemma, we can give the following definition.
Definition 2.8. Let Tqbe the category having as objects the objects of Eqand wi*
*th
morphisms Hom Tq(V, W ) = Hom ^Tq(V, W )= ~.
For convenience we will use the same notation for the morphisms of Tq as for
those of ^Tq.
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: THE FUNCTOR CATEGORY Fquad7
2.1.3. Properties of the category Tq. We have the following important property.
Proposition 2.9. For f : V ! W a morphism in the category Eq, we have the
following relation in Tq
[W -Id!W- f V ] O [V -f!W -IdW ] = IdV.
In particular, [W -Id!W- f V ] is a retraction of [V -f!W -IdW ].
Proof.It is a direct consequence of the definition of Tq.
To close this paragraph we give two useful constructions in the category Tq.
Definition 2.10 (Transposition). The transposition functor, tr : Tqop ! Tq, is
defined on objects by tr(V ) = V and on morphisms by:
tr(f) = tr([V ! X W ]) = [W ! X V ],
for V and W objects of Tqopand f an element of Hom Tqop(W, V ).
Observe that the transposition functor is involutive.
Proposition 2.11 (Orthogonal sum). There exists a bifunctor ? : Tq x Tq! Tq,
called the orthogonal sum, defined on objects by:
?(V, W ) = V ?W
and on morphisms by:
0 g0 f?f0 g?g0
?([V -f!X- gW ], [V 0f-!X0- W 0]) = [V ?V 0---! X?X0--- W ?W 0].
This bifunctor gives Tq the structure of a symmetric monoidal category, with un*
*it
{0}.
Proof.It is straightforward to verify that ? is a well-defined bifunctor, and t*
*hat it
is associative, symmetric and that the object {0} of Tqis a unit for ?.
2.2. Definition and properties of the category Fquad. As the category Tq is
essentially small, we can give the following definition.
Definition 2.12. The category Fquadis the category of functors from Tq to E.
Remark 2.13. By analogy with the classical definition of Mackey functors, given
in [12] and due, originally, to Dress [3], and with the work of Lindner in [8],*
* we
can view the category Fquadas the category of generalized Mackey functors over *
*Eq.
Note that, in our definition, we consider all the functors from Tqto E and not *
*only
the additive functors, unlike [12].
By classical results about functor categories and by the Yoneda lemma, we ob*
*tain
the following theorem.
Theorem 2.14. (1) The category Fquadis abelian.
(2)The tensor product of vector spaces induces a structure of symmetric mon*
*oidal
category on Fquad.
(3)For any object V of Tq, the functor PV = F2[Hom Tq(V, -)] is a projective
object and there is a natural isomorphism:
Hom Fquad(PV , F ) ' F (V )
for all objects F of Fquad.
The set of functors {PV |V 2 S} is a set of projective generators of F*
*quad,
where S is a set of representatives of isometry classes of non-degenerate
quadratic spaces. In particular, the category Fquad has enough projecti*
*ve
objects.
The transposition functor of Tqallows us to give the following definition.
8 CHRISTINE VESPA
Definition 2.15. The duality functor of Fquadis the functor D : Fquadop! Fquad
given by:
DF = -* O F O trop
for F an object of Fquad, -* the duality functor from Eop to E and tr the trans*
*po-
sition functor of Tq defined in 2.10.
The following proposition summarizes the basic properties of the duality func*
*tor
D.
Proposition 2.16. (1)The functor D is exact.
(2)The functor D is right adjoint to the functor Dop, i.e. we have a natural
isomorphism:
Hom Fquad(F, DG) ' Hom Fquadop(DopF, G) ' Hom Fquad(G, DF ).
(3)For F an object of Fquad with values in finite dimensional vector spaces,
the unit of the adjunction between Fquadand Fquadop, F ! DDopF , is an
isomorphism.
A straightforward consequence of the second point of the last proposition and
theorem 2.14 is:
Corollary 2.17. The category Fquadhas enough injective objects.
3.Connection between F and Fquad
Recall that F is the category of functors from Ef to E, where Ef is the full
subcategory of E having as objects the finite dimensional spaces. The main resu*
*lt
of this section is the following theorem.
Theorem 3.1. There is a functor
' : F ! Fquad
which satisfies the following properties:
(1)' is exact;
(2)' preserves tensor products;
(3)' is fully-faithful;
(4)if S is a simple object of F, '(S) is a simple object of Fquad.
To define the functor ' of the last theorem we need to define the forgetful *
*functor
ffl : Tq! Ef which can be viewed as an object of the category Fquad, by composi*
*ng
with Ef ,! E.
3.1. The forgetful functor ffl of Fquad.
3.1.1. Definition.
Notation 3.2. We denote by O : Eq! Ef the functor which forgets the quadratic
form.
Proposition 3.3. There exists a functor ffl : Tq! Ef defined by ffl(V ) = O(V )*
* and
ffl([V -f!W ?W 0g-W ]) = pg O O(f)
where pg is the orthogonal projection from W ?W 0to W .
The proof of this proposition relies on the following straightforward proper*
*ty of
the pseudo push-out of Eq.
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: THE FUNCTOR CATEGORY Fquad9
Lemma 3.4. For a pseudo push-out diagram in Eq
g
V ________//_V ?V 00
f|| |l|
fflffl| fflffl|
V ?V 0_k__//V ?V 0?V 00
we have the following relation in Ef:
O(g) O pf = plO O(k)
where pf and pl are the orthogonal projections associated to, respectively, f a*
*nd l.
Proof of the proposition.It is straightforward to check that the functor ffl is*
* well
defined on the classes of morphisms of Tq. To show that ffl is a functor, we ve*
*rify
that
ffl([V -Id!V -IdV ]) = Id
and the relation for the composition is a direct consequence of the above lemma.
3.1.2. The fullness of ffl. The aim of this section is to prove the following p*
*roposition:
Proposition 3.5. The functor ffl : Tq! Ef is full.
Proof.Let (V, qV ) and (W, qW ) be two objects of Tqand f 2 Hom Ef(ffl(V, qV ),*
* ffl(W, qW ))
be a linear map from V to W . We prove, by induction on the dimension of V , th*
*at
there is a morphism in Tq: T = [V -'!X = W ?Y- i W ] such that ffl(T ) = f. The
proof is based on the idea that, for a sufficiently large space X, we can obtai*
*n all
the linear maps.
As the quadratic space V is non-degenerate, we know that it has even dimensi*
*on.
To start the induction, let (V, qV ) be a non-degenerate quadratic space of *
*dimen-
sion two, with symplectic basis {a, b} and f : V ! W be a linear map. We verify
that the following linear map:
g1 : V ! W ?H1?H0 ' W ?Vect(a1, b1)?Vect(a0, b0)
a 7-! f(a) + (q(a) + q(f(a)))a1 + a0
b 7-! f(b) + (q(b) + q(f(b)))a1 + (1 + B(f(a), f(b)))b0
preserves the quadratic form. Consequently, the morphism:
T = [V -g1!W ?H1?H0 - W ]
is a morphism of Tqsuch that ffl(T ) = f.
Let Vn be a non-degenerate quadratic space of dimension 2n, {a1, b1, . .,.an*
*, bn}
be a symplectic basis of Vn and fn : Vn ! W be a linear map. By induction, there
exists a map :
gn : Vn ! W ?Y
a1 7-! f(a1) + y1
b1 7-! f(b1) + z1
. . . . . .
an 7-! f(an) + yn
bn 7-! f(bn) + zn
where yiand zi, for all integers i between 1 and n, are elements of Y , which p*
*reserves
the quadratic form and such that:
ffl([Vn gn-!W ?Y - W ]) = fn.
Let Vn+1 be a non-degenerate quadratic space of dimension 2(n + 1),
{a1, b1, . .,.an, bn, an+1, bn+1} a symplectic basis of Vn+1 and fn+1 : Vn+1 ! *
*W a
linear map. To define the map gn+1, we will consider the restriction of fn+1 ov*
*er
10 CHRISTINE VESPA
Vn and extend the map gn given by the inductive assumption. For that, we need
the following space: E ' W ?Y ?H?n0?H?n0?H1?H0, for which we specify the
notations for a basis:
E ' W ?Y ?(?ni=1Vect(ai0, bi0))?(?ni=1Vect(Ai0, Bi0))?Vect(A1, B1)?Vect(C0, D0).
We verify that the following map:
V gn+1---!W ?Y ?H?n0?H?n0?H1?H0
a1 7-! f(a1) + y1 + a10
b1 7-! f(b1) + z1 + A10
. . . . . .
ai 7-! f(ai) + yi+ ai0
bi 7-! f(bi) + zi+ Ai0
. . . . . .
an 7-! f(an) + yn + an0
bn 7-! f(bn) + zn + An0 P
an+1 7-! f(an+1)P+ (q(an+1) + q(f(an+1)))A1 + C0 + ni=1B(f(ai), f(an+1))b*
*i0
+ ni=1B(f(bi), f(an+1))Bi0
bn+1 7-! f(bn+1)P+ (q(bn+1) + q(f(bn+1)))A1P+ (1 + B(f(an+1), f(bn+1)))D0
+ ni=1B(f(ai), f(bn+1))bi0+ ni=1B(f(bi), f(bn+1))Bi0
preserves the quadratic form and we have
ffl([V -gn+1--!W ?Y ?H?n0?H?n0?H1?H0 - W ]) = f
which completes the inductive step.
3.2. Proof of theorem 3.1. Theorem 3.1 is a consequence of a general result abo*
*ut
functor categories which we recall in Proposition A.2 of the appendix. But, as *
*the
functor ffl is not essentially surjective, we can not apply directly the propos*
*ition to
the category F. Consequently, we introduce a category F0 equivalent to F.
Definition 3.6. The category Ef-(even)is the full subcategory of Ef having as
objects the F2-vector spaces of even dimension.
Notation 3.7. We denote by F0 the category of functors from Ef-(even)to E.
We have the following result:
Proposition 3.8. The categories F and F0 are equivalent.
The proof of this proposition relies on the following standard lemma.
Lemma 3.9. [7] Let n > 0 be a natural integer. Any idempotent linear map
e2n-1 : F22n! F22nof rank 2n - 1, verifies:
PFF22n-1' PFF22n. e2n-1
where PFF2n(-) = F2[Hom Ef(F2n, -)] is the standard projective object of F give*
*n by
the Yoneda lemma.
Proof of proposition 3.8.Let V be an object of Ef.
If the dimension of V is even, V is an object of Ef-(even).
If the dimension of V is odd dim(V ) = 2n - 1, by the previous lemma, PFF22n*
*-1
is a direct summand of PFF22n.
As the category Ef-(even)is a full subcategory of Ef, we obtain by propositi*
*on
A.4 of the appendix, the theorem.
Proof of theorem 3.1.The functor ' of theorem 3.1 is, by definition, the precom*
*posi-
tion functor by the functor ffl : Eq! Ef. The two first points of the theorem a*
*re clear.
As the objects of Eqare spaces of even dimension, the functor ffl : Eq! Ef fact*
*orizes
through the inclusion Ef-(even),! Ef. This induces a functor ffl0: Eq! Ef-(even)
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: THE FUNCTOR CATEGORY Fquad11
which is full and essentially surjective. We deduce from proposition A.2 that t*
*he
functor - O ffl0: F0 ! Fquadis fully-faithful and, for a simple object S of F0,*
* S O ffl0
is simple in Fquad. The theorem follows from proposition 3.8.
3.3. Duality. In this section, we prove that the duality defined over Fquadin 2*
*.15
is an extension of the duality over F, D : Fop ! F, given by DF = -*O F O (-*)op
for F an object of F and -* the duality functor from Eop to E.
Proposition 3.10. We have the following commutative diagram, up to natural
isomorphism
op op
Fop __'__//Fquad
D || D||
fflffl| fflffl|
F ____'_//_Fquad.
Proof.This relies on the following commutative diagram:
op
Tqop__ffl//_(Ef)op
tr|| -*||
fflffl| fflffl|
Tq___ffl_//Ef.
The commutativity is a consequence of classical results of linear algebra about*
* the
duality of vector spaces and the fact that a non-singular bilinear form on a ve*
*ctor
space V determines a privileged isomorphism between V and V *.
4.The category Fiso
In this section, we define a subcategory Fisoof Fquadwhich is, by theorem 4.*
*7, an
abelian symmetric monoidal category with enough projective objects. The category
Fisois related to Fquadby the following theorem.
Theorem 4.1. There is a functor
~ : Fiso! Fquad
which satisfies the following properties:
(1)~ is exact;
(2)~ preserves tensor products;
(3)~ is fully-faithful;
(4)if S is a simple object of Fiso, ~(S) is a simple object of Fquad.
We obtain a classification of the simple objects of Fisofrom the following t*
*heorem.
Theorem 4.2. There is a natural equivalence
Y
Fiso' F2[O(V )] - mod
V 2S
where S is a set of representatives of isometry classes of objects of Edegq.
4.1. Definition of the category Fiso.
4.1.1. The category Sp(Edegq). Example 2.4 implies that the category Sp(Edegq) *
*is
defined. In this section, we give some properties of this category which are si*
*milar
to those given for the category Tq.
Definition 4.3 (Transposition). The transposition functor, tr : Sp (Edegq)op !
Sp(Edegq), is defined on objects by tr(V ) = V and on morphisms by:
tr(f) = tr([V X ! W ]) = [W X ! V ],
for f an element of Hom Sp(Edegq)op(W, V ).
12 CHRISTINE VESPA
Proposition 4.4 (Orthogonal sum). There exists a bifunctor ? : Sp (Edegq) x
Sp(Edegq) ! Sq, called the orthogonal sum, defined on objects by:
?(V, W ) = V ?W
and on morphisms by:
0 g0 f?f0 g?g0
?([V- f X g-!W ], [V 0f- X0-! W 0]) = [V ?V 0--- X?X0---! W ?W 0].
This bifunctor gives Sp(Edegq) the structure of a symmetric monoidal category, *
*with
unit {0}.
4.1.2. The category Fiso. As the category Sp(Edegq) is essentially small, we ca*
*n give
the following definition.
Definition 4.5. The category Fisois the category of functors from Sp(Edegq) to *
*E.
Remark 4.6. The category Fisois equivalent to the category of Mackey functors
from Edegqto E, by the paper [8].
As for the category Fquad, we obtain the following theorem.
Theorem 4.7. (1) The category Fisois abelian.
(2)The tensor product of vector spaces induces a structure of symmetric mon*
*oidal
category on Fiso.
(3)For an object V of Sp(Edegq), the functor QV = F2[Hom Sp(Edegq)(V, -)] *
*is a
projective object and there is a natural isomorphism:
Hom Fiso(QV , F ) ' F (V )
for all objects F of Fiso.
The set of functors {QV |V 2 S} is a set of projective generators of
Fiso, where S is a set of representatives of isometry classes of degener*
*ate
quadratic spaces. In particular, the category Fiso has enough projective
objects.
Definition 4.8. The duality functor of Fisois the functor D : Fisoop! Fisogiven
by:
DF = -* O F O trop
for F an object of Fiso, -* the duality functor from Eop to E and tr is the tra*
*nspo-
sition functor of Sp(Edegq) defined in 4.3.
The following proposition summarizes the basic properties of the duality fun*
*ctor
D.
Proposition 4.9. (1)The functor D is exact.
(2)The functor D is right adjoint to the functor Dop, i.e. we have a natural
isomorphism:
Hom Fiso(F, DG) ' Hom Fisoop(DopF, G) ' Hom Fiso(G, DF ).
(3)For F an object of Fiso with values in finite dimensional vector spaces,
the unit of the adjunction between Fiso and Fisoop: F ! DDopF is an
isomorphism.
A straightforward consequence of the first point of the last proposition and*
* the-
orem 4.7 is:
Corollary 4.10. The category Fisohas enough injective objects.
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: THE FUNCTOR CATEGORY Fquad13
4.2. An equivalent definition of Fiso. In order to apply proposition A.2 in the
appendix to prove theorem 4.1, we will use the same strategy as for theorem 3.1*
*. In
other words, we will introduce a category equivalent to Fisosuch that ~ will be*
* the
precomposition functor by a full and essentially surjective functor. First, we *
*give
the following definition.
Definition 4.11. The category Sq is the full subcategory of Sp(Edegq) having as
objects the non-degenerate quadratic spaces.
Remark 4.12. A morphism of Sq is represented by a diagram V f- D -g!W
where V and W are non-degenerate quadratic spaces and D is a possibly degenerate
quadratic space.
The transposition functor and the orthogonal sum defined in the previous sec*
*tion
for the category Sp(Edegq) induce, by restriction, a transposition functor and *
*an
orthogonal sum for the category Sq.
As the category Sq is, by definition, a full subcategory of the category Sp(*
*Edegq),
we have the existence of a functor ~0from Fisoto Func(Sq, E) induced by the inc*
*lu-
sion Sq,! Sp(Edegq). The aim of this section is to show that ~0is an isomorphis*
*m.
Theorem 4.13. There exists a natural isomorphism:
Fiso' Func(Sq, E)
where Func(Sq, E) is the category of functors from Sq to E.
To prove the theorem, we require some results about the idempotents of the
category Sp(Edegq).
4.2.1. The idempotents of Sp(Edegq). We begin this section with the following n*
*ota-
tion.
Notation 4.14. Let V be an object of Edegq, ff and fi be morphisms of Hom Edegq*
*(D, V ).
We denote by fff,fithe following morphism of Hom Sp(Edegq)(V, V ):
[V -ffD fi-!V ].
Proposition 4.15. (1)An idempotent of Hom Sp(Edegq)(V, V ) is of the form
eff= [V -ffD ff-!V ]
where ff is an element of Hom Edegq(D, V ), for some D.
(2)For ff and fi two morphisms of Edegqwith range V , the idempotents effand
eficommute.
(3)For ff and fi two morphisms of Edegqwith range V , the elements 1 + effa*
*nd
1 + efiare idempotents of F2[Hom Sp(Edegq)(V, V )] which commute.
Proof.By definition of Sp(Edegq), fff,fiO fff,fi= [V D0 ! V ] where D0 is the
pullback in Edegqof the diagram D fi-!V -ffD and fff,fiO fff,fi= fff,fiif and o*
*nly
if there exists an isomorphism g : D ! D0 such that the following diagram is
14 CHRISTINE VESPA
commutative:
D AUUUUU+
+AA'AUUUUUfiUUUA++
+gAA__A+ -1UUUUU
++ 0_g___// _UUUU**//_
+ D D fi V
++ | |
ff++ g-1| |ff
++ fflffl|fflffl|
++D __fi_//V
++|
++ff|
~~+fflffl|
V
This implies that ff = fi. Consequently fff,fiO fff,fi= fff,fiif and only if ff*
*f,fi= eff.
The second point is straightforward and the last point is a direct consequence
of the second one by a standard result about idempotents (noting that -1 = 1 in
F2).
4.2.2. Proof of theorem 4.13. The proof of theorem 4.13 relies on the following
crucial lemma.
Lemma 4.16. For V an object of Span(Edegq) and ff : A ,! V a subobject of V in
Edegq,
QV . eff' QA .
Proof.Let ff* : QA ! QV (respectively ff* : QV ! QA ) be the morphism of Fiso
which corresponds by the Yoneda lemma to the element [V -ffA Id-!A] of QV (A)
(respectively [A -IdA ff-!V ] of QA (V )).
As [V ff-A -Id!A] O [A -Id A -ff!V ] = Id we have ff* O ff* = Id and as
[A -IdA ff-!V ] O [V -ffA Id-!A] = effwe have ff* O ff* = .eff.
Proof of theorem 4.13.Let A be an object of Sp(Edegq), there exists an object V*
* of
Sq such that A is a subobject of V . By the previous lemma, we deduce that QA is
a direct summand of QV . As the category Sq is a full subcategory of Sp(Edegq) *
*we
obtain, by proposition A.4 of the appendix, the theorem.
4.3. Relation between Fisoand Fquad. The main result of this section is theorem
4.1, which gives the existence of an exact, fully-faithful functor ~ : Fiso! Fq*
*uad
which preserves the simple objects. To define the functor ~ of this theorem we *
*need
to define and study the functor oe : Tq! Sq.
4.3.1. Definition of oe : Tq! Sq.
Proposition 4.17. There exists a monoidal functor oe : Tq! Sqdefined by oe(V ) =
V and
oe([V ! X W ]) = [V V x W ! W ]
X
where V x W is the pullback in Edegq.
X
The proof of this proposition relies on the following important lemma.
Lemma 4.18. For V , W and X objects of Eqand V -f!W , V -g!X morphisms of
Eq, we have:
X x W ' V
X?VW
where - x - is the pullback over A in Edegqand -? - is the pseudo push-out over
A B
B in Eq.
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: THE FUNCTOR CATEGORY Fquad15
Proof.It is a straightforward consequence of the definitions of the pseudo push*
*-out
and the pullback.
Remark 4.19. The result of the previous lemma explains why we have imposed
that the morphisms of the category Edegqare monomorphisms.
Proof of proposition 4.17.The functor oe is well defined on the classes of morp*
*hisms
of Tq. To show that oe is a functor, we verify that
oe([V -Id!V -IdV ]) = Id
and, that oe respects composition, which is a direct consequence of the above l*
*emma.
Finally, the following consequence of the definition of the pullback and of the*
* or-
thogonal sum:
for T1 = [V1 ! X1 W1] and T2 = [V2 ! X2 W2], we have:
(V1 x W1)?(V2 x W2) ' (V1?V2) x (W1?W2)
X1 X2 X1?X2
implies that oe preserves the monoidal structures.
4.3.2. The fullness of oe. The aim of this section is to prove the following pr*
*oposi-
tion.
Proposition 4.20. The functor oe is full.
The proof of this proposition relies on the following technical lemma.
Lemma 4.21. Let D be a degenerate quadratic space of dimension r which has the
following decomposition
D = (x1, ffl1)? . .?.(xr, fflr)
where ffli2 {0, 1}, H a non-degenerate quadratic space and f an element of
Hom Edegq(D, H). Then there exists elements k1, . .,.kr in H and a non-degenera*
*te
quadratic space H0 such that
H = Vect(f(x1), k1)? . .?.Vect(f(xr), kr)?H0
and B(f(xi), ki) = 1 where B is the underlying bilinear form.
Proof.We prove this lemma by induction on the dimension r of the space D.
For r = 1, we have f : (x1, ffl1) ! H and f(x1) = h1. As H is, by hypothesis,
non-degenerate there exists an element k1 in H such that B(h1, k1) = 1. Then, t*
*he
space K = Vect(h1, k1) is a non-degenerate subspace of H, so we have H = K?K? ,
with K? non-degenerate.
Suppose that the result is true for r = n. Let (x1, . .,.xn, xn+1) be linear*
*ly inde-
pendent vectors and f : (x1, ffl1)? . .?.(xn, ffln)?(xn+1, ffln+1) ! H. By rest*
*riction,
we have
f O i : (x1, ffl1)? . .?.(xn, ffln) ! H
and by the inductive assumption, we have the existence of k1, . .,.kn in H such
that:
H = Vect(f(x1), k1)? . .V.ect(f(xn), kn)?H0.
We decompose f(xn+1) over this basis to obtain the following decomposition
f(xn+1) = ni=1(ffif(xi) + fiiki) + h0
where ffi and fii are elements of F2 and h0 is an element of H0. As f preserves
the quadratic form and, consequently, the underlying bilinear form, we have fii*
* =
B(f(xn+1), f(xi)) = B(xn+1, xi) = 0 for all i. Hence
(4.21.1) f(xn+1) = ni=1ffif(xi) + h0.
16 CHRISTINE VESPA
As the vectors (f(x1), . .,.f(xn), f(xn+1)) are linearly independent, by the *
*in-
jectivity of f, we have h0 6= 0 and, as H0 is non-degenerate, there exists an e*
*le-
ment k0 in H0 such that B(h0, k0) = 1. We deduce the following decomposition
H0= Vect(h0, k0)?H00.
In the equality (4.21.1), after reordering, we can suppose that
ff1 = . .=.ffp = 1
and
ffp+1 = . .=.ffn = 0.
Then we have,
B(f(xn+1), ki) = 1 pouri = 1, . .,.p
and
B(f(xn+1), ki) = 0 pouri = p + 1, . .,.n.
Consequently, by the following decomposition of H
H = ?pi=1Vect(f(xi), ki+ k0)?nj=p+1Vect(f(xj), kj)?Vect(f(xn+1), k0)?H00
we obtain the result.
Proof of proposition 4.20.We prove that, for a morphism S = [V- f D g-!W ] of
Sq, there exists a morphism T in Hom Tq(V, W ) such that oe(T ) = S.
First we decompose the morphism S as an orthogonal sum of more simple mor-
phisms. By theorem 1.10 we have D ' H?Rad (V ) such that H is non-degenerate
and Rad(V ) is isometric to either (x, 0)?r or (x, 1)?r, where r is the dimensi*
*on of
Rad(V ). By the previous lemma, we can decompose the morphisms f and g in the
following form:
f : H?Rad (D) ! H?D0?V 0' V
and
g : H?Rad (D) ! H?D00?W 0' W
where D0(respectively D00) is one of the non-degenerate spaces constructed in l*
*emma
4.21 and V 0(respectively W 0) is the orthogonal of H?D0(respectively H?D00). We
deduce that :
S = [H H ! H]?[V 0 0 ! W 0]?[D0 Rad(D) ! D00].
Furthermore, again by theorem 1.10
[D0 Rad(D) ! D00] = ?i[Vi (xi, ffli) ! Wi]
where, by lemma 4.21, Vi and Wi are spaces of dimension two, therefore:
S = [H H ! H]?[V 0 0 ! W 0]?[V1 (x1, ffl1) ! W1]? . .?.[Vr (xr, fflr) ! *
*Wr].
According to proposition 4.17, it is enough to prove that for each morphism
Sff, which appears as a factor in the previous decomposition of S, there exists*
* a
morphism of Tq, Tff, such that oe(Tff) = Sff.
Obviously, we have
oe([H ! H H]) = [H H ! H]
and
oe([V ! V ?W W ]) = [V 0 ! W ].
For the morphisms [V (x, ffl) ! W ], we have to consider several cases.
In the case ffl = 0, as all the non-zero element x of H1 verify q(x) = 1, we *
*have
Hom Eqdeg((x, 0), H1) = ;. Consequently, we have to consider only the following
morphism:
S1 = [H0 f-(x, 0) g-!H0].
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: THE FUNCTOR CATEGORY Fquad17
After composing by an element of O+2= O(H0), we can suppose that f(x) =
g(x) = a0. The morphism
0 g0
T1 = [H0 f-!H0?H0 - H0]
where, f0 is defined by:
f0(a0) = a0 andf0(b0) = b0 + a00
and g0 is defined by:
g0(a0) = a0 andg0(b0) = b0 + b00
satisfies oe(T1) = S1.
In the case ffl = 1, we have to consider three kinds of morphisms.
For
S2 = [H0 f-(x, 1) g-!H0],
as only the element a0+b0 of H0 verify q(a0+b0) = 1, we have f(x) = g(x) = a0+b*
*0.
The morphism
0 g0
T2 = [H0 f-!H0?H0 - H0]
where, f0 is defined by:
f0(a0) = a0 + a00andf0(b0) = b0 + a00
and g0 is defined by:
g0(a0) = a0 + b00andg0(b0) = b0 + b00
satisfies oe(T2) = S2.
For
S3 = [H1 f-(x, 1) g-!H1],
after composing by an element of O-2= O(H1), we can suppose that f(x) = g(x) =
a1. The morphism
0 g0
T3 = [H1 f-!H1?H0 - H1]
where, f0 is defined by:
f0(a1) = a1 andf0(b1) = b1 + b0
and g0 is defined by:
g0(a1) = a1 andg0(b1) = b1 + a0
satisfies oe(T3) = C3.
For
S4 = [H0 f-(x, 1) g-!H1],
we have f(x) = a0 + b0 and, after composing by an element of O-2= O(H1), we
can suppose that g(x) = a1. The morphism
0 g0
T4 = [H0 f-!H1?H0 - H1]
where, f0 is defined by:
f0(a0) = a1 + b1 + a0 + b0 etf0(b0) = b1 + a0 + b0
and g0 is defined by:
g0(a1) = a1 etg0(b1) = b1
satisfies oe(T4) = C4.
The final possibility results from the previous one by transposition.
18 CHRISTINE VESPA
4.3.3. Proof of theorem 4.1. The functor ~ of the theorem 4.1 is, by definition,
the precomposition functor by oe : Tq! Sq. The two first point of the theorem a*
*re
clear. By the proposition 4.17 the functor oe is full and, by definition, it is*
* essentially
surjective. So, the two last points of the theorem 4.1 are direct consequences *
*of the
proposition A.2 given in the appendix.
4.3.4. Duality.
Proposition 4.22. We have the following commutative diagram, up to natural
isomorphism
op op
Fisoop__~__//Fquad
D || |D|
fflffl| |fflffl
Fiso___~__//_Fquad.
Proof.This relies on the following commutative diagram:
op
Tqop_oe__//(Sp(Edegq))op
tr|| tr||
fflffl| fflffl|
Tq ___oe__//_Sp(Edegq)
which is a direct consequence of the definitions.
4.4. The isotropic functors. In this section we are interested in an important
family of functors of Fisonamed the isotropic functors; the choice of terminolo*
*gy
will be explained below. After giving the definition of these functors we prove*
* that
they are self-dual. We will show in the following sections that these functors *
*give
rise to a family of projective generators of the category Fiso.
4.4.1. Definition. Let (IdV)* be the element of
DQV (V ) = QV (V )* = Hom (F2[End Sp(Edegq)(V )], F2)
defined by:
(IdV)*([IdV]) = 1 and (IdV)*([f]) = 0 for allf 6= IdV.
We denote by aV : QV ! DQV the morphism of Fisowhich corresponds by the
Yoneda lemma to the element (IdV)* of DQV (V ).
Definition 4.23. The isotropic functor isoV : Sp(Edegq) ! E of Fisois the image
of QV by the morphism aV .
Notation 4.24. We denote by KV the kernel of aV . We have the following short
exact sequence:
0 ! KV ! QV -aV-!isoV ! 0.
In the following lemma, we give, an explicit description of the vector spaces
KV (W ) and isoV (W ), which are elementary consequences of the definition.
Lemma 4.25. For an object W of Sp(Edegq), we have that:
o KV (W ) is the subvector space of QV (W ) generated by the elements [V
H ! W ] where H 6' V ;
o as a vector space, isoV (W ) is isomorphic to the subspace of QV (W ) ge*
*ner-
ated by the elements [V -IdV ! W ]. Consequently isoV (W ) has basis the
set Hom Edegq(V, W ).
Remark 4.26. Observe that the isomorphism given in the second point is not nat-
ural.
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: THE FUNCTOR CATEGORY Fquad19
Remark 4.27. The terminology "isotropic functor" was motivated by the first case
considered by the author. For the quadratic space (x, 0), we have:
iso(x,0)(V ) ' F2[IV ]
where IV = {v 2 V \ {0}| q(v) = 0} is the isotropic cone of the quadratic space*
* V .
4.4.2. Self-duality.
Remark 4.28. For simplicity, we will denote in this section, Dop by D.
To begin we recall several definitions.
Definition 4.29. (1)A morphism b : F ! DF is self-adjoint if b = Db O jF ,
where jF : F ! D2F is the unit of the adjunction between D and Dop.
(2)A functor F is self-dual if there exists an isomorphism
fl : F -'!DF
which is self-adjoint.
The main result of this section is the following proposition.
Proposition 4.30. The isotropic functors of Fquadare self-dual.
The proof of this proposition relies on the following lemma.
Lemma 4.31. Let F be a functor which takes finite dimensional values and a :
F ! DF a self-adjoint morphism, then im(a) is self-dual.
Remark 4.32. This lemma and its proof are direct consequences of lemma 1.2.4
in [10].
Proof.The morphism a admits the following factorisation by im(a)
p O "j
a : F____////_im(a)_//_DF.
We deduce the existence of "awhich makes the following diagram commutative
p
0______//Ker(a)_____//F_______//_im(a)_____//0
| | |
| |jF |"a
fflffl| fflffl|Dj fflffl|
0_____//Ker(Dj)____//D2F_____//D(im(a))___//_0.
As F has finite dimensional values, the unit of the adjunction jF : F ! D2F is *
*an
isomorphism. Consequently, by the following commutative diagram
p O "j
F _______////_im(a)___//DF
' jF|| |"a| ||||
fflffl| fflffl|O " ||
D2F __Dj////_D(im(a))Dp//_DF
we obtain that "ais an isomorphism.
If we dualize the first commutative diagram, we obtain:
Dp O D"a= DjF O D2j.
So
Dp O D"aO jim(a)O p = DjF O D2j O jim(a)O p = j O p = a.
We have also
Dp O "aO p = Dp O Dj O jF = Da O jF = a.
We deduce that D"aO jim(a)= "a.
20 CHRISTINE VESPA
Proof of proposition 4.30.By the previous lemma, it is enough to prove that the
morphism aV : QV ! DQV is self-dual. By the Yoneda lemma we have the
following commutative diagram:
Hom Fiso(QV , DQW )__f___//_HomFiso(QW , DQV )
' || |'|
fflffl| fflffl|
F2[Hom Sp(Edegq)(W, V_)]*//_F2[Hom Sp(Edeg)(V, W )]*
F2[tr]* q
where tr : Sp(Edegq)op ! Sp(Edegq) is the transposition functor defined in 4.3 *
*and
f is the natural isomorphism given in proposition 4.9. By definition, aV : QV !
DQV corresponds, by the Yoneda lemma, to the element (IdV)* of DQV (V ). Since
tr(IdV) = IdV, we deduce that aV = DaV O jQV .
4.5. Decomposition of the projective objects QV of Fiso. The aim of this
section is to prove the following theorem.
Theorem 4.33. For V an object of Edegq, we have:
M
QV ' isoA
A 2 SV
where SV is the set of subobjects of V in Edegq, represented by a morphism ff :*
* A ,!
V .
The proof of this theorem relies on the following proposition.
Proposition 4.34. For an object V of Edegq, the element EV of F2[Hom Sp(Edegq)(*
*V, V )]
Y
EV = (1 + eff)
ff : A ,! V
A 2 SV \ V
verifies:
(1)EV . EV = EV
(2)QV . EV ' isoV . In particular Hom (isoV , F ) ' F (EV ) . F (V ).
Proof.The first point is a direct consequence of proposition 4.15 (3) and (1).
For the second point, we consider the following split short exact sequence:
0 ! QV . (1 + EV ) ! QV ! QV . EV ! 0
and we recall that we have by notation 4.24 the following short exact sequence:
0 ! KV ! QV -aV-!isoV ! 0.
By the 5-lemma, to obtain the result, it is sufficient, to prove that:
QV . (1 + EV ) ' KV .
By expanding 1 + EV , we obtain that
Y
1 + EV = efl
fl : A ,! V
A 2 RV
where RV is a subset of SV \ V . Consequently
X
QV . (1 + EV ) QV . efl= KV
fl : A ,! V
A 2 SV \ V
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: THE FUNCTOR CATEGORY Fquad21
where the last equality is a direct consequence of lemma 4.25.
On the other side, for fl : A ,! V with A 2 SV \ V , we have
Y
(QV . efl) . EV = QV . efl. (1 + efl) (1 + eff) = 0
ff : A ,! V
ff 6= fl
A 2 SV \ V
where the first equality follows from proposition 4.15 (3).
Consequently QV . efl QV . (1 + EV ) and
X
KV = QV . efl QV . (1 + EV ).
fl : A ,! V
A 2 SV \ V
We deduce that
QV . (1 + EV ) ' KV .
An important consequence of the previous proposition is given in the following
corollary.
Corollary 4.35. For V and W objects of Edegq, we have:
ae
Hom Fiso(isoV , isoW ) ' F2[O(V0)] ifWo'tVherwise.
Proof.By lemma 4.25 an element of isoW (V ) is represented by a linear sum of
[W W -f!V ].
o If W 6' V . We have
(1 + ef)[W W -f!V ] = [W W -f!V ] + [W W -f!V ] = 0
as the idempotents (1 + ef) commute by proposition 4.15, we deduce that:
Hom (isoV , isoW ) = isoW (EV ) isoW (V ) = 0
where the first equality is given by proposition 4.34 (2).
o If W ' V . For ff : A ,! V with A 2 SV \ V we have:
eff[W W ! V ] = [W A ! V ] 2 KW (V ).
So isoW (eff)[W W ! V ] = 0 and
Hom (isoV , isoW ) = isoW (EV ) isoW (V ) = isoW (V ) ' Hom Edegq(W, V ) ' F2[*
*O(V )]
by lemma 4.25.
Proposition 4.36. For ff : A ,! V a subobject of V , the idempotent Effdefined
by: X
Eff= eff (1 + efi)
fi : B ,! A
B 2 SA \ A
verifies
QV . Eff' isoA .
Proof.By the proposition 4.15, Effis clearly an idempotent. The result follows
from proposition 4.34 and lemma 4.16.
22 CHRISTINE VESPA
Proof of theorem 4.33.By the proof of 4.34, there exists an exact sequence
M
QV . efl! QV ! isoV ! 0
fl : A ,! V
A 2 SV
hence a complex M
isoA ! QV ! isoV ! 0
fl : A ,! V
A 2 SV
by proposition 4.36. We deduce from proposition 4.15 that the idempotents given
in the proposition 4.36 are orthogonal. Consequently the map
M
isoA ! QV
fl : A ,! V
A 2 SV
is injective. Furthermore, by lemma 4.25, for an object X of Sp(Edegq)
i M j
QV (X) ' isoV (X) isoW (X) .
ff : W ,! V
W 2 SV \ V
We deduce that the previous complex is a short exact sequence, which is split by
proposition 4.34.
By theorem 4.7 and the self-duality of the isotropic functors given in propos*
*ition
4.30, we obtain the following corollary of theorem 4.33.
Corollary 4.37. The set of functors {isoV |V 2 S} is a set of projective genera*
*tors
(resp. injective cogenerators) of Fiso, where S is a set of representatives of *
*isometry
classes of possibly degenerate quadratic spaces.
4.6. Proof of theorem 4.2. In this section we prove the equivalence between the
category Fisoand the product of categories of modules over the orthogonal group*
*s.
Proof of theorem 4.2.In [11] we have the following result (Corollary 6.4, p. 10*
*3).
For any abelian category C the following assertions are equivalent:
(1)The category C has arbitrary direct sums and {Pi}i2Iis a set of projecti*
*ve
generators of finite type of C.
(2)The category C is equivalent to the subcategory Funcadd(Pop, Ab) of
Func(Pop, Ab) having as objects the functors satisfying F (f + g) = F (f*
*) +
F (g) where f and g are morphisms of Hom Pop(V, W ) and P is the full
subcategory of C having as objects {Pi | i 2 I}.
Let C be the category Fiso. By corollary 4.37, the set of functors {isoV |V *
*2 S} is
a set of projective generators of Fiso, where S is a set of representatives of *
*isometry
classes of degenerate quadratic spaces. By proposition 4.34 (2) isoV is a dire*
*ct
summand of QV ; as QV is of finite type, we deduce that isoV is of finite type.
Consequently, by the previous result we obtain that
Fiso' Funcadd(Pop, Ab)
where P is the full subcategory of Fisohaving as objects the isotropic functors*
*. By
corollary 4.35, Hom Fiso(isoV , isoW ) = 0 if V 6' W . Consequently
Y op
Funcadd(Pop, Ab) ' Funcadd(IsoV , Ab)
V 2O(V )
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: THE FUNCTOR CATEGORY Fquad23
where IsoV is the full subcategory of Fisowith one object, the functor isoV . By
corollary 4.35, Hom Fiso(isoV , isoV ) = F2[O(V )]; we deduce that
Y
Fiso' F2[O(V )] - mod.
V 2S
Appendix A. Properties of the precomposition functor
In this appendix we list several results about functor categories. We have c*
*hosen
to provide proofs since, even if these results are well-known, they are not eas*
*y to
find in the literature.
We are interested in the following question:
Let C and D be two categories, A be an abelian category, F : C ! D be a
functor and -OF : Func(D, A) ! Func(C, A) be the precomposition functor, where
Func(C, A) is the category of functors from C to A.
When the functor F has a property P, what can we deduce for the precompositi*
*on
functor?
Before giving three answers in the following propositions, we recall that, b*
*y [6],
Func(C, A) and Func(D, A) are abelian categories as the category A is abelian.
Furthermore, we remark that the precomposition functor is exact.
Proposition A.1. If F is essentially surjective, then - O F is faithful.
Proof.As the precomposition functor is exact, it is sufficient to prove that, i*
*f H is
an object of Func(D, A) such that H O F = 0, then H = 0. For an object D of D,
there exists an object C of C such that F (C) ' D as F is essentially surjectiv*
*e. So
H(D) ' H(F (C)) = H O F (C) = 0.
Proposition A.2. If F is full and essentially surjective, then:
(1)the precomposition functor is fully-faithful;
(2)any subobject of an object in the image of the precomposition functor is
isomorphic to an object in the image of the precomposition functor;
(3)the image by the precomposition functor of a simple functor of Func(D, A)
is a simple functor of Func(C, A).
Proof. (1)According to proposition A.1, the functor - O F is faithful.
For the fullness, we consider two objects G and H of Func(D, A) and ff
an element of Hom Func(C,A)(G O F, H O F ). We want to prove that there
exists a morphism fi of Hom Func(D,A)(G, H) such that fi O F = ff.
Let D be an object of D, as F is essentially surjective, we can chose *
*an
object C of C such that there exists an isomorphism:
OE : F (C) ! D.
We define an element fi(D) of Hom A(G(D), H(D)) by the following com-
position
-1) ff(C) H(OE)
G(D) G(OE-----!G O F (C) ---! H O F (C) ---! H(D).
Now, we show that fi defines a natural transformation from G to H.
Let f : D ! D0 be an element of Hom D(D, D0) and OE : F (C) ! D and
OE0: F (C0) ! D0be the isomorphisms associated to the choices of C and C0
by the essential surjectivity of F . As the functor F is full, there exi*
*sts an
element g of Hom C(C, C0) such that
(A.2.1) F (g) = OE0-1O f O OE.
24 CHRISTINE VESPA
Moreover, the following diagram is commutative
fi(D)
G(D) ________//_H(D)
G(OE-1)|| |H(OE-1)|
fflffl|ff(C) |fflffl
G O F (C)_____//_H O F (C)
GOF(g)|| |HOF(g)|
fflffl|ff(C0) |fflffl
G O F (C0)____//H O F (C0)
G(OE0)|| |H(OE0)|
fflffl|fi(D0) |fflffl
G(D0)________//H(D0)
because the higher square (respectively lower) commute by the definition
of fi(D) (respectively fi(D0)) and the commutativity of the square in the
center is a consequence of the naturality of ff. Since
G(OE0) O GF (g) O G(OE-1) = G(OE0O F (g) O OE-1) = G(f)
and
H(OE0) O HF (g) O H(OE-1) = H(OE0O F (g) O OE-1) = H(f)
where the first equalities rise from the functoriality of G and H and the
second ones rise from the relation A.2.1, we deduce that fi is natural.
(2)Let G be an object of Func(D, A) and H be a subobject of G O F . As in
the first point, for an object D of D, by the essential surjectivity of *
*F , we
can chose an object C of C such that there exists an isomorphism:
OE : F (C) ! D.
We define an object of A by:
K(D) := H(C).
As in the first point, for f : D ! D0 a morphism of Hom D(D, D0), we
denote by OE : F (C) ! D and OE0: F (C0) ! D0the isomorphisms associated
to the choices of C and C0 by the essential surjectivity of F . Since t*
*he
functor F is full, there exists an element g of Hom C(C, C0) such that
(A.2.2) F (g) = OE0-1O f O OE.
We define a morphism
K(f) : K(D) ! K(D0)
by K(f) = H(g). We obtain the following commutative diagram:
" G(OE)
K(D) _______H(C)O_____//G O F (C)___//_G(D)
K(f)|| H(g)|| GOF(g)|| G(f)||
fflffl| fflffl|" fflffl|G(OE0)fflffl|
K(D0) ______H(C0)O___//_G O F (C0)_//_G(D0).
Since the horizontal arrows of the diagram are monomorphisms, the def-
inition of K(f) does not depend of the choice of the morphism g in A.2.2.
The functor K, thus defined, satisfies K O F ' H. Moreover, we deduce
from the commutativity of the above diagram that K is a subfunctor of G.
GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: THE FUNCTOR CATEGORY Fquad25
(3)Let S be a simple functor of Func(D, A) and (- O F )(S) = S O F its image
by - O F . Suppose that S O F is not simple, so that there exists a prop*
*er
subfunctor H of S O F . By the proof of the point (2), of the propositio*
*n,
there exists a subfunctor K of S such that:
H ' K O F.
Since H is a proper subfunctor of S O F , K is a proper subfunctor of S.
This contradicts the simplicity of S.
Remark A.3. The constructions used in the proposition are closely related to the
left Kan extension. We have prefered to give an explicit proof rather than a sl*
*icker
one using the Kan extension.
In the following proposition, we take A = ModA the category of right A-modul*
*es
where A is a commutative ring with unit. By the Yoneda lemma, we know that the
functor
PXC: C Hom-C(X,-)-------!Ens A[-]---!ModA
is a projective object of Func(C, ModA ), where A[-] is the linearisation funct*
*or.
We have, in this case, the following proposition.
Proposition A.4. If F : C ! D is full and, for all objects D of D, there exists*
* an
object C of C such that PDD is a direct summand of PFDCthen:
(1)the precomposition functor - O F is faithful;
(2)a morphism oe of Func(D, ModA ) such that (- O F )(oe) is an isomorphism,
is an isomorphism;
(3)the categories Func(D, ModA ) and Func(C, ModA ) are equivalent.
Proof. (1)As the precomposition functor is exact, it is sufficient to prove *
*that,
for H an object of Func(D, A) such that H O F = 0, H = 0. Let D be
an object of D, then H(D) = Hom(PDD, H) by the Yoneda lemma. By
hypothesis, this is a direct summand of
Hom(PFDC, H) ' H O F (C) = 0.
Consequently H(D) = 0.
(2)Let oe : F ! G be a morphism of Func(D, ModA ). We have the following
exact sequence:
0 ! Ker(oe) ! F -oe!G ! Coker(oe) ! 0.
As (- O H)(oe) is an isomorphism, Ker(oe) O H = 0 and Coker(oe) O H = 0.
Consequently, Ker(oe) = Coker(oe) = 0 by the argument of the previous
point. Hence, oe is an isomorphism.
(3)By general results, the functor (- O F ) : Func(D, ModA ) ! Func(C, ModA*
* )
admits a right adjoint, given by Kan extension. We will denote by (-")
this adjoint. We will prove that (-") and (- O F ) define an equivalence*
* of
categories.
As F is full, we have, for all objects W of C
PFDC(F W ) = A[HomD (F C, F W )] = A[HomC(C, W )] = PCC(W ).
It follows that PFDCOF = PCC. As PDDOF is a direct summand of PFDCOF by
hypothesis, we deduce that PDDO F is a direct summand of PCD and hence
projective. Consequently, (-") is an exact functor since
H"(D) ' Hom (PDD, "H) ' Hom (PDDO F, H).
26 CHRISTINE VESPA
Let H be an object of Func(C, ModA ), then there is a sequence of iso-
morphisms:
(- O F ) O (-")(H) = "H(F Z) ' Hom(PFDZO F, H) ' Hom(PZC, H) ' H(Z).
So, the counit of the adjunction
(A.4.1) (- O F ) O (-") ! Id
is an isomorphism.
Consequently, the unit of the adjunction
(A.4.2) Id ! (-") O (- O F )
induces, by composition with (- O F ), an isomorphism
(A.4.3) (- O F ) ! (- O F ) O (-") O (- O F ).
So, by the previous point of the proposition, the unit of the adjuncti*
*on
Id ! (-") O (- O F ) is an isomorphism.
References
1.Cahit Arf, Untersuchungen "uber quadratische Formen in K"orpern der Charakte*
*ristik 2. I, J.
Reine Angew. Math. 183 (1941), 148-167.
2.Jean B'enabou, Introduction to bicategories, Reports of the Midwest Category*
* Seminar,
Springer, Berlin, 1967, pp. 1-77.
3.Andreas W. M. Dress, Contributions to the theory of induced representations,*
* Algebraic K-
theory, II: "Classical" algebraic K-theory and connections with arithmetic (*
*Proc. Conf., Bat-
telle Memorial Inst., Seattle, Wash., 1972), Springer, Berlin, 1973, pp. 183*
*-240. Lecture Notes
in Math., Vol. 342.
4.Vincent Franjou, Eric M. Friedlander, Teimuraz Pirashvili, and Lionel Schwar*
*tz, Rational rep-
resentations, the Steenrod algebra and functor homology, Panoramas et Synth`*
*eses [Panoramas
and Syntheses], vol. 16, Soci'et'e Math'ematique de France, Paris, 2003.
5.Vincent Franjou, Eric M. Friedlander, Alexander Scorichenko, and Andrei Susl*
*in, General
linear and functor cohomology over finite fields, Ann. of Math. (2) 150 (199*
*9), no. 2, 663-
728.
6.Pierre Gabriel, Des cat'egories ab'eliennes, Bull. Soc. Math. France 90 (196*
*2), 323-448.
7.Nicholas J. Kuhn, Generic representations of the finite general linear group*
*s and the Steenrod
algebra. II, K-Theory 8 (1994), no. 4, 395-428.
8.Harald Lindner, A remark on Mackey-functors, Manuscripta Math. 18 (1976), no*
*. 3, 273-278.
9.Albrecht Pfister, Quadratic forms with applications to algebraic geometry an*
*d topology, Lon-
don Mathematical Society Lecture Note Series, vol. 217, Cambridge University*
* Press, Cam-
bridge, 1995. _
10.Laurent Piriou, Sous-objets de I n dans la cat'egorie des foncteurs entre *
*F2-espaces vecto-
riels, J. Algebra 194 (1997), no. 1, 53-78.
11.N. Popescu, Abelian categories with applications to rings and modules, Acade*
*mic Press, Lon-
don, 1973, London Mathematical Society Monographs, No. 3.
12.Jacques Th'evenaz and Peter Webb, The structure of Mackey functors, Trans. A*
*mer. Math.
Soc. 347 (1995), no. 6, 1865-1961.
13.Christine Vespa, La cat'egorie Fquaddes foncteurs de Mackey g'en'eralis'es p*
*our les formes
quadratiques sur F2, Ph.D. thesis, Universit'e Paris 13, 2005.
Laboratoire Analyse, G'eom'etrie et Applications, UMR 7539, Institut Galil'e*
*e, Uni-
versit'e Paris 13, 93430 Villetaneuse, France
E-mail address: vespa@math.univ-paris13.fr