# a note on polynomial functors and ext 1 groups

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A note on polynomial functors and ext1 GroupsGeoffrey M. L. Powell aa L'institut galilee, universite de parisnord , Villetaneuse, FrancePublished online: 27 Jun 2007.

To cite this article: Geoffrey M. L. Powell (1997) A note on polynomial functors and ext1 Groups, Communications inAlgebra, 25:3, 979-987, DOI: 10.1080/00927879708825903

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COMMUNICATIONS IN ALGEBRA, 25(3), 979-987 (1997)

A Note on Polynomial Functors and ~ x t ' Groups

Geoffrey M.L. Powell

L'INSTITUT GALILEE, UNIVERSITE DE PARIS-NORD, VILLETANEUSE, FRANCE

1 Introduction

Let F be a finite field of characteristic p and let F = F ( F ) denote the category of functors Ef + E , where E is the category of IF-vector spaces and Ef is the full subcategory of finite dimensional vector spaces.

3 is an abelian category; an object is szmple if it has no non-trivial subobjects and is finite if it has a finite composition series. It is known that the isomorphism classes of simple objects in F are indexed by the q-regular partitions; however, the details of this are not required for this paper. References for the study of the category 3 are [Kl, K2, K3, S].

No ta t ion 1.1 Suppose that F E 3 takes values in E f . Put dF(n) := d i m F ( F n ) .

Kuhn has shown that, for a functor F E F which takes finite-dimensional values, there exist uniquely-determined integral coefficients dk = dk(F) , such that:

for all natural numbers n (see [Kl] ) .

Definit ion 1.2 A functor F E 3 is polynomial if it takes finite-dimensional values and there exists n such that d k ( F ) = 0 for k > n. If d,(F) # 0, then F is said to have polynomial degree exactly n. 0

The category 3 comes equipped with the dzfterence functor A : F -+ F; this may be defined on objects as the kernel A F ( V ) = ker {F(V@IF) + F ( V ) ) , of the map induced by the natural projection: V @IF + V .

It is known (see [Kl]) that a functor is polynomial if and only if it is finite; moreover F is polynomial of degree exactly n if and only if An+'(F) = 0 but An(F) # 0. The full subcategory of functors of degree < d is denoted by Fd.

Copyr~gh t C 1997 by Marcel Dekker, Inc

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980 POWELL

This defines a filtration F , C Fl C . . . c Fd C . . .3, by subcategories of the category 3, where the category F , is defined as the colimit of this filtration by the subcategories Fd and is called the category of analytic functors. 3 is a natural object of study in algebraic K-theory; it is of particular importance since the calculation of Ezt groups in F is equivalent to the calculation of MacLane cohomology [FLS]. For F = F , the prime field, it is of interest to topologists due to the connection established in [HLS] between F and the category U of unstable modules over the mod-p Steenrod algebra. Namely there are functors 1 : U + F and r : 3 + U, with 1 left adjoint to T , which induce an equivalence of categories U I N i l Z F , ( F ) , where N i l is the localizing subcategory of nilpotent unstable modules.

No ta t ion 1.3 Let M, denote the category of modules over M,(F) , the semi-group of n x n matrices with entries in F. 0

This note studies polynomial functors; the main results are the following:

T h e o r e m 1 If F zs a functor of polynomial degree 5 d and G is any functor, then the natural map Hornr(F, G ) 4 H o ~ ~ , + , ( F ( F ~ + ' ) , G ( F ~ + ' ) ) is an isomorphism.

Definit ion 1.4 Suppose that F E F is a functor; the connectivity of F is the smallest integer c such that F(IFC) # 0. If FA is a simple functor, indexed by a partition A, write c(X) for the connectivity of FA. 0

The next result is an immediate corollary of the preceding theorem:

T h e o r e m 2 Suppose that G zs a polynomial functor of degree d and that F is a functor with connectivity c ( F ) > d + 1; then Ext;(F, G ) = 0 = E&G, F ) . The methods used to prove these results also permit the proof of the following

T h e o r e m 3 If H zs a functor which takes finite-dzmensional values and d n ( H ) = 0, for some n , then H splzts as H % G @ F, where d,(F) = 0 for s 5 n and dt (G) = 0 for t 2 n.

R e m a r k 1.5 The method of proof relies on constructing a 'model' for the zntermediate extension of F ( I F ~ ) , as explained in the next section; the results may become clearer to the reader when viewed in terms of the embedding theorem of [Kl], together with the understanding of the polynomial filtration of the symmetric powers given in [KK]. 0

2 Polynomial functors

This section provides an approach to the study of polynomial functors; the results will probably be clear to the experts and many are implicit in [Kl].

There is a natural functor en : F -+ M,, given by evaluation: F - F(IFn). The functor en admits left and right adjoints: r,, 1, : M , -+ T (see [K2]) which are exten- sions: the unit M 2 e,l,M and the co-unit e,r,M 2 M of the respective adjunctions are isomorphisms for all M E M,.

The zntennedzate extenszon is the functor c, : M , -, 3 which is defined on objects, M E M,, as the image h M of the map y~ : 1,M + rnM which is adjoint to

: enlnM + M . This is an extension, since e,c,,M M and is the smallest such extension, in that c,M is a sub-quot,ient of any functor F such that e,F Z M . The following holds:

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POLYNOMIAL FUNCTORS 98 1

Propos i t i on 2.1 If F is a polynomzal functor of degree d and n 2 d, write N := F(Fn) There is an zsomorphzsm F c,,N.

Proof : By [K2], c,,N appears as a subquotient of F ; it is therefore polynomial of degree 5 d. However, since c,,(N) is an extension, c,,N(IFk) F(IFk) for k 5 n. The dimen- sions of these spaces 0 5 k 5 d determine d ~ ( * ) and d % ~ ( * ) , which are equal, since these are polynomial functions of degree 5 d which agree on d + 1 points. Hence, we may conclude that the functors are equal. 0

This paper essentially gives an explicit model for the intermediate extension in the situation of the proposition. The aim of this section is to give a description of the coefficients dk and a choice of basis of the vector spaces F ( F ~ ) , when F E Fd and k > d. The methods require the choice of a standard skeleton for Ef, together with a choice of basis for each functor.

No ta t ion 2.2 Fix a basis {xl , . . . , xn, . . .) for IFm and embed IFn -+ IFm as the subspace spanned by the first n vectors. Let I ( n , k ) denote the set of 'standard' embeddings F' -+ .Fk, which are induced by monomorphisms a : [n] -3 [k], via x, I-+ x,,. (Here, [N] denotes the set { I , . . . , N)). Thus; if C,v denotes the symmetric group acting on the left of [N] as the group of permutations, then I ( n , k) admits a right &,-action and a left Ck-action.

Define S ( n , k) c T(n , k) to be the set of standard embeddings induced by increuzng functions; S ( n , k) gives a set of coset representatives of I ( n , k) under the right En-action. Thus, for any a' E T ( n , k) , there are unique elements a E S(n. k ) and p E C, such that a' = ap . 0

For a fixed k, let {q,(1 5 z 5 k) be the set of standard projections IFk -+ pk onto k - 1 dimensional subspaces, which factor through elements of S ( k - 1, k). Choose the numbering of these elements so that ker(q,) = ( s t ) .

Definit ion 2.3 Suppose that F is a functor in F; define the following subspaces of F ( F k

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