Generic representations of orthogonal groups: projective functors in the category Fquad

TL;DR

This paper advances the understanding of the category Fquad, associated with quadratic forms over F_2, by decomposing projective objects and characterizing polynomial functors, thus enriching the algebraic structure analysis.

Contribution

It introduces a filtration and decomposition of standard projective objects in Fquad, providing a detailed description of polynomial functors within this category.

Findings

Decomposition of the first two standard projective objects in Fquad.

Complete description of polynomial functors in Fquad.

Refined understanding of the algebraic structure of Fquad.

Abstract

In this paper, we continue the study of the category of functors Fquad, associated to F_2-vector spaces equipped with a nondegenerate quadratic form, initiated in two previous papers of the author. We define a filtration of the standard projective objects in Fquad; this refines to give a decomposition into indecomposable factors of the two first standard projective objects in Fquad. As an application of these two decompositions, we give a complete description of the polynomial functors of the category Fquad.

Equations

$$0\subset P_V^{(0)}\subset P_V^{(1)}\subset \dots \subset P_V^{(\dim (V)-1)}\subset P_V^{(\dim (V))}=P_V.$$

$$P_V/P_V^{(\dim (V)-1)}\simeq \kappa ({\mathrm{iso}}_V)$$

$$P_{H_0}=\iota (P_{{{\mathbb{F}}_2}^{\oplus 2}}^{\mathcal{F}})\oplus ({{\mathrm{Mix}}_{0,1}}^{\oplus 2}\oplus {\mathrm{Mix}}_{1,1})\oplus \kappa (is{o}_{H_0})$$

$$P_{H_1}=\iota (P_{{{\mathbb{F}}_2}^{\oplus 2}}^{\mathcal{F}})\oplus {{\mathrm{Mix}}_{1,1}}^{\oplus 3}\oplus \kappa (is{o}_{H_1})$$

$$\iota ({\Lambda }^1),\iota ({\Lambda }^2),\iota (S_{(2,1)}),\kappa (is{o}_{(x,0)}),\kappa (is{o}_{(x,1)}),R_{H_0},R_{H_1},S_{H_1}$$

$${\mathrm{Ext}}_{{\mathcal{F}}_{quad}}^n(R_{H_0},R_{H_0})\simeq {\mathbb{F}}_2\mathrm{and}{\mathrm{Ext}}_{{\mathcal{F}}_{quad}}^n(R_{H_1},R_{H_1})\simeq {\mathbb{F}}_2$$

$$ϵ([V\stackrel{f}{\to }W\perp W^{\prime }\stackrel{g}{\leftarrow }W])=p_g\circ \mathcal{O}(f)$$

$${\mathcal{F}}_{iso}\simeq \prod _{V\in \mathcal{S}}{\mathbb{F}}_2[O(V)]-mod$$

$$0\to R_{H_0}\to \kappa (is{o}_{H_0})\to R_{H_0}\to 0$$

$$\kappa (is{o}_{H_1})=F_{H_1}\oplus (S_{H_1}{)}^{\oplus 2}$$

$$0\to R_{H_1}\to F_{H_1}\to R_{H_1}\to 0$$

$${\mathrm{Mix}}_{\alpha ,1}(V)={\mathbb{F}}_2[S_V]$$

$${\mathrm{Mix}_{\alpha,1}}([V\xrightarrow{f}W\bot L\xleftarrow{i_{W}}W])[(v_{1},v_{2})]=\left\{\begin{array}[]{cc}[(p_{W}\circ f(v_{1}),p_{W}\circ f(v_{2}))]&\mathrm{if\ }f(v_{1}+v_{2})\in W\\ 0&\mathrm{otherwise}\end{array}\right.$$

$$0\to {\mathrm{\Sigma }}_{\alpha ,1}\to {\mathrm{Mix}}_{\alpha ,1}\to {\mathrm{\Sigma }}_{\alpha ,1}\to 0.$$

$$\dots \subset k_d{\mathrm{\Sigma }}_{\alpha ,1}\subset \dots \subset k_1{\mathrm{\Sigma }}_{\alpha ,1}\subset k_0{\mathrm{\Sigma }}_{\alpha ,1}={\mathrm{\Sigma }}_{\alpha ,1}.$$

$$k_d{\mathrm{\Sigma }}_{\alpha ,1}/k_{d+1}{\mathrm{\Sigma }}_{\alpha ,1}\simeq L_{\alpha }^{d+1}$$

$$P_V^{(i)}(W)={\mathbb{F}}_2[{\mathrm{Hom}}_{{\mathcal{T}}_q}^{(i)}(V,W)]$$

$$P^{\prime }\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$0\subset P_V^{(0)}\subset P_V^{(1)}\subset \dots \subset P_V^{(\dim (V)-1)}\subset P_V^{(\dim (V))}=P_V.$$

$$\begin{array}[]{cccl}g_{1}:&V&\rightarrow&W\bot H_{1}\bot H_{0}\bot H_{0}\simeq W\bot\mathrm{Span}(a_{1},b_{1})\bot\mathrm{Span}(a_{0},b_{0})\bot\mathrm{Span}(a^{\prime}_{0},b^{\prime}_{0})\\ &a&\longmapsto&f(a)+(q(a)+q(f(a)))a_{1}+a_{0}\\ &b&\longmapsto&f(b)+(q(b)+q(f(a)))a_{1}+(1+B(f(a),f(b)))b_{0}+a^{\prime}_{0}.\end{array}$$

$$\begin{array}[]{cccc}g_{n}:&V_{n}&\rightarrow&W\bot Y\\ &a_{i}&\longmapsto&f_{n}(a_{i})+y_{i}\\ &b_{i}&\longmapsto&f_{n}(b_{i})+z_{i}\\ \end{array}$$

$$E\simeq W\perp W^{\prime }\perp ({\perp }_{i=1}^n\mathrm{Span}(a_0^i,b_0^i))\perp ({\perp }_{i=1}^n\mathrm{Span}(A_0^i,B_0^i))\perp \mathrm{Span}(A_1,B_1)$$

$$\perp \mathrm{Span}(C_0,D_0)\perp \mathrm{Span}(E_0,F_0).$$

$$\begin{array}[]{cccl}g_{n+1}:&V&\rightarrow&W\bot W^{\prime}\bot H_{0}^{\bot n}\bot H_{0}^{\bot n}\bot H_{1}\bot H_{0}\bot H_{0}\\ &a_{i}&\longmapsto&f_{n+1}(a_{i})+y_{i}+a_{0}^{i}\quad\mathrm{for}\ i\ \mathrm{between}\ 1\ \mathrm{and}\ n\\ &b_{i}&\longmapsto&f_{n+1}(b_{i})+z_{i}+A_{0}^{i}\\ &a_{n+1}&\longmapsto&f_{n+1}(a_{n+1})+(q(a_{n+1})+q(f_{n+1}(a_{n+1})))A_{1}+C_{0}\\ &&&+\sum_{i=1}^{n}B(f_{n+1}(a_{i}),f_{n+1}(a_{n+1}))b_{0}^{i}\\ &&&+\sum_{i=1}^{n}B(f_{n+1}(b_{i}),f_{n+1}(a_{n+1}))B_{0}^{i}\\ &b_{n+1}&\longmapsto&f_{n+1}(b_{n+1})+(q(b_{n+1})+q(f_{n+1}(b_{n+1})))A_{1}\\ &&&+(1+B(f_{n+1}(a_{n+1}),f_{n+1}(b_{n+1})))D_{0}\\ &&&+\sum_{i=1}^{n}B(f_{n+1}(a_{i}),f_{n+1}(b_{n+1}))b_{0}^{i}\\ &&&+\sum_{i=1}^{n}B(f_{n+1}(b_{i}),f_{n+1}(b_{n+1}))B_{0}^{i}+E_{0}\end{array}$$

$$T=[V_{n+1}\stackrel{g_{n+1}}{\to }W\perp W^{\prime }\perp H_0^{\perp n}\perp H_0^{\perp n}\perp H_1\perp H_0\perp H_0\hookleftarrow W]$$

$$V\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\mathrm{Hom}}_{{\mathcal{T}}_q}^{(0)}(V,-)\to {\mathrm{Hom}}_{{\mathcal{E}}^f}(ϵ(V),ϵ(-))$$

$$p_W\circ \mathcal{O}(\alpha )=p_W^{\prime }\circ \mathcal{O}({\alpha }^{\prime }),$$

$$\alpha (a_i)=w_i+w_i^{\prime },\alpha (b_i)=x_i+x_i^{\prime }$$

$${\alpha }^{\prime }(a_i)=w_i+w_i^{\prime \prime },{\alpha }^{\prime }(b_i)=x_i+x_i^{\prime \prime }$$

Full text

  

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

Keywords: functor categories; quadratic forms over ${\mathbb{F}_{2}}$; Mackey functors; representations of orthogonal groups over ${\mathbb{F}_{2}}$.

Introduction

In the paper [math.AT/0606484] we defined the category of functors ${\mathcal{F}_{quad}}$ from a category having as objects the nondegenerate ${\mathbb{F}_{2}}$-quadratic spaces to the category ${\mathcal{E}}$ of $\mathbb{F}_{2}$-vector spaces, where ${\mathbb{F}_{2}}$ 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 ${\mathbb{F}_{2}}$, to that which exists for the general linear groups. We recall that the category ${\mathcal{F}}$ of functors from the category ${\mathcal{E}}^{f}$ of finite dimensional $\mathbb{F}_{2}$-vector spaces to the category ${\mathcal{E}}$ of all $\mathbb{F}_{2}$-vector spaces is a very useful tool for the study of the stable cohomology of the general linear groups with suitable coefficients (see [FFSS]). Another motivation, in topology, for the study of the category ${\mathcal{F}}$ is the connection which exists between this category and unstable modules over the Steenrod algebra (see [Sch]). In order to have a good understanding of the category ${\mathcal{F}_{quad}}$, we seek to classify its simple objects. We constructed in [math.AT/0606484] two families of simple objects in ${\mathcal{F}_{quad}}$. The first one is obtained by the fully-faithful, exact functor $\iota:{\mathcal{F}}\rightarrow{\mathcal{F}_{quad}}$, defined in [math.AT/0606484], which preserves simple objects. By [KuhnII], the simple objects in ${\mathcal{F}}$ are in one-to-one correspondence with the irreducible representations of finite general linear groups over ${\mathbb{F}_{2}}$. The second family is obtained by the fully-faithful, exact functor $\kappa:{\mathcal{F}_{iso}}\rightarrow{\mathcal{F}_{quad}}$, which preserves simple objects, where ${\mathcal{F}_{iso}}$ is equivalent to the product of the categories of modules over the orthogonal groups of possibly degenerate quadratic forms. In [Vespa2], we constructed two families of simple objects in the category ${\mathcal{F}_{quad}}$ which are neither in the image of $\iota$ nor in the image of $\kappa$. These simple objects are subfunctors of the tensor product between an object in the image of $\iota$ and an object in the image of $\kappa$. We proved that these simple objects in ${\mathcal{F}_{quad}}$ are the composition factors of two particular mixed functors, defined in [Vespa2].

The aim of this paper is to begin a programme to obtain a complete classification of the simple objects in ${\mathcal{F}_{quad}}$. 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 ${\mathcal{F}_{quad}}$. 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 ${\mathcal{F}_{quad}}$. In work in progress, we obtain a general decomposition of standard projective object $P_{H}$ of ${\mathcal{F}_{quad}}$ which 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 rôle in the category ${\mathcal{F}_{quad}}$ (for example, for the description of the polynomial functors of ${\mathcal{F}_{quad}}$). Furthermore, recall that the decompositions of the injective standard $I_{{\mathbb{F}_{2}}}^{{\mathcal{F}}}$ of the category ${\mathcal{F}}$ and thus, by duality, that of the projective standard $P_{{\mathbb{F}_{2}}}^{{\mathcal{F}}}$, is fundamental for the comprehension of the other injective standards of ${\mathcal{F}}$. Hence, the decompositions of the two smaller projective standard of ${\mathcal{F}_{quad}}$ represent an important step in the understanding of the category ${\mathcal{F}_{quad}}$.

We briefly summarize the contents of this paper. After some recollections on the category ${\mathcal{F}_{quad}}$, where we recall the definitions of the isotropic functors and the mixed functors, we define a filtration of the standard projective objects $P_{V}$ in ${\mathcal{F}_{quad}}$:

[TABLE]

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

Theorem**.**

Let $V$ be a nondegenerate ${\mathbb{F}_{2}}$-quadratic space.

1. (1)

There is a natural equivalence: $P_{V}^{(0)}\simeq\iota(P^{{\mathcal{F}}}_{\epsilon(V)})$, where $\iota:{\mathcal{F}}\rightarrow{\mathcal{F}_{quad}}$, $\epsilon$ is the functor that forgets the quadratic form and $P^{{\mathcal{F}}}_{\epsilon(V)}$ is the standard projective object in ${\mathcal{F}}$ associated to the vector space $\epsilon(V)$. 2. (2)

The functor $P_{V}^{(0)}$ is a direct summand of $P_{V}$.

Proposition**.**

Let $V$ be a nondegenerate ${\mathbb{F}_{2}}$-quadratic space, we have a natural equivalence:

[TABLE]

where $\kappa:{\mathcal{F}_{iso}}\rightarrow{\mathcal{F}_{quad}}$ and $\mathrm{iso}_{V}$ is an isotropic functor in ${\mathcal{F}_{iso}}$.

An important consequence of the Theorem concerning the functor $P_{V}^{(0)}$ is given in following result.

Theorem**.**

The category $\iota({\mathcal{F}})$ is a thick subcategory of ${\mathcal{F}_{quad}}$.

Then by an explicit study of the filtration of the functors $P_{H_{0}}$ and $P_{H_{1}}$ we obtain the following fundamental decompositions of these two standard projective functors.

Theorem**.**

1. (1)

The standard projective object $P_{H_{0}}$ admits the following decomposition:

[TABLE]

where ${\mathrm{Mix}_{0,1}}$ and ${\mathrm{Mix}_{1,1}}$ are two mixed functors and $iso_{H_{0}}$ is an isotropic functor. 2. (2)

The standard projective object $P_{H_{1}}$ admits the following decomposition:

[TABLE]

where ${\mathrm{Mix}_{1,1}}$ is a mixed functor and $iso_{H_{1}}$ is an isotropic functor.

These decompositions have several interesting consequences. Firstly, thanks to this theorem we can complete the study of the functors $\mathrm{Mix}_{0,1}$ and $\mathrm{Mix}_{1,1}$ started in [Vespa2] by the following result.

Proposition**.**

The functors $\mathrm{Mix}_{0,1}$ and $\mathrm{Mix}_{1,1}$ are indecomposable.

We want to emphasize that the complete structure of the direct summands of the decompositions of $P_{H_{0}}$ and $P_{H_{1}}$ is understood. The structure of the isotropic functors is given in [math.AT/0606484], those of the mixed functors $\mathrm{Mix}_{0,1}$ and $\mathrm{Mix}_{1,1}$ is the main result of [Vespa2] and is completed by the previous proposition and those of $P^{{\mathcal{F}}}_{{\mathbb{F}_{2}}^{\oplus 2}}$ follows from [KuhnII]. Then, these decompositions give rise to a classification of the simple functors $S$ of ${\mathcal{F}_{quad}}$ such that $S(H_{0})\neq 0$ or $S(H_{1})\neq 0$.

Proposition**.**

The isomorphism classes of non-constant simple functors of ${\mathcal{F}_{quad}}$ such that either $S(H_{0})\neq 0$ or $S(H_{1})\neq 0$ are:

[TABLE]

where $R_{H_{0}},\ R_{H_{1}}$ and $S_{H_{1}}$ are the simple functors introduced in Corollary 1.7.

These decompositions also allow us to derive some homological calculations in the category ${\mathcal{F}_{quad}}$.

Proposition**.**

For $n$ a natural number, we have:

[TABLE]

where $R_{H_{0}}$ and $R_{H_{1}}$ are the simple functors introduced in Corollary 1.7.

Finally, after having introduced the notion of polynomial functor for the category ${\mathcal{F}_{quad}}$, which generalizes that for ${\mathcal{F}}$, we obtain the following result as an application of the classification of the simple functors $S$ of ${\mathcal{F}_{quad}}$ such that $S(H_{0})\neq 0$ or $S(H_{1})\neq 0$ and of the thickness of the subcategory $\iota({\mathcal{F}})$ of ${\mathcal{F}_{quad}}$.

Theorem**.**

The polynomial functors of ${\mathcal{F}_{quad}}$ are in the image of the functor $\iota:{\mathcal{F}}\rightarrow{\mathcal{F}_{quad}}$.

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

1. The category ${\mathcal{F}_{quad}}$: some recollections

We recall in this section some definitions and results about the category ${\mathcal{F}_{quad}}$ obtained in [math.AT/0606484].

Let ${\mathcal{E}_{q}}$ be the category having as objects finite dimensional ${\mathbb{F}_{2}}$-vector spaces equipped with a non degenerate quadratic form and with morphisms linear maps that preserve the quadratic forms. By the classification of quadratic forms over the field ${\mathbb{F}_{2}}$ (see, for instance, [Pfister]) we know that only spaces of even dimension can be nondegenerate and, for a fixed even dimension, there are two non-equivalent nondegenerate spaces, which are distinguished by the Arf invariant. We will denote by $H_{0}$ (resp. $H_{1}$) the nondegenerate quadratic space of dimension two such that $\mathrm{Arf}(H_{0})=0$ (resp. $\mathrm{Arf}(H_{1})=1$). The orthogonal sum of two nondegenerate quadratic spaces $(V,q_{V})$ and $(W,q_{W})$ is, by definition, the quadratic space $(V\oplus W,q_{V\oplus W})$ where $q_{V\oplus W}(v,w)=q_{V}(v)+q_{W}(w)$. Recall that the spaces $H_{0}\bot H_{0}$ and $H_{1}\bot H_{1}$ are isomorphic. Observe that the morphisms of ${\mathcal{E}_{q}}$ are injective linear maps and this category does not admit push-outs or pullbacks. There exists a pseudo push-out in ${\mathcal{E}_{q}}$ that allows us to generalize the construction of the category of co-spans of Bénabou [Benabou] and thus to define the category ${\mathcal{T}_{q}}$ in which there exist retractions.

Definition 1.1**.**

The category ${\mathcal{T}_{q}}$ is the category having as objects those of ${\mathcal{E}_{q}}$ and, for $V$ and $W$ objects in ${\mathcal{T}_{q}}$, $\mathrm{Hom}_{{\mathcal{T}_{q}}}(V,W)$ is the set of equivalence classes of diagrams in ${\mathcal{E}_{q}}$ of the form $V\xrightarrow{f}X\xleftarrow{g}W$ for the equivalence relation generated by the relation $\mathcal{R}$ defined as follows: $V\xrightarrow{f}X_{1}\xleftarrow{g}W\quad\mathcal{R}\quad V\xrightarrow{u}X_{2}\xleftarrow{v}W$ if there exists a morphism $\alpha$ of ${\mathcal{E}_{q}}$ such that $\alpha\circ f=u$ and $\alpha\circ g=v$. The composition is defined using the pseudo push-out. The morphism of $\mathrm{Hom}_{{\mathcal{T}_{q}}}(V,W)$ represented by the diagram $V\xrightarrow{f}X\xleftarrow{g}W$ will be denoted by $[V\xrightarrow{f}X\xleftarrow{g}W]$.

Remark 1.2**.**

A morphism of $\mathrm{Hom}_{{\mathcal{T}_{q}}}(V,W)$ is represented by a diagram of the form: $V\xrightarrow{g}W\bot W^{\prime}\xleftarrow{i_{W}}W$, where $i_{W}$ is the canonical inclusion. In the following, we will use this representation of a morphism, without further comment.

By definition, the category ${\mathcal{F}_{quad}}$ is the category of functors from ${\mathcal{T}_{q}}$ to ${\mathcal{E}}$. Hence ${\mathcal{F}_{quad}}$ is abelian and has enough projective objects. By the Yoneda lemma, for any object $V$ of ${\mathcal{T}_{q}}$, the functor $P_{V}={\mathbb{F}_{2}}[\mathrm{Hom}_{{\mathcal{T}_{q}}}(V,-)]$ is a projective object and there is a natural isomorphism: $\mathrm{Hom}_{{\mathcal{F}_{quad}}}(P_{V},F)\simeq F(V)$, for all objects $F$ of ${\mathcal{F}_{quad}}$. The set of functors $\{P_{V}|V\in\mathcal{S}\}$, named the standard projective objects in ${\mathcal{F}_{quad}}$, is a set of projective generators of ${\mathcal{F}_{quad}}$, where $\mathcal{S}$ is a set of representatives of isometry classes of nondegenerate quadratic spaces.

There is a forgetful functor $\epsilon:{\mathcal{T}_{q}}\rightarrow{\mathcal{E}}^{f}$ in ${\mathcal{F}_{quad}}$, defined by $\epsilon(V)=\mathcal{O}(V)$ and

[TABLE]

where $p_{g}$ is the orthogonal projection from $W\bot W^{\prime}$ to $W$ and $\mathcal{O}:{\mathcal{E}_{q}}\rightarrow{\mathcal{E}}^{f}$ is the functor which forgets the quadratic form. By the fullness of the functor $\epsilon$ and an argument of essential surjectivity, we obtain the following theorem.

Theorem 1.3**.**

[math.AT/0606484]* There is a functor $\iota:{\mathcal{F}}\rightarrow{\mathcal{F}_{quad}}$, which is exact, fully faithful and preserves simple objects.*

In order to define another subcategory of ${\mathcal{F}_{quad}}$, we consider the category ${\mathcal{E}_{q}^{\mathrm{deg}}}$ having as objects finite dimensional ${\mathbb{F}_{2}}$-vector spaces equipped with a (possibly degenerate) quadratic form and with morphisms injective linear maps which preserve the quadratic forms. The category ${\mathcal{E}_{q}^{\mathrm{deg}}}$ admits pullbacks; consequently the category of spans $\mathrm{Sp}({\mathcal{E}_{q}^{\mathrm{deg}}})$ ([Benabou]) is defined. By definition, the category ${\mathcal{F}_{iso}}$ is the category of functors from $\mathrm{Sp}({\mathcal{E}_{q}^{\mathrm{deg}}})$ to ${\mathcal{E}}$. As in the case of the category ${\mathcal{F}_{quad}}$, the category ${\mathcal{F}_{iso}}$ is abelian and has enough projective objects: by the Yoneda lemma, for any object $V$ of $\mathrm{Sp}({\mathcal{E}_{q}^{\mathrm{deg}}})$, the functor $Q_{V}={\mathbb{F}_{2}}[\mathrm{Hom}_{\mathrm{Sp}({\mathcal{E}_{q}^{\mathrm{deg}}})}(V,-)]$ is a projective object in ${\mathcal{F}_{iso}}$. We define a particular family of functors of ${\mathcal{F}_{iso}}$, the isotropic functors, which form a set of projective generators and injective cogenerators of ${\mathcal{F}_{iso}}$. The category ${\mathcal{F}_{iso}}$ is related to ${\mathcal{F}_{quad}}$ by the following theorem.

Theorem 1.4**.**

[math.AT/0606484]* There is a functor $\kappa:{\mathcal{F}_{iso}}\rightarrow{\mathcal{F}_{quad}}$, which is exact, fully-faithful and preserves simple objects.*

We obtain the classification of the simple objects of the category ${\mathcal{F}_{iso}}$ from the following theorem.

Theorem 1.5**.**

[math.AT/0606484]* There is a natural equivalence of categories*

[TABLE]

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

The object of ${\mathcal{F}_{iso}}$ which corresponds, by this equivalence, to the module ${\mathbb{F}_{2}}[O(V)]$ is the isotropic functor $iso_{V}$, defined in [math.AT/0606484]. Recall that, as a vector space, $iso_{V}(W)$ is isomorphic to the subspace of $Q_{V}(W)$ generated by the elements $[V\xleftarrow{\mathrm{Id}}V\rightarrow W]$.

A straightforward consequence of the classification of simple objects of ${\mathcal{F}_{iso}}$ given in Theorem 1.5 is given in the following corollary. Recall that, by definition, an object $F$ of ${\mathcal{F}_{quad}}$ is finite if it has a finite composition series with simple subquotients.

Corollary 1.6**.**

The isotropic functors $\kappa(iso_{V})$ are finite in the category ${\mathcal{F}_{quad}}$.

In section 3, we will require the composition series for the isotropic functors associated to some small quadratic spaces. For $\alpha\in\{0,1\}$, let $(x,\alpha)$ be the degenerate quadratic space of dimension one generated by $x$ such that $q(x)=\alpha$. Since the orthogonal groups $O(x,0)$ and $O(x,1)$ are trivial and $O(H_{0})\simeq\mathfrak{S}_{2}$ and $O(H_{1})\simeq\mathfrak{S}_{3}$, we deduce from Theorem 1.5 and 1.4, the following corollary.

Corollary 1.7**.**

1. (1)

The functors $\kappa(iso_{(x,0)})$ and $\kappa(iso_{(x,1)})$ are simple in ${\mathcal{F}_{quad}}$. 2. (2)

The functor $\kappa(iso_{H_{0}})$ is indecomposable. We have the following non-split short exact sequence:

[TABLE]

where $R_{H_{0}}$ is the functor obtained from the trivial representation of $O(H_{0})$. 3. (3)

The functor $\kappa(iso_{H_{1}})$ admits the following decomposition:

[TABLE]

where $S_{H_{1}}$ is the functor obtained from the natural representation of $O(H_{1})$ and $F_{H_{1}}$ is an indecomposable functor for which we have the following non-split short exact sequence:

[TABLE]

where $R_{H_{1}}$ is the functor obtained from the trivial representation of $O(H_{1})$.

In [Vespa2], we define a new family of functors of ${\mathcal{F}_{quad}}$, named the mixed functors and we decompose two particular functors of this family: the functors $\mathrm{Mix}_{0,1}$ and $\mathrm{Mix}_{1,1}$. We recall the following description of these functors.

Proposition 1.8**.**

[Vespa2]* For $\alpha\in\{0,1\}$, the functors ${\mathrm{Mix}_{\alpha,1}}:{\mathcal{T}_{q}}\rightarrow{\mathcal{E}}$ are defined by*

[TABLE]

where $S_{V}=\{(v_{1},v_{2})\ |v_{1}\in V,v_{2}\in V,\ q(v_{1}+v_{2})=\alpha,\ B(v_{1},v_{2})=1\}$ and

[TABLE]

where $p_{W}$ is the orthogonal projection.

In [Vespa2], for a positive integer $n$, we defined subfunctors $L^{n}_{\alpha}$ of $\iota(\Lambda^{n})\otimes\kappa(iso_{(x,\alpha)})$, where $\Lambda^{n}$ is the $n$th exterior power and we proved that these functors are simple. The functor $L^{1}_{\alpha}$ is equivalent to the functor $\kappa(iso_{(x,\alpha)})$. We obtain the following result.

Theorem 1.9**.**

[Vespa2]* Let $\alpha$ be an element in $\{0,1\}$.*

1. (1)

The functor ${\mathrm{Mix}_{\alpha,1}}$ is infinite. 2. (2)

There exists a subfunctor ${\mathrm{\Sigma}_{\alpha,1}}$ of ${\mathrm{Mix}_{\alpha,1}}$ such that we have the following short exact sequence

[TABLE] 3. (3)

The functor ${\mathrm{\Sigma}_{\alpha,1}}$ is uniserial with unique composition series given by the decreasing filtration given by the subfunctors $k_{d}{\mathrm{\Sigma}_{\alpha,1}}$ of ${\mathrm{\Sigma}_{\alpha,1}}$:

[TABLE]

1. (a)

The head of ${\mathrm{\Sigma}_{\alpha,1}}$ (i.e. ${\mathrm{\Sigma}_{\alpha,1}}/k_{1}{\mathrm{\Sigma}_{\alpha,1}}$) is isomorphic to the functor $\kappa(iso_{(x,\alpha)})$ where $iso_{(x,\alpha)}$ is a simple object in ${\mathcal{F}_{iso}}$. 2. (b)

For $d>0$

[TABLE]

where $L^{d+1}_{\alpha}$ is a simple object of the category ${\mathcal{F}_{quad}}$ that is neither in the image of $\iota$ nor in the image of $\kappa$.

*The functor $L^{d+1}_{\alpha}$ is a subfunctor of $\iota(\Lambda^{d+1})\otimes\kappa(iso_{(x,\alpha)})$, where $\Lambda^{d+1}$ is the *$(d+1)$st exterior power functor.

2. Filtration of the standard projective functors $P_{V}$ of

${\mathcal{F}_{quad}}$

In this section, we define a filtration of the standard projective functors $P_{V}$ of ${\mathcal{F}_{quad}}$. This construction gives rise to an essential tool to obtain, in section 3, the direct decompositions of the projective objects $P_{H_{0}}$ and $P_{H_{1}}$ of ${\mathcal{F}_{quad}}$, into indecomposable summands.

After defining this filtration, we will deduce general results about the projective $P_{V}$ of ${\mathcal{F}_{quad}}$. 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 $\iota({\mathcal{F}})$ is a thick subcategory of ${\mathcal{F}_{quad}}$. We will also show that the top quotient of this filtration is isomorphic to $\kappa(iso_{V})$, where $iso_{V}$ is the isotropic functor.

2.1. Definition of the filtration

We recall that a morphism in ${\mathcal{T}_{q}}$ from $V$ to $W$, where $V$ and $W$ are nondegenerate quadratic spaces, is represented by a diagram $V\rightarrow X\leftarrow W.$

Definition 2.1**.**

A morphism $[V\rightarrow X\leftarrow W]$ in ${\mathcal{T}_{q}}$ has rank equal to $i$ if the pullback in ${\mathcal{E}_{q}^{\mathrm{deg}}}$ of the diagram $V\rightarrow X\leftarrow W$ is a quadratic space of dimension $i$.

Notation 2.2**.**

We denote by $\mathrm{Hom}^{(i)}_{{\mathcal{T}_{q}}}(V,W)$ the subset of $\mathrm{Hom}_{{\mathcal{T}_{q}}}(V,W)$ of morphisms of rank less than or equal to $i$.

We have the following proposition:

Proposition 2.3**.**

For $W$ an object in ${\mathcal{T}_{q}}$, the following subvector space of $P_{V}(W)$:

[TABLE]

defines a subfunctor $P_{V}^{(i)}$ of $P_{V}$.

Proof.

It is sufficient to verify that for all morphisms $f=[W\rightarrow Y\leftarrow Z]$ of ${\mathcal{T}_{q}}$ and $g=[V\rightarrow X\leftarrow W]$ of $\mathrm{Hom}^{(i)}_{{\mathcal{T}_{q}}}(V,W)$, the composition $f\circ g$ has rank less than or equal to $i$. The composition $f\circ g$ is represented by the following commutative diagram:

[TABLE]

where $P$ and $P^{\prime}$ are the pullbacks and $X\underset{W}{\bot}Y$ is the pseudo push-out defined in [math.AT/0606484]. Consequently: $f\circ g=[V\rightarrow X\underset{W}{\bot}Y\leftarrow Z].$ Since $[V\rightarrow X\leftarrow W]$ 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 ${\mathcal{E}_{q}^{\mathrm{deg}}}$, that $P^{\prime}$ 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: $P_{V}^{(\mathrm{dim}(V))}\simeq P_{V}.$

We deduce the following proposition.

Proposition 2.5**.**

The functors $P_{V}^{(i)}$, for $i=0,\ldots,\mathrm{dim}(V)$, define an increasing filtration of the functor $P_{V}$.

Proof.

The inclusion of vector spaces $P_{V}^{(i)}(W)\subset P_{V}^{(i+1)}(W)$ is clear, for $W$ an object in ${\mathcal{T}_{q}}$. 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 obtained, for all objects $V$ of ${\mathcal{T}_{q}}$, the following filtration of the functor $P_{V}$:

[TABLE]

The aim of this section is to study the two extremities of this filtration, namely, the functor $P_{V}^{(0)}$ and the quotient $P_{V}/P_{V}^{(\mathrm{dim}(V)-1)}$.

2.2.1. The functor $P_{V}^{(0)}$

For $V$ an object in ${\mathcal{T}_{q}}$, we recall that the functor $P^{{\mathcal{F}}}_{\epsilon(V)}$ of ${\mathcal{F}}$ defined by $P^{{\mathcal{F}}}_{\epsilon(V)}(-)={\mathbb{F}_{2}}[\mathrm{Hom}_{{\mathcal{E}}^{f}}(\epsilon(V),-)]$, where $\epsilon:{\mathcal{T}_{q}}\rightarrow{\mathcal{E}}^{f}$ is the forgetful functor of ${\mathcal{F}_{quad}}$, is projective, by the Yoneda lemma. The aim of this paragraph is to prove the following theorem:

Theorem 2.6**.**

Let $V$ be an object of ${\mathcal{T}_{q}}$.

1. (1)

There is a natural equivalence: $P_{V}^{(0)}\simeq\iota(P^{{\mathcal{F}}}_{\epsilon(V)})$, where $\iota:{\mathcal{F}}\rightarrow{\mathcal{F}_{quad}}$ is the functor given in Theorem 1.3. 2. (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 in ${\mathcal{T}_{q}}$. A morphism $T=[V\xrightarrow{\alpha}W\bot W^{\prime}\xleftarrow{i_{W}}W]$ has rank zero if and only if $p_{W^{\prime}}\circ\alpha$ is an injective linear map, where $p_{W^{\prime}}$ is the orthogonal projection from $W\bot W^{\prime}$ to $W^{\prime}$.

For $V$ and $W$ objects in ${\mathcal{T}_{q}}$, the forgetful functor $\epsilon:{\mathcal{T}_{q}}\rightarrow{\mathcal{E}}^{f}$ gives rise to a map $\mathrm{Hom}_{{\mathcal{T}_{q}}}(V,W)\rightarrow\mathrm{Hom}_{{\mathcal{E}}^{f}}(\epsilon(V),\epsilon(W))$. By passage to the vector spaces freely generated by these sets and by functoriality of $\epsilon$, we deduce the existence of a morphism from $P_{V}$ to $\iota(P^{{\mathcal{F}}}_{\epsilon(V)})$. As the functors $P_{V}^{(0)}$ are subfunctors of $P_{V}$, we obtain a morphism $f$ from $P_{V}^{(0)}$ to $\iota(P^{{\mathcal{F}}}_{\epsilon(V)})$. Consequently, to prove Theorem 2.6, it is sufficient to prove the following proposition.

Proposition 2.8**.**

The map $P_{V}^{(0)}(W)\xrightarrow{f_{W}}P^{{\mathcal{F}}}_{\epsilon(V)}(\epsilon(W))$ is an isomorphism for $V$ and $W$ objects in ${\mathcal{T}_{q}}$.

The surjectivity of $f_{W}$ relies on the following lemma, which is an improved version of the fullness of the forgetful functor $\epsilon$ given in [math.AT/0606484].

Lemma 2.9**.**

Let $(V,q_{V})$ and $(W,q_{W})$ be two objects of ${\mathcal{T}_{q}}$ and $f\in\mathrm{Hom}_{{\mathcal{E}}^{f}}(\epsilon(V,q_{V}),\epsilon(W,q_{W}))$ a linear map, then there exists a morphism $T=[V\xrightarrow{\varphi}W\bot Y\xleftarrow{i_{W}}W]$ of rank zero such that $\epsilon(T)=f$.

Proof.

As the quadratic space $V$ is nondegenerate, we know that it has even dimension. We write $\mathrm{dim}(V)=2n$. We prove the result by induction on $n$.

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

[TABLE]

Consequently, the morphism: $T=V\xrightarrow{g_{1}}W\bot H_{1}\bot H_{0}\hookleftarrow W$, is a morphism of rank zero of ${\mathcal{T}_{q}}$ such that $\epsilon(T)=f$.

Let $V_{n}$ be a nondegenerate quadratic space of dimension $2n$, $\{a_{1},b_{1},\ldots,a_{n},b_{n}\}$ be a symplectic basis of $V_{n}$ and $f_{n}:V_{n}\rightarrow W$ be a linear map. By induction, there exists a map:

[TABLE]

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 $T=[V_{n}\xrightarrow{g_{n}}W\bot Y\hookleftarrow W]$ is of rank zero and verifies $\epsilon([V_{n}\xrightarrow{g_{n}}W\bot Y\hookleftarrow W])=f_{n}.$

Let $V_{n+1}$ be a nondegenerate quadratic space of dimension $2(n+1)$, $\{a_{1},b_{1},\ldots,a_{n},b_{n},a_{n+1},b_{n+1}\}$ a symplectic basis of $V_{n+1}$ and $f_{n+1}:V_{n+1}\rightarrow W$ a linear map. To define the map $g_{n+1}$, 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: $E\simeq W\bot W^{\prime}\bot H_{0}^{\bot n}\bot H_{0}^{\bot n}\bot H_{1}\bot H_{0}\bot H_{0}$ for which we specify the notations for a basis:

[TABLE]

[TABLE]

The following map:

[TABLE]

preserves the quadratic form. Furthermore, the morphism

[TABLE]

is of rank zero and satisfies: $\epsilon(T)=f_{n+1}$, which completes the inductive step.

∎

The proof of the injectivity of $f_{W}$ 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 $D^{\prime}$ subquadratic spaces (possibly degenerate) of $V$ and $\underline{f}:D\rightarrow D^{\prime}$ an isometry between these two quadratic spaces. Then, there exists an isometry $f:V\rightarrow V$ such that the following diagram is commutative:

[TABLE]

Proof.

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

Proof of the injectivity of $f_{W}$.

The natural map $f$ is induced by the natural map

[TABLE]

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 $T=[V\xrightarrow{\alpha}W\bot W^{\prime}\xleftarrow{i_{W}}W]$ and $T^{\prime}=[V\xrightarrow{\alpha^{\prime}}W\bot W^{\prime\prime}\xleftarrow{i^{\prime}_{W}}W]$ two generators of $P_{V}^{(0)}(W)$ such that

[TABLE]

we have $T=T^{\prime}$.

Let $\{a_{1},b_{1},\ldots,a_{n},b_{n}\}$ be a symplectic basis of $V$. We deduce from 2.10.1 that, for all $i\in\{1,\ldots,n\}$, we have:

[TABLE]

and

[TABLE]

where, for all $i\in\{1,\ldots,n\}$, $w_{i}$ and $x_{i}$ are in $W$, $w^{\prime}_{i}$ and $x^{\prime}_{i}$ are in $W^{\prime}$ and $x^{\prime\prime}_{i}$ and $w^{\prime\prime}_{i}$ are in $W^{\prime\prime}$. By Lemma 2.7, since the morphisms are of rank zero, $\{w^{\prime}_{1},x^{\prime}_{1},\ldots,w^{\prime}_{n},x^{\prime}_{n}\}$ and $\{w^{\prime\prime}_{1},x^{\prime\prime}_{1},\ldots,w^{\prime\prime}_{n},x^{\prime\prime}_{n}\}$ are two linearly independent families of vectors.

We will denote by $\underline{W^{\prime}}=\mathrm{Span}(w^{\prime}_{1},x^{\prime}_{1},\ldots,w^{\prime}_{n},x^{\prime}_{n})$ (respectively $\underline{W^{\prime\prime}}=\mathrm{Span}(w^{\prime\prime}_{1},x^{\prime\prime}_{1},\ldots,w^{\prime\prime}_{n},x^{\prime\prime}_{n})$) the subquadratic space (possibly degenerate), of $W^{\prime}$ (respectively $W^{\prime\prime}$) and we define the linear map $\underline{f}:\underline{W^{\prime}}\rightarrow\underline{W^{\prime\prime}}$ by $\underline{f}(w^{\prime}_{i})=w^{\prime\prime}_{i}$ and $\underline{f}(x^{\prime}_{i})=x^{\prime\prime}_{i}$ for all $i\in\{1,\ldots,n\}$.

Since $\alpha$ and $\alpha^{\prime}$ preserve the quadratic forms, we deduce from the relations 2.10.2 and 2.10.3 that $\underline{f}$ preserves the quadratic form. Hence, we can apply Theorem 2.10 to the nondegenerate space $W^{\prime}\bot W^{\prime\prime}$, which gives a morphism $f:W^{\prime}\bot W^{\prime\prime}\rightarrow W^{\prime}\bot W^{\prime\prime}$ of ${\mathcal{E}_{q}}$, such that, the restriction of this morphism to $W^{\prime}$ coincides with $\underline{f}$. We deduce the commutativity of the following diagram:

[TABLE]

where $\tilde{\alpha}=i_{W\bot W^{\prime}}\circ\alpha$ and $\tilde{\alpha^{\prime}}=i_{W\bot W^{\prime\prime}}\circ\alpha^{\prime}$. Consequently, we obtain the equality $T=T^{\prime}$ since, by inclusion, we have

[TABLE]

and

[TABLE]

∎

Notation 2.11**.**

For $V$ and $W$ two objects of $\mathcal{E}_{q}$, and $f$ a morphism of $\mathrm{Hom}_{{\mathcal{E}}^{f}}(\epsilon(V),\epsilon(W))$, we denote by $t_{f}$ the morphism of $\mathrm{Hom}^{(0)}_{{\mathcal{T}_{q}}}(V,W)$ corresponding to $f$ and by $[t_{f}]$ the canonical generator of $P_{V}^{(0)}(W)$ obtained from $t_{f}$. To simplify the notation, we will denote the morphism $t_{\mathrm{Id}_{V}}$ of $\mathrm{Hom}^{(0)}_{{\mathcal{T}_{q}}}(V,V)$ 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 $\mathcal{E}_{q}$, $f:\epsilon(W)\rightarrow\epsilon(X)$ and $g:\epsilon(V)\rightarrow\epsilon(W)$ morphisms of ${\mathcal{E}}^{f}$, we have:

[TABLE]

where $t_{f},\ t_{g}$ and $t_{f\circ g}$ are respectively the morphisms of $\mathrm{Hom}^{(0)}_{{\mathcal{T}_{q}}}(W,X)$, $\mathrm{Hom}^{(0)}_{{\mathcal{T}_{q}}}(V,W)$ and $\mathrm{Hom}^{(0)}_{{\mathcal{T}_{q}}}(V,X)$ associated to the linear maps $f,\ g$ and $f\circ g$.

We can apply this result to the idempotents of the ring of endomorphisms $\mathrm{End}(P_{V})$, to obtain the following proposition.

Proposition 2.13**.**

The canonical generator $[e_{V}]$ of $P_{V}^{(0)}$ is an idempotent of the ring of endomorphisms $\mathrm{End}(P_{V})$ such that $P_{V}.[e_{V}]\simeq P_{V}^{(0)}.$

Proof.

The canonical generator $[e_{V}]$ is an idempotent of $\mathrm{End}(P_{V})$ by Corollary 2.12. By definition of the rank filtration $P_{V}.[e_{V}]\subset P_{V}^{(0)}$ and, for a canonical generator $[t_{f}]$ of $P_{V}^{(0)}$, we have $[t_{f}]=[t_{f}]\cdot[e_{V}]$. ∎

The idempotent $[e_{V}]$ plays a central rôle in the proof of the thickness of the subcategory $\iota({\mathcal{F}})$ in ${\mathcal{F}_{quad}}$, which is the subject of the following paragraph. For that, the following result is necessary.

Lemma 2.14**.**

Let $V$ and $W$ be objects of ${\mathcal{T}_{q}}$, the functor $\iota$ induces an isomorphism:

[TABLE]

where $\epsilon:{\mathcal{T}_{q}}\rightarrow{\mathcal{E}}$ is the forgetful functor.

Proof.

By Proposition 2.13 and Theorem 2.6 we have the following equivalences:

[TABLE]

[TABLE]

∎

To conclude this paragraph, we give the following property of $e_{V}$ which will be useful in section 4 concerning the polynomial functors of ${\mathcal{F}_{quad}}$.

Lemma 2.15**.**

For $V$ and $W$ two objects of ${\mathcal{T}_{q}}$, we have: $e_{V\bot W}=e_{V}\bot e_{W}$, where $\bot:{\mathcal{T}_{q}}\times{\mathcal{T}_{q}}\rightarrow{\mathcal{T}_{q}}$ is the functor induced by the orthogonal sum.

Proof.

This is a straightforward consequence of Proposition 2.8. ∎

2.2.2. The category $\iota({\mathcal{F}})$ is a thick subcategory of ${\mathcal{F}_{quad}}$

The aim of this paragraph is to prove the following result.

Theorem 2.16**.**

The category $\iota({\mathcal{F}})$ is a thick subcategory of ${\mathcal{F}_{quad}}$, where $\iota:{\mathcal{F}}\rightarrow{\mathcal{F}_{quad}}$ 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 [math.AT/0606484].

Proposition 2.17**.**

Let ${\mathcal{C}}$ and ${\mathcal{D}}$ be two small categories, ${\mathcal{A}}$ be an abelian category, $F:{\mathcal{C}}\rightarrow{\mathcal{D}}$ be a functor and $-\circ F:\mathrm{Func}({\mathcal{D}},{\mathcal{A}})\rightarrow\mathrm{Func}({\mathcal{C}},{\mathcal{A}})$ be the precomposition functor, where $\mathrm{Func}({\mathcal{C}},{\mathcal{A}})$ is the category of functors from ${\mathcal{C}}$ to ${\mathcal{A}}$. If $F$ is full and essentially surjective, then any subobject (respectively 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 $\iota({\mathcal{F}})$ of ${\mathcal{F}_{quad}}$ is full by Theorem 1.3.

• •

Let $F^{{\mathcal{F}}}$ be an object in ${\mathcal{F}}$ and $G$ a subobject of $\iota(F^{{\mathcal{F}}})$. Let ${\mathcal{F}}^{\prime}$ be the category of functors from ${\mathcal{E}}^{f-(even)}$ to ${\mathcal{E}}$, where ${\mathcal{E}}^{f-(even)}$ is the full subcategory of ${\mathcal{E}}^{f}$ having as objects the ${\mathbb{F}_{2}}$-vector spaces of even dimension. The categories ${\mathcal{F}}$ and ${\mathcal{F}}^{\prime}$ are equivalent [math.AT/0606484]. The functor $\epsilon:{\mathcal{E}_{q}}\rightarrow{\mathcal{E}}^{f}$ factorizes through the inclusion ${\mathcal{E}}^{f-(even)}\hookrightarrow{\mathcal{E}}^{f}$. This induces a functor $\epsilon^{\prime}:{\mathcal{E}_{q}}\rightarrow{\mathcal{E}}^{f-(even)}$ which is full and essentially surjective. Consequently, we can use Proposition 2.17 to obtain: $G\simeq\iota(G^{{\mathcal{F}}})$. Similarly, we obtain the result for the quotient.

• •

Let $G^{{\mathcal{F}}}$ and $H^{{\mathcal{F}}}$ be objects of ${\mathcal{F}}$, we set $G=\iota(G^{{\mathcal{F}}})$ and $H=\iota(H^{{\mathcal{F}}})$. For a short exact sequence: $0\rightarrow G\rightarrow F\rightarrow H\rightarrow 0,$ we have to prove that there exists a functor $F^{{\mathcal{F}}}$ in ${\mathcal{F}}$ such that $F=\iota(F^{{\mathcal{F}}})$.

Let $P_{1}\rightarrow P_{0}\rightarrow G^{{\mathcal{F}}}\rightarrow 0$ and $Q_{1}\rightarrow Q_{0}\rightarrow H^{{\mathcal{F}}}\rightarrow 0$ be projective presentations of $G^{{\mathcal{F}}}$ and $H^{{\mathcal{F}}}$ in ${\mathcal{F}}$, we have the following commutative diagram

[TABLE]

where the columns are projective resolutions in ${\mathcal{F}_{quad}}$, by the horseshoe lemma. By Lemma 2.14, the morphism $\iota(P_{1})\oplus\iota(Q_{1})\rightarrow\iota(P_{0})\oplus\iota(Q_{0})$ is induced by a morphism of ${\mathcal{F}}$ denoted by $f$. Consequently, $F\cong\iota(\mathrm{Coker}(f))\in\iota({\mathcal{F}})$.

∎

By Theorem 2.16, we deduce from Lemma 2.14, the following characterization of the simple functors of ${\mathcal{F}}$ in ${\mathcal{F}_{quad}}$ which will be used in section 4 of this paper concerning the polynomial functors of ${\mathcal{F}_{quad}}$.

Lemma 2.18**.**

1. (1)

Let $F$ be a functor of ${\mathcal{F}_{quad}}$, then $F$ is in the image of the functor $\iota:{\mathcal{F}}\rightarrow{\mathcal{F}_{quad}}$ if and only if, for all objects $V$ in ${\mathcal{T}_{q}}$,

[TABLE] 2. (2)

Let $S$ be a simple object in ${\mathcal{F}_{quad}}$, then $S$ is in the image of the functor $\iota:{\mathcal{F}}\rightarrow{\mathcal{F}_{quad}}$ if and only if there exists an object $W$ in ${\mathcal{T}_{q}}$ such that

[TABLE]

Proof.

1. (1)

The forward implication is a consequence of the following fact: for a functor $F$ in the image of $\iota$, $\mathrm{Hom}_{{\mathcal{F}_{quad}}}(P_{V}^{(0)},F)=F(e_{V})F(V)=F(V).$