GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS IN THE CATEGORY Fquad

GENERIC REPRESENTATIONS OF ORTHOGONAL GROUPS: PROJECTIVE FUNCTORS IN THE CATEGORY

F_quad

CHRISTINE VESPA

Abstract. In this paper, we continue the study of the category of functors Fquad, associated to F2-vector spaces equipped with a nonde- generate quadratic form, initiated in [12] and [11]. We define a filtration of the standard projective objects inFquad; this refines to give a decom- position into indecomposable factors of the two first standard projective objects inFquad: PH₀ andPH₁. As an application of these two decom- positions, we give a complete description of the polynomial functors of the categoryFquad.

Mathematics Subject Classification: 18A25, 16D90, 20C20.

Keywords: functor categories; quadratic forms overF2; Mackey functors;

representations of orthogonal groups overF2.

Introduction

In the paper [12] we defined the category of functorsF_quadfrom a category having as objects the nondegenerate F2-quadratic spaces to the categoryE ofF2-vector spaces, whereF2 is the field with two elements. The motivation for the construction of this category is to obtain an analogous framework for the orthogonal groups over F², to that which exists for the general linear groups. We recall that the category F of functors from the category E^f of finite dimensional F2-vector spaces to the categoryE of allF2-vector spaces is a very useful tool for the study of the stable cohomology of the general linear groups with suitable coefficients (see [4]). Another motivation, in topology, for the study of the category F is the connection which exists between this category and unstable modules over the Steenrod algebra (see [9]). In order to have a good understanding of the category F_quad, we seek to classify its simple objects. We constructed in [12] two families of simple objects inF_quad. The first one is obtained by the fully-faithful, exact functor ι : F → Fquad, defined in [12], which preserves simple objects. By [6], the simple objects in F are in one-to-one correspondence with the irreducible representations of finite general linear groups over F2. The second family is obtained by the fully-faithful, exact functor κ : F_iso → F_quad, which preserves simple objects, where F_iso is equivalent to the product of the categories of modules over the orthogonal groups of possibly degenerate

Date: April 11, 2008.

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quadratic forms. In [11], we constructed two families of simple objects in the category F_quad which are neither in the image of ι nor in the image of κ. These simple objects are subfunctors of the tensor product between an object in the image of ι and an object in the image of κ. We proved that these simple objects in F_quad are the composition factors of two particular mixed functors, defined in [11].

The aim of this paper is to begin a programme to obtain a complete clas- sification of the simple objects in F_quad. Accordingly, we seek to decompose the projective generators of this category into indecomposable factors and to obtain the simple factors of these indecomposable factors. This paper begins the study of the standard projective objects in the category F_quad. Although explicit decompositions of all the projective generators are not provided in this paper, we give several useful tools, results and examples for the realization of this programme. Furthermore, we deduce from the results contained in this paper several interesting consequences for the structure of the category F_quad. In work in progress, we obtain a general decomposition of standard projective object PH ofFquadwhich is indexed by the subspaces of H. Here we present explicit decompositions of the standard projective objects associated to “small” quadratic spaces, since these decompositions play a fundamental rˆole in the category F_quad (for example, for the de- scription of the polynomial functors of F_quad). Furthermore, recall that the decompositions of the injective standardI_F^F₂ of the category F and thus, by duality, that of the projective standard P_F^F₂, is fundamental for the compre- hension of the other injective standards ofF. Hence, the decompositions of the two smaller projective standard of F_quad represent an important step in the understanding of the category F_quad.

We briefly summarize the contents of this paper. After some recollections on the category F_quad, where we recall the definitions of the isotropic func- tors and the mixed functors, we define a filtration of the standard projective objects P_V inF_quad:

0⊂P_V⁽⁰⁾ ⊂P_V⁽¹⁾ ⊂. . .⊂P_V^(dim(V)−1) ⊂P_V^(dim(V)) =P_V.

We obtain a general description of the two extremities of this filtration.

Theorem. Let V be a nondegenerate F²-quadratic space.

(1) There is a natural equivalence: P_V⁽⁰⁾ ' ι(P_(V^F ₎), where ι : F → Fquad, is the functor that forgets the quadratic form and P_(V^F ₎ is the standard projective object in F associated to the vector space (V).

(2) The functor P_V⁽⁰⁾ is a direct summand of P_V.

Proposition. Let V be a nondegenerate F2-quadratic space, we have a nat- ural equivalence:

P_V/P_V^(dim(V)−1) 'κ(iso_V)

where κ:F_iso → F_quad and iso_V is an isotropic functor in F_iso.

An important consequence of the Theorem concerning the functor P_V⁽⁰⁾ is given in following result.

Theorem. The category ι(F) is a thick subcategory of F_quad.

Then by an explicit study of the filtration of the functors P_H0 and P_H1 we obtain the following fundamental decompositions of these two standard projective functors.

Theorem. (1) The standard projective object P_H0 admits the following decomposition:

P_H0 =ι(P^F

F2⊕2)⊕(Mix_0,1^⊕2⊕Mix_1,1)⊕κ(iso_H0)

where Mix_0,1 and Mix_1,1 are two mixed functors and iso_H0 is an isotropic functor.

(2) The standard projective object P_H1 admits the following decomposi- tion:

P_H1 =ι(P^F

F2⊕2)⊕Mix_1,1^⊕3⊕κ(iso_H1)

where Mix_1,1 is a mixed functor and iso_H1 is an isotropic functor.

These decompositions have several interesting consequences. Firstly, thanks to this theorem we can complete the study of the functors Mix_0,1 and Mix_1,1 started in [11] by the following result.

Proposition. The functors Mix_0,1 and Mix_1,1 are indecomposable.

We want to emphasize that the complete structure of the direct summands of the decompositions of P_H0 and P_H1 is understood. The structure of the isotropic functors is given in [12], those of the mixed functors Mix_0,1 and Mix_1,1is the main result of [11] and is completed by the previous proposition and those of P^F

F2⊕2 follows from [6]. Then, these decompositions give rise to a classification of the simple functors S of F_quad such that S(H₀)6= 0 or S(H1)6= 0.

Proposition. The isomorphism classes of non-constant simple functors of F_quad such that either S(H₀)6= 0 or S(H₁)6= 0 are:

ι(Λ¹), ι(Λ²), ι(S_(2,1)), κ(iso_(x,0)), κ(iso_(x,1)), R_H0, R_H1, S_H1

where R_H0, R_H1 and S_H1 are the simple functors introduced in Corollary 1.7.

These decompositions also allow us to derive some homological calcula- tions in the category F_quad.

Proposition. For n a natural number, we have:

Extⁿ_F

quad(R_H0, R_H0)'F2 and Extⁿ_F

quad(R_H1, R_H1)'F2

where RH0 and RH1 are the simple functors introduced in Corollary 1.7.

Finally, after having introduced the notion of polynomial functor for the category Fquad, which generalizes that for F, we obtain the following result as an application of the classification of the simple functors S of F_quad such that S(H₀)6= 0 or S(H₁)6= 0 and of the thickness of the subcategory ι(F) of F_quad.

Theorem. The polynomial functors of F_quadare in the image of the functor ι :F → F_quad.

Most of the results of this paper are contained in the Ph.D. thesis of the author [10].

1. The category F_quad: some recollections

We recall in this section some definitions and results about the category F_quad obtained in [12].

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 that preserve the quadratic forms. By the classification of qua- dratic forms over the field F2 (see, for instance, [7]) we know that only spaces of even dimension can be nondegenerate and, for a fixed even dimen- sion, there are two non-equivalent nondegenerate spaces, which are distin- guished by the Arf invariant. We will denote by H₀ (resp. H₁) the non- degenerate quadratic space of dimension two such that Arf(H₀) = 0 (resp.

Arf(H₁) = 1). The orthogonal sum of two nondegenerate quadratic spaces (V, q_V) and (W, q_W) is, by definition, the quadratic space (V ⊕W, q_V⊕W) where qV⊕W(v, w) = qV(v) +qW(w). Recall that the spaces H0⊥H0 and H₁⊥H₁ are isomorphic. Observe that the morphisms ofE_q are injective lin- ear maps and this category does not admit push-outs or pullbacks. There exists a pseudo push-out in E_q that allows us to generalize the construction of the category of co-spans of B´enabou [1] and thus to define the category Tq in which there exist retractions.

Definition 1.1. The category T_q is the category having as objects those of E_q and, for V and W objects in T_q, HomTq(V, W) is the set of equiva- lence classes of diagrams in E_q of the form V −→^f X ←−^g W for the equiva- lence relation generated by the relation R defined as follows: V −→^f X1

←−g

W R V −→^u X₂ ←−^v W if there exists a morphismαofE_qsuch thatα◦f =u and α◦g =v. The composition is defined using the pseudo push-out. The morphism of HomTq(V, W) represented by the diagram V −→^f X ←−^g W will be denoted by [V −→^f X ←−^g W].

Remark 1.2. A morphism of HomT_q(V, W) is represented by a diagram of the form: V −→^g W⊥W⁰ ←−^iW W, where i_W is the canonical inclusion. In the following, we will use this representation of a morphism, without further comment.

By definition, the categoryF_quad is the category of functors fromT_q toE. Hence F_quad is abelian and has enough projective objects. By the Yoneda lemma, for any objectV ofT_q, the functorP_V =F2[HomTq(V,−)] is a projec- tive object and there is a natural isomorphism: HomF_quad(P_V, F)' F(V), for all objectsF ofF_quad. The set of functors {P_V|V ∈ S}, named the stan- dard projective objects in F_quad, is a set of projective generators of F_quad, where S is a set of representatives of isometry classes of nondegenerate quadratic spaces.

There is a forgetful functor :T_q→ E^f inF_quad, defined by(V) = O(V) and

([V −→^f W⊥W⁰ ←−^g W]) =p_g◦ O(f)

wherep_g is the orthogonal projection fromW⊥W⁰ toW andO :E_q → E^f is the functor which forgets the quadratic form. By the fullness of the functor and an argument of essential surjectivity, we obtain the following theorem.

Theorem 1.3. [12] There is a functor ι :F → F_quad, which is exact, fully faithful and preserves simple objects.

In order to define another subcategory ofFquad, we consider the category E_q^deg having as objects finite dimensional F2-vector spaces equipped with a (possibly degenerate) quadratic form and with morphisms injective lin- ear maps which preserve the quadratic forms. The category E_q^deg admits pullbacks; consequently the category of spans Sp(E_q^deg) ([1]) is defined. By definition, the category Fiso is the category of functors from Sp(E_q^deg) to E. As in the case of the category F_quad, the category F_iso is abelian and has enough projective objects: by the Yoneda lemma, for any object V of Sp(E_q^deg), the functor Q_V = F2[Hom_Sp(E^deg

q )(V,−)] is a projective object in F_iso. We define a particular family of functors ofF_iso, the isotropic functors, which form a set of projective generators and injective cogenerators of Fiso. The category F_iso is related to F_quad by the following theorem.

Theorem 1.4. [12] There is a functor κ : F_iso → F_quad, which is exact, fully-faithful and preserves simple objects.

We obtain the classification of the simple objects of the category F_iso from the following theorem.

Theorem 1.5. [12] There is a natural equivalence of categories F_iso ' Y

V∈S

F2[O(V)]−mod

where S is a set of representatives of isometry classes of quadratic spaces (possibly degenerate) and O(V) is the orthogonal group.

The object ofF_iso which corresponds, by this equivalence, to the module F²[O(V)] is the isotropic functor isoV, defined in [12]. Recall that, as a vector space, iso_V(W) is isomorphic to the subspace of Q_V(W) generated by the elements [V ←^Id−V →W].

A straightforward consequence of the classification of simple objects of F_iso given in Theorem 1.5 is given in the following corollary. Recall that, by definition, an object F of F_quad is finite if it has a finite composition series with simple subquotients.

Corollary 1.6. The isotropic functors κ(isoV) are finite in the category F_quad.

In section 3, in order to obtain the classification of simple objects S of F_quad such that eitherS(H₀)6= 0 orS(H₁)6= 0, we require the composition factors of the isotropic functors associated to some small quadratic spaces.

For α ∈ {0,1}, let (x, α) be the degenerate quadratic space of dimension one generated byxsuch thatq(x) =α. Since the orthogonal groupsO(x,0) and O(x,1) are trivial and O(H₀)'S₂ and O(H₁)'S₃, we deduce from Theorem 1.5 and 1.4, the following corollary.

Corollary 1.7. (1) The functors κ(iso_(x,0)) and κ(iso_(x,1)) are simple in Fquad.

(2) The functorκ(iso_H0)is indecomposable. We have the following non- split short exact sequence:

0→R_H0 →κ(iso_H0)→R_H0 →0

where R_H0 is the functor obtained from the trivial representation of O(H₀).

(3) The functor κ(isoH1) admits the following decomposition:

κ(iso_H1) =F_H1 ⊕(S_H1)^⊕2

where S_H1 is the functor obtained from the natural representation of O(H₁) and F_H1 is an indecomposable functor for which we have the following non-split short exact sequence:

0→RH1 →FH1 →RH1 →0

where R_H1 is the functor obtained from the trivial representation of O(H₁).

In [11], we define a new family of functors ofF_quad, named the mixed func- tors and we decompose two particular functors of this family: the functors Mix_0,1 and Mix_1,1. We recall the following description of these functors.

Proposition 1.8. [11] For α ∈ {0,1}, the functors Mix_α,1 : T_q → E are defined by

Mix_α,1(V) = F2[S_V]

where S_V ={(v₁, v₂) |v₁ ∈V, v₂ ∈V, q(v₁+v₂) =α, B(v₁, v₂) = 1} and Mix_α,1([V −→^f W⊥L←−^iW W])[(v₁, v₂)] =

[(p_W ◦f(v₁), p_W ◦f(v₂))] if f(v₁+v₂)∈W

0 otherwise

where pW is the orthogonal projection.

In [11], for a positive integer n, we defined subfunctors Lⁿ_α of ι(Λⁿ) ⊗ κ(iso_(x,α)), where Λⁿ is the nth exterior power and we proved that these functors are simple. The functor L¹_α is equivalent to the functor κ(iso_(x,α)).

We obtain the following result.

Theorem 1.9. [11] Let α be an element in {0,1}.

(1) The functor Mix_α,1 is infinite.

(2) There exists a subfunctor Σ_α,1 of Mix_α,1 such that we have the fol- lowing short exact sequence

0→Σα,1 →Mixα,1 →Σα,1 →0.

(3) The functor Σα,1 is uniserial with unique composition series given by the decreasing filtration given by the subfunctors k_dΣ_α,1 of Σ_α,1:

. . .⊂k_dΣ_α,1 ⊂. . .⊂k₁Σ_α,1 ⊂k₀Σ_α,1 = Σ_α,1.

(a) The head ofΣ_α,1 (i.e. Σ_α,1/k₁Σ_α,1) is isomorphic to the functor κ(iso_(x,α)) where iso_(x,α) is a simple object in F_iso.

(b) For d >0

k_dΣ_α,1/k_d+1Σ_α,1 'L^d+1_α

whereL^d+1_α is a simple object of the categoryF_quadthat is neither in the image of ι nor in the image of κ.

The functor L^d+1_α is a subfunctor of ι(Λ^d+1)⊗κ(iso_(x,α)), where Λ^d+1 is the (d+ 1)st exterior power functor.

2. Filtration of the standard projective functors P_V of F_quad In this section, we define a filtration of the standard projective functors P_V of F_quad. This construction gives rise to an essential tool to obtain, in section 3, the direct decompositions of the projective objects P_H0 and P_H1 of F_quad, into indecomposable summands.

After defining this filtration, we will deduce general results about the projective P_V of F_quad. In Theorem 2.6 we prove that the rank zero part is a direct summand of P_V and we identify this functor. This result allows us to prove that ι(F) is a thick subcategory of F_quad. We will also show that the top quotient of this filtration is isomorphic to κ(iso_V), where iso_V is the isotropic functor.

2.1. Definition of the filtration. We recall that a morphism in T_q from V toW, whereV andW are nondegenerate quadratic spaces, is represented by a diagram V →X ←W.

Definition 2.1. A morphism [V → X ← W] in T_q has rank equal to i if the pullback in E_q^deg of the diagram V → X ← W is a quadratic space of dimension i.

Notation 2.2. We denote by Hom⁽ⁱ⁾_T

q(V, W) the subset of HomT_q(V, W) of morphisms of rank less than or equal to i.

We have the following proposition:

Proposition 2.3. For W an object in T_q, the following subvector space of P_V(W):

P_V⁽ⁱ⁾(W) =F2[Hom⁽ⁱ⁾_Tq(V, W)]

defines a subfunctor P_V⁽ⁱ⁾ of P_V.

Proof. It is sufficient to verify that for all morphisms f = [W → Y ← Z] of T_q and g = [V → X ← W] of Hom⁽ⁱ⁾_T

q(V, W), the composition f ◦g has rank less than or equal to i. The composition f ◦g is represented by the following commutative diagram:

P^{0 //}

Z

P ^//

W ^//

Y

V ^//X ^//X⊥

WY whereP and P⁰ are the pullbacks andX⊥

WY is the pseudo push-out defined in [12]. Consequently: f ◦g = [V → X⊥

WY ← Z]. Since [V → X ← W]

is an element of P_V⁽ⁱ⁾(W), we know that the dimension of P is less than or equal to i. We deduce from the injectivity of the morphisms of E_q^deg, that

P⁰ has dimension smaller than or equal to i.

The following lemma is a straightforward consequence of Definition 2.1.

Lemma 2.4. There exists a natural equivalence: P_V^(dim(V)) 'PV. We deduce the following proposition.

Proposition 2.5. The functors P_V⁽ⁱ⁾, for i = 0, . . . ,dim(V), define an in- creasing filtration of the functor P_V.

Proof. The inclusion of vector spaces P_V⁽ⁱ⁾(W) ⊂ P_V⁽ⁱ⁺¹⁾(W) is clear, for W an object in T_q. Consequently, P_V⁽ⁱ⁾ is a subfunctor of P_V⁽ⁱ⁺¹⁾ by the

proposition 2.3.

2.2. Extremities of the filtration. In the previous section, we have ob- tained, for all objects V of T_q, the following filtration of the functorP_V:

0⊂P_V⁽⁰⁾ ⊂P_V⁽¹⁾ ⊂. . .⊂P_V^(dim(V)−1) ⊂P_V^(dim(V)) =PV.

The aim of this section is to study the two extremities of this filtration, namely, the functor P_V⁽⁰⁾ and the quotientP_V/P_V^(dim(V)−1).

2.2.1. The functor P_V⁽⁰⁾. For V an object in T_q, we recall that the functor P_(V^F ₎ of F defined by P_(V^F ₎(−) =F2[Hom_E^f((V),−)], where :T_q → E^f is the forgetful functor ofFquad, is projective, by the Yoneda lemma. The aim of this section is to prove the following theorem:

Theorem 2.6. Let V be an object of T_q.

(1) There is a natural equivalence: P_V⁽⁰⁾ 'ι(P_(V^F ₎), where ι:F → F_quad is the functor given in Theorem 1.3.

(2) The functor P_V⁽⁰⁾ is a direct summand of P_V.

Before proving this result, we give the following useful characterization of the morphisms of rank zero, which is a straightforward consequence of the definition of the rank of a morphism.

Lemma 2.7. Let V be an object inT_q. A morphism T = [V −→^α W⊥W⁰ ←−^iW W] has rank zero if and only if p_W⁰ ◦α is an injective linear map, where pW⁰ is the orthogonal projection from W⊥W⁰ to W⁰.

ForV andW objects inT_q, the forgetful functor :T_q → E^f gives rise to a map

HomT_q(V, W)→Hom_E^f((V), (W)). By passage to the vector spaces freely generated by these sets and by functoriality of , we deduce the existence of

a morphism fromP_V toι(P_(V^F ₎). As the functorsP_V⁽⁰⁾are subfunctors ofP_V, we obtain a morphism f from P_V⁽⁰⁾ to ι(P_(V^F ₎). Consequently, to prove the first part of Theorem 2.6, it is sufficient to prove the following proposition.

Proposition 2.8. The map P_V⁽⁰⁾(W) −^f−^W→ P_(V^F ₎((W)) is an isomorphism for V and W objects in T_q.

The surjectivity off_W relies on the following lemma, which is an improved version of the fullness of the forgetful functor given in [12].

Lemma 2.9. Let (V, q_V) and (W, q_W) be two objects of T_q and f ∈ Hom_E^f((V, q_V), (W, q_W)) a linear map, then there exists a morphism T = [V −→^ϕ W⊥Y ←−^iW W] of rank zero such that (T) =f.

Proof. As the quadratic spaceV is nondegenerate, we know that it has even dimension. We write dim(V) = 2n. We prove the result by induction onn.

To start the induction, let (V, qV) be a nondegenerate quadratic space of dimension two, with symplectic basis {a, b} and f : V → W be a linear map. The following linear map preserves the quadratic form:

g₁ : V → W⊥H₁⊥H₀⊥H₀ 'W⊥Span(a₁, b₁)⊥Span(a₀, b₀)⊥Span(a⁰₀, b⁰₀) 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 +a⁰₀. Consequently, the morphism: T =V −^g→¹ W⊥H₁⊥H₀ ←- W, is a morphism of rank zero of T_q such that(T) =f.

LetV_nbe a nondegenerate quadratic space of dimension 2n,{a₁, b₁, . . . , a_n, b_n} be a symplectic basis ofV_nandf_n:V_n→W be a linear map. By induction, there exists a map:

g_n : V_n → W⊥Y a_i 7−→ f_n(a_i) +y_i b_i 7−→ f_n(b_i) +z_i

where y_i and z_i, for all integers i between 1 and n, are elements of Y. The map g_n preserves the quadratic form and the morphism T = [V_n −→^gn W⊥Y ←- W] is of rank zero and verifies ([V_n −→^gn W⊥Y ←- W]) =f_n.

Let V_n+1 be a nondegenerate quadratic space of dimension 2(n + 1), {a₁, b₁, . . . , a_n, b_n, a_n+1, b_n+1} a symplectic basis of V_n+1 and f_n+1 :V_n+1 → W a linear map. To define the mapg_n+1, we will consider the restriction of f_n+1 to V_n and extend the map g_n given by the inductive assumption. For that, we need the following space: E ' W⊥Y⊥H₀^⊥n⊥H₀^⊥n⊥H₁⊥H₀⊥H₀ for which we specify the notations for a basis:

E 'W⊥Y⊥(⊥ⁿ_i=1Span(aⁱ₀, bⁱ₀))⊥(⊥ⁿ_i=1Span(Aⁱ₀, B₀ⁱ))⊥Span(A₁, B₁)

⊥Span(C0, D0)⊥Span(E0, F0).

The following map:

g_n+1 : V → W⊥Y⊥H₀^⊥n⊥H₀^⊥n⊥H₁⊥H₀⊥H₀

a_i 7−→ f_n+1(a_i) +y_i+aⁱ₀ fori between 1 and n b_i 7−→ f_n+1(b_i) +z_i+Aⁱ₀

a_n+1 7−→ f_n+1(a_n+1) + (q(a_n+1) +q(f_n+1(a_n+1)))A₁+C₀ +Pn

i=1B(f_n+1(a_i), f_n+1(a_n+1))bⁱ₀ +Pn

i=1B(f_n+1(b_i), f_n+1(a_n+1))B₀ⁱ

b_n+1 7−→ f_n+1(b_n+1) + (q(b_n+1) +q(f_n+1(b_n+1)))A₁ +(1 +B(f_n+1(a_n+1), f_n+1(b_n+1)))D₀ +Pn

i=1B(f_n+1(a_i), f_n+1(b_n+1))bⁱ₀ +Pn

i=1B(f_n+1(b_i), f_n+1(b_n+1))B₀ⁱ +E₀ preserves the quadratic form. Furthermore, the morphism

T = [Vn+1 gn+1

−−→W⊥Y⊥H₀^⊥n⊥H₀^⊥n⊥H1⊥H0⊥H0 ←- W]

is of rank zero and satisfies: (T) = fn+1, which completes the inductive step.

The proof of the injectivity off_W relies on the following result, which can be regarded as Witt’s theorem for degenerate quadratic forms.

Theorem 2.10. Let V be a nondegenerate quadratic space, D and D⁰ sub- quadratic spaces (possibly degenerate) of V andf :D→D⁰ an isometry be- tween these two quadratic spaces. Then, there exists an isometry f :V →V such that the following diagram is commutative:

V ^{f //}V

D^?

OO

f

//D⁰.^?

OO

Proof. For a proof of this result, we refer the reader to [2]§4, theorem 1.

Proof of the injectivity of f_W. The natural mapf is induced by the natural map

Hom⁽⁰⁾_T

q(V,−)→Hom_E^f((V), (−))

by passage to the vector spaces freely generated by these sets. So, f is injective if and only if this natural map is injective. Consequently, it is sufficient to verify that, for T = [V −→^α W⊥W⁰ ←−^iW W] and T⁰ = [V ^α

0

−→ W⊥W^{00 i}

0

←−W W] two generators of P_V⁽⁰⁾(W) such that (2.10.1) pW ◦ O(α) = p⁰_W ◦ O(α⁰), we have T =T⁰.

Let{a₁, b₁, . . . , a_n, b_n}be a symplectic basis ofV. We deduce from 2.10.1 that, for all i∈ {1, . . . , n}, we have:

(2.10.2) α(a_i) =w_i+w⁰_i, α(b_i) = x_i+x⁰_i and

(2.10.3) α⁰(a_i) =w_i+w_i⁰⁰, α⁰(b_i) =x_i+x⁰⁰_i

where, for all i∈ {1, . . . , n}, wi and xi are in W, w⁰_i and x⁰_i are in W⁰ and x⁰⁰_i and w_i⁰⁰are in W⁰⁰. By Lemma 2.7, since the morphisms are of rank zero, {w⁰₁, x⁰₁, . . . , w⁰_n, x⁰_n} and {w₁⁰⁰, x⁰⁰₁, . . . , w⁰⁰_n, x⁰⁰_n} are two linearly independent families of vectors.

We will denote by W⁰ = Span(w₁⁰, x⁰₁, . . . , w⁰_n, x⁰_n) (respectively W⁰⁰ = Span(w₁⁰⁰, x⁰⁰₁, . . . , w⁰⁰_n, x⁰⁰_n)) the subquadratic space (possibly degen- erate), of W⁰ (respectivelyW⁰⁰) and we define the linear mapf :W⁰ →W⁰⁰ by f(w⁰_i) =w⁰⁰_i and f(x⁰_i) = x⁰⁰_i for all i∈ {1, . . . , n}.

Sinceαandα⁰ preserve the quadratic forms, we deduce from the relations 2.10.2 and 2.10.3 that f preserves the quadratic form. Hence, we can apply Theorem 2.10 to the nondegenerate spaceW⁰⊥W⁰⁰, which gives a morphism f : W⁰⊥W⁰⁰ → W⁰⊥W⁰⁰ of E_q, such that, the restriction of this morphism to W⁰ coincides with f. We deduce the commutativity of the following diagram:

W_{ _}

iW

q

i⁰_W

""

EE EE EE EE EE EE EE EE EE EE EE

V ^{α˜ //}

α˜W⁰WWWWWWWWWW++

WW WW WW WW WW WW WW

WW W⊥(W⁰⊥W⁰⁰)

Id⊥f

((R

RR RR RR RR RR RR

W⊥(W⁰⊥W⁰⁰)

where ˜α = iW⊥W⁰ ◦α and ˜α⁰ = i_W⊥W⁰⁰ ◦α⁰. Consequently, we obtain the equality T =T⁰ since, by inclusion, we have

T = [V −→^α W⊥W⁰ ←−^iW W] = [V −→^α˜ W⊥W⁰⊥W⁰⁰ ←−^iW W] and

T⁰ = [V ^α

0

−→W⊥W^{00 i}

0

←−W W] = [V −^α→^˜0 W⊥W⁰⊥W^{00 i}

0

←−W W].

Notation 2.11. For V and W two objects of Eq, and f a morphism of Hom_E^f((V), (W)), we denote by t_f the morphism of Hom⁽⁰⁾_T

q(V, W) cor- responding to f and by [tf] the canonical generator of P_V⁽⁰⁾(W) obtained from t_f. To simplify the notation, we will denote the morphism t_IdV of Hom⁽⁰⁾_T

q(V, V) by e_V.

We deduce from the first point of Theorem 2.6 the following corollary.

Corollary 2.12. For V, W and X objects of Eq, f : (W) → (X) and g :(V)→(W) morphisms of E^f, we have:

t_f ◦t_g =t_f◦g

wheret_f, t_g andt_f◦gare respectively the morphisms ofHom⁽⁰⁾_T

q(W, X),Hom⁽⁰⁾_T

q(V, W) and Hom⁽⁰⁾_T

q(V, X) associated to the linear maps f, g and f◦g.

We can apply this result to the idempotents of the ring of endomorphisms End(P_V), to obtain the following proposition.

Proposition 2.13. The canonical generator[e_V]ofP_V⁽⁰⁾ is an idempotent of the ring of endomorphisms End(P_V)such that P_V.[e_V]'P_V⁽⁰⁾. Consequently P_V⁽⁰⁾ is a direct summand of PV.

Proof. The canonical generator [e_V] is an idempotent of End(P_V) by Corol- lary 2.12. By definition of the rank filtration PV.[eV] ⊂ P_V⁽⁰⁾ and, for a canonical generator [t_f] of P_V⁽⁰⁾, we have [t_f] = [t_f]·[e_V].

Since the idempotents [e_V] and 1−[e_V] of End(P_V) are orthogonal and satisfy [e_V] + (1−[e_V]) = 1 we obtain the following decomposition of P_V into direct summands: P_V = P_V.[e_V]⊕P_V.(1−[e_V]). So we deduce that

P_V⁽⁰⁾ is a direct summand of PV.

The idempotent [e_V] plays a central rˆole in the proof of the thickness of the subcategory ι(F) inF_quad, which is the subject of the following section.

For that, the following result is necessary.

Lemma 2.14. Let V and W be objects of T_q, the functor ι induces an isomorphism:

HomF(P_(V^F ₎, P_(W)^F )−→^' HomF_quad(P_V⁽⁰⁾, P_W⁽⁰⁾), where :T_q → E is the forgetful functor.

Proof. By Proposition 2.13 and Theorem 2.6 we have the following equiva- lences:

HomF_quad(P_V⁽⁰⁾, P_W⁽⁰⁾)'P_W⁽⁰⁾(eV)P_W⁽⁰⁾(V)'P_W⁽⁰⁾(V) 'ι(P_(W^F ₎)(V)'P_(W^F ₎((V))'HomF(P_(V^F ₎, P_(W)^F ).

To conclude this section, we give the following property of e_V which will be useful in section 4 concerning the polynomial functors of Fquad.

Lemma 2.15. For V and W two objects of T_q, we have: e_V⊥W =e_V⊥e_W, where ⊥:Tq× Tq → Tq is the functor induced by the orthogonal sum.

Proof. This is a straightforward consequence of Proposition 2.8.

2.2.2. The category ι(F) is a thick subcategory of F_quad. Recall that a sub- category C of an abelian categoryA is thick ifC is a full subcategory which contains all subobjects and quotient objects of its objects and is closed un- der formation of extensions. The aim of this section is to prove the following result.

Theorem 2.16. The category ι(F) is a thick subcategory of F_quad, where ι :F → F_quad is the functor defined in Theorem 1.3.

To prove this theorem, we need the following general result about the precomposition functor which is proved in the Appendix of [12].

Proposition 2.17. Let C and D be two small categories, A be an abelian category, F : C → D be a functor and − ◦F : Func(D,A) → Func(C,A) be the precomposition functor, where Func(C,A) is the category of functors from C to A. If F is full and essentially surjective, then any subobject (re- spectively quotient) of an object in the image of the precomposition functor is isomorphic to an object in the image of the precomposition functor.

Proof of Theorem 2.16. • The subcategory ι(F) of F_quad is full by Theorem 1.3.

• Let F^F be an object in F and G a subobject of ι(F^F). Let F⁰ be the category of functors from E^f−(even) to E, where E^f−(even) is the full subcategory ofE^f having as objects theF2-vector spaces of even dimension. The categoriesF andF⁰ are equivalent [12]. The functor : E_q → E^f factorizes through the inclusion E^f−(even) ,→ E^f. This induces a functor ⁰ : Eq → E^f−(even) which is full and essentially surjective. Consequently, we can use Proposition 2.17 to obtain:

G'ι(G^F). Similarly, we obtain the result for the quotient.

• LetG^F andH^F be objects ofF, we setG=ι(G^F) andH =ι(H^F).

For a short exact sequence: 0 → G → F → H → 0, we have to prove that there exists a functor F^F in F such that F =ι(F^F).

Let P₁ →P₀ →G^F →0 and Q₁ →Q₀ →H^F → 0 be projective presentations of G^F and H^F inF, we have the following commuta- tive diagram

0 ^//ι(P1) ^//

ι(P1)⊕ι(Q1) ^//

ι(Q1) ^//

0

0 ^//ι(P₀) ^//

ι(P₀)⊕ι(Q₀) ^//

ι(Q₀) ^//

0

0 ^//G

//F

// H

//0

0 0 0

where the columns are projective resolutions in F_quad, by the horse- shoe lemma. By Lemma 2.14, the morphismι(P₁)⊕ι(Q₁)→ι(P₀)⊕ ι(Q₀) is induced by a morphism of F denoted by f. Consequently, F ∼=ι(Coker(f))∈ι(F).

By Theorem 2.16, we deduce from Lemma 2.14, the following character- ization of the simple functors of F inF_quad which will be used in section 4 of this paper concerning the polynomial functors of F_quad.

Lemma 2.18. (1) Let F be a functor of F_quad, then F is in the image of the functor ι:F → F_quad if and only if, for all objects V in T_q,

F(e_V)F(V) =F(V).

(2) Let S be a simple object in F_quad, then S is in the image of the functor ι : F → F_quad if and only if there exists an object W in T_q such that

S(e_W)S(W)6= 0.

Proof. (1) The forward implication is a consequence of the following fact: for a functorF in the image of ι, HomF_quad(P_V⁽⁰⁾, F) =F(e_V)F(V) = F(V). The reverse implication relies on the fact that the condition F(e_V)F(V) = F(V) implies that F is a quotient of a sum of pro- jective objects of the form P_V⁽⁰⁾. Since the category ι(F) is thick in F_quad by Theorem 2.16, we obtain the result.

(2) Observe that, if S(e_W)S(W) 6= 0, we have HomF_quad(P_W⁽⁰⁾, S) 6= 0, thus S is a quotient of P_W⁽⁰⁾ by simplicity ofS. Lemma 2.14 implies that there exists a one-to-one correspondance between the indecom- posable factors ofP_V⁽⁰⁾and those ofP_(V^F ₎. We deduce that the simple quotients of P_V⁽⁰⁾ arise from F. Consequently, S is in the image of the functor ι.

2.2.3. The quotient P_V/P_V^(dim(V)−1). The aim of this section is to prove the following result:

Proposition 2.19. Let V be an object inT_q, we have a natural equivalence:

P_V/P_V^(dim(V)−1) 'κ(iso_V)

where isoV is an isotropic functor and κ:Fiso → Fquad is the functor given in Theorem 1.4.

To prove this proposition, we need the following notation and result:

Notation 2.20. Denote by σ_f the natural map P_V −→^σf κ(iso_V) which corre- sponds to the canonical generator [V ←^Id−V −→^f V] of iso_V(V) by the equiva- lence Hom(P_V, κ(iso_V))'iso_V(V)'F2[O(V)] given by the Yoneda lemma.

Lemma 2.21. The functor κ(isoV) of Fquad is a quotient of the functor P_V =F2[HomTq(V,−)].

Proof. The natural map P_V −→^σf κ(iso_V) is surjective: a pre-image of the canonical generator [V ←^Id− V −→^g W] of κ(iso_V)(W) by (σ_f)_W, is the mor- phism

[V ^g◦f

−1

−−−→W ←^Id−W].

A formal consequence of the previous lemma is given in the following result.

Lemma 2.22. The functor κ(iso_V) of F_quad is a quotient of the functor PV/P_V^(dim(V)−1).

Proof. By definition of the filtration and by the previous lemma, we have the diagram:

0 ^//P_V^{(dim(V)−1) i //}P_V

σId

// //P_V/P^(dim(V)−1)_V ^//

wwww τ

0

κ(iso_V)

where i is the canonical inclusion of P_V^(dim(V)−1) inP_V. By definition ofσ_Id, we have σ_Id◦i = 0, from which we deduce the existence of the surjection

τ :P_V/P_V^(dim(V)−1) →κ(iso_V).

We will prove below that this natural map is an isomorphism. It is sufficient to prove the following result.

Proposition 2.23. For V and W two objects of T_q, we have an isomor- phism

(P_V/P_V^(dim(V)−1))(W)'κ(iso_V)(W).

The proof of this proposition relies on the following lemma.

Lemma 2.24. For a non-zero canonical generator of(P_V/P_V^(dim(V)−1))(W) represented by the morphism T = [V −→^g W⊥W⁰ ←−^iW W] of T_q, we have g(V)⊂W and T = [V −→^f W ←^Id−W], where g =i_W ◦f.

Proof. By definition of the filtration, for V and W two objects of T_q, the vector space (P_V/P_V^(dim(V)−1))(W) is generated by Hom^[dim(V_Tq ^)](V, W) where Hom^[dim(V_Tq ^)](V, W) is the set of morphisms fromV toW whose the pullback DinE_q^deg is a quadratic space such that dim(D) = dim(V). We deduce from the existence of a monomorphism from DtoV and from the equality of the dimensions, that DandV are isometric. Consequently, for the morphismT of the statement, we have, by definition of the pullback, g(V)⊂W. Thus, we haveT = [V −→^f W ←^Id−W], whereg =i_W◦f, by the equivalence relation defined over the morphisms of Tq in Definition 1.1.

Proof of Proposition 2.23. The natural mapτ obtained in the proof of Lemma 2.22 defines, for W an object in T_q, the linear map

τ_W : (P_V/P_V^(dim(V)−1))(W) → κ(iso_V)(W) T = [V −→^f W ←^Id−W] 7−→ [V ←^Id−V −→^f W]

which is clearly an isomorphism.

3. Decomposition of the standard projective functors P_H0 and P_H1

On abelian categories, the decompositions into direct summands of a func- tor F ofF_quadcorrespond to decompositions into orthogonal idempotents of 1 in the ring EndF_quad(F) (see for example [3]). One of the difficulties of the categoryFquadlies in the fact that the rings of endomorphisms of projectives P_V and their representations are not well-understood. The decompositions of projectives P_V, obtained in work in preparation, using a refinement of the rank filtration will allow us to understand the structure of these rings better.

In this section, we obtain the decompositions into indecomposable factors of the projective objects P_H0 and P_H1 by an explicit study of the filtration defined in section 2. This section concludes by several consequences of these decompositions. In particular, we give a classification of the “small” simple functors of F_quad, which is an essential ingredient in the following section about the polynomial functors of Fquad.

3.1. Decomposition of P_H0. To obtain the decomposition of the functor P_H0 into indecomposable factors, we give an explicit description of the sub- quotients of the filtration; then, we prove that the filtration splits for this functor and we identify the factors of this decomposition.

3.1.1. Explicit description of the subquotients of the filtration. The aim of this section is to give a basis of the vector spacesP_H⁽⁰⁾

0(V),P_H⁽¹⁾

0/P_H⁽⁰⁾

0(V) and PH0/P_H⁽¹⁾₀(V) forV a given object in Tq.

We deduce from Theorem 2.6 and Notation 2.11, the following result.

Lemma 3.1. A basis B_H⁽⁰⁾

0 of P_H⁽⁰⁾

0(V) is given by the set:

B_H⁽⁰⁾

0 ={[t_f] for f ∈Hom_Ef(F2⊕2

, (V))}.

By definition of the filtration, a canonical generator of P_H⁽¹⁾

0/P_H⁽⁰⁾

0(V), rep- resented by the morphism T = [H₀ −→^f V⊥L ←−^iV V] of T_q, satisfies the following property: I =f(H₀)∩i(V) is a quadratic space of dimension one.

Lemma 3.2. Let T = [H₀ −→^f V⊥L←−^iV V] be a morphism of T_q which rep- resents a canonical generator of P_H⁽¹⁾

0/P_H⁽⁰⁾

0(V), and {a₀, b₀} be a symplectic basis of H0, then the map f in T has one of the three following forms.

(1) If I = (f(a₀),0), the map f :H₀ →V⊥L is defined by:

f(a₀) = v and f(b₀) =w+l

for v and w elements of V satisfying q(v) = 0 and B(v, w) = 1 and l a non-zero element of L.

(2) If I = (f(b₀),0), the map f :H₀ →V⊥L is defined by:

f(a₀) = v+l and f(b₀) =w

for v and w elements of V satisfying q(w) = 0 andB(v, w) = 1 and l a non-zero element of L.

(3) If I = (f(a₀+b₀),1), the map f :H₀ →V⊥L is defined by:

f(a₀) =v+l and f(b₀) = w+l

for v and w elements of V satisfying q(v+w) = 1 and B(v, w) = 1 and l a non-zero element of L.

Proof. The quadratic spaceH₀ has three subspaces of dimension one which are: Span(a0) and Span(b0) isometric to (x,0) and Span(a0+b0) isometric to (x,1). These three subspaces give rise to each one of the maps f defined

in the statement.

Notation 3.3. The morphisms [H₀ −→^f V⊥L ←−^iV V], where f is one of the morphisms described in the point (1) (respectively (2) and (3)) of the previous lemma, will be known as type A(respectivelyB andC) morphisms.