Some Finiteness Results in the Category U

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Viemam I Math (2015)43,181-192 DOI 10 U ' n 7 / s l 0 0 l 3 - 0 l 4 - 0 l l 6 - 9

Some Finiteness Results in the Category U

Nguyen The Cuong. Lionel Schwartz

Received: 30 January 2014 / Accepted. 10 June 2014/Pubhshed online: 21 January 2015

Abstract In this note, we investigate some fmiteness properties of the category U of unstable modules over the Steenrod algebra We show a finiteness property for the injective resolution of finitely generated unstable modules, we show that such a module has an mjective resolution, sucb that each term is a finite direct sum of indecomposable injective unstable modules. We also show a stabilization result under Frobenius twist for Exti^-groups.

Keywords Unstable modules • Minimal resolutions - MacLane homology Mathematics Subject Classification (2010) 55S10 • 18G10

1 Introduction

ThisnoteinvestigatessomefinitenesspropertiesofthecategoryiV of unstable modules over the Steenrod algebra. For simplicity, only the case of tbe prime 2 will be considered. The main finiteness property of W is to be locally Noetherian ([18], see also [17]). This means that if W IS a finitely generated unstable module, each submodule is also finitely generated.

Injective objects of tbe category are described in [16], It is natural and useful to ask for some control on injective resolutions Tbe first result of the paper is

Theorem 5 Let M be an unstable finitely generated module. It has an injective reso- lution I " such that each J * is a finite direct sum of indecomposable injective unstable modules.

N T, Cuong - L Schwartz O )

UMR 7539 CNRS. USPC. Universite Pans 13 and LIA CNRS Formath Vieman VUletaneuse, 93430 Paris, Fiance

e-mail- schw J i i z @ u n n - p a n s l 3 . ( r N. T Cuong

e-mail [[email protected]

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N.T Cuong, L Schwartz This is to be compared with an analogous theorem in tbe category T of functors from finite dimensional Fj-vector spaces to Fj-vector spaces [4, 6, 21]:

Theorem 1 Let F be a polynomial functor taking finite dimensional values. There exists an injective resolution T* of M such that each I * is a finite direct sum of indecomposable injective fun ctors

In the sequel, such injective resolutions (for both categories) will be called of finite type.

The proof of Theorem 5 uses Theorem 1 but is not a direct consequence.

The hnk between Theorem 5 and Theorem 1 is given by tbe functor f \1A ^f T [9].

This functor induces an equivalence t//A/(7 = T^ of the quodent category of i / b y the full subcategory Mil of nilpotent modules to the category of analydc functors. The functor / has a right adjoint m, the composition £ = mo f is the localization functor away from Mil.

The natural map M —> t(M) is initial among morphisms M ^* L,L being A^iV-closed, i.e., such that Ef.%f(N, L) ^ {0} for i = 0, 1, all A' e Mil [7].

The functor i is left exact and admits right derived functors £'. In some interesting cases described later, the functors £' are computed by the MacLane homology. Here is a general result'

Theorem 6 If M is a finitely generated unstable module, the unstable modules V (M) are finitely generated

This is related to Theorem 5 but not (at least directly) equivalent.

Recall the Frobenius twist functor * on tbe category U [21]. This functor is also called

"double fimctor": <^(M)^" ^ M" and <t(jW)-"+' ^ (0). But because of [8], the terminology "Frobenius functor" looks better. The stabilization result which follows is a direct corollary of Theorem 5:

Theorem 7 Let M and N be nvo finitely generated unstable modules. Consider the direct system induced by the maps ^^~^^ M —> 4)*A/:

>Ext[^(<I>*M, A?) — » - E x t [ ^ ( * * + ' A f , N ) —> •-• . For k large enough, this map is an isomorphism, and the stabilized terms are isomorphic to E x t V ( / ( M ) , / ( A ' ) ) .

A similar stabilization result holds in the category of strict polynomial functors for the map induced by the Frobenius twist [5], F and G denoting stnct polynomial functors of the same degree, and F ' * ' being the k-th Frobenius twist, the system

> Ext'-p ( F ' * > , G ' * ' ) - ^ Ext^ ( F ' * + " , G < * + " ) —> . . ,

stabilizes for k large. In general, the colJmit is not isomorphic to Eyit'j^(0{F). 0(G)), O : V -^ J^ is the forgetful functor However, in interesting cases die isomorphism holds, it holds m particular when F and G are the canonical liftings of the simple objects of V in the category V. Another difference is that on the nght-band side of the Ext-group, die module does come wiih a Frobenius twist: we have just A' instead of O* (Af) Again in some mleresfing cases, it is possible to replace N by **(Af). The main reason to keep N is that die cohmit considered in the corollary is more natural for topological applications

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aome l-mneness Results in the Category U 183

It is also worth notmg that it is nol clear to the authors how to prove the preceding result using projective resolutions. The category U is locally Noethenan. Thus, given a fimtely generated module M, it is possible to construct a projective resolution of M such that each term of the resolution Is a finite direct sum of indecomposable projective modules.

However, when replacmg M by **(A/), control is lost on the size of the i-th term in the resolution.

hi particular, if E x t ^ ( / ( W ) , f(N)) = {Oj, flien Ext[^(**(M). N) ^ (Oj for k large enough. This case applies for i = 1 in the following situation. Let Un, n E N , be the Krull filtration on U (see the next section). Let M be a finitely generated unstable module, assume M € Un\ Un-\. Let R be the smallest submodule in tin such dial M/R e Un-\. The following is a corollary of Theorem 7:

Proposition 1 Fork targe enough, the ¥2-vector space Ext^ (ij>*(fi ® R), M) IS trivial.

This is applied in [2]. This statement can be proved direcdy using only Steenrod operations; however, it is much more tedious.

The followmg comment was added just before sending the revised version to the journal.

Steven Sam has announced a proof of the Artinian conjecture. This shows that Theorem 1 holds for a polynomial functor F whose injective hull is a finite direct sum of indecomposable polynomial injective functors. It is therefore natural to ask if the same holds for unstable modules. Simple examples show it cannot extend without modifications However, the followmg looks to be likely:

Conjecture 1 Let M be an unstable module whose injective hull is a finite direct sum of indecomposable injective unstable modules. Let T* be its minimal resolution. Let A be a given integer, then ( I * ) ' is finite dimensional for each i.

The module X* could be an infinite direct sum, however, in a given degree, only a fimte number of terms of tbe direct sum should be nontrivial. This is supported by v examples and facts.

2 Backgrounds: the Nilpotent Filtration, the KruII Filtration, and Functors This section recalls briefly facts about the category U. We refer mostly to [21] and [12] for all of this material. The subcategory Mils, s > 0, of Z^ is tbe smallest thick subcategory stable under colimits and containing ail A--suspensions of unstable modules.

U - Milo 3 Mih DMil2D---^MihD---

Proposition 2 An unstable module M has a convergent decreasing filtration {M^ }j>o with M,(Ms-\.\ = S* Rsi^) where Rs(M) is a reduced unstable module, i.e., which does not contain a nontrivial suspension.

An unstable module is mlpotent if it belongs to Mil]. The following is proved in [21, Lemma 6.1.4]; see also [12]:

Proposition 3 Let M be a finitely generated unstable module. Then the R, (Af) are finitely geneniied and trivial ifi is large enough.

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N T. Cuong, L Schwartz The category of unstable modules U, as an Abelian category, has a Krull filtration. These are thick subcategories which are stable under colimits

UoCUiCU2<Z---CU defined as follows.

The category Uo is the largest thick subcategory generated by simple objects and stable under colimits. It is the subcategory of locally fimte modules, M eU is locally finite if tbe span over A2 of each x E M \s finite. Having defmed by induction Ifn, we define U„+\ as follows. We introduce the quotient category U/U„ whose objects are the same of those as U but where morphisms in U that have kernel and cokemel in ll„ are formally inverted. Then (W/^n)o is defined as above, and U„+\ is the pre-image of this subcategory in U via the canonical projection functor; see [7] for details We have

Theorem 2 Let M e U and K„(M) be the largestsubobjectof M belonging toU„, then M = \JK„(M).

Let us give some examples:

I ) ' F ( n ) s ltn\ U„-\, the unstable modules F{n) are the canonical generators of W, generated in degree n by i„ and as an F2-vector space, it has a basis forming by Sq' where / runs through the set of multi-indices of excess less than or equal to n;

the«-diextenorpower A " ( f (I)), F(l)®" eUn \U„-\ etc.

There is a charactenzation of the Krull filtration m terms of the functor f introduced by Jean Lannes. The functor Ty, V being an elementary Abehan p-group, is left adjoint to M \-^ H*BV ®M.lfV = F2, Ty is denoted by T. As H-'B'Lj2 splits up, in U, as F2 ® Ii' 6 Z / 2 the functor T is naturally equivalent \.old®T. Below are die main properties of Tv-

Theorem 3 ([15,21]) The functor Ty commutes with colitnits (as a left adjoint). It is exact Moreover, there is a canonical isomorphism

TviM ®N)^ TV(M) ® 7"v(A').

If J W ^ S F 2 , i t w n t e s a s 7 ' v ( E M ) = HTyiM)

Below is the characterization of the Krull filtration alluded to the above:

Theorem 4 The following two conditions are equivalent:

M eU„, f''+HM) = {0].

A very nice proof of this result is in [14], as well as in Nguyen The Cuong's thesis [1].

Corollary 1 IfM e U„, and N e U„. then M ® A' e Um+„.

Let J^ be die category of functors from finite dimensional F2-vector spaces to all vector spaces. Define a functor, [9], / • U —»• J" by

fiM)iV) = Homu(M, H'(BV))' = TviM)^.

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aomt;! ii.ueness Results in the Category U !85

Let J^„ be the subcategory of polynomial functors of degree less than or equal to n, It is defined as follows. Let f e JP, let A(F) £ J^ be defmed by

A(F)(V) ^ Ker(F(l^ ©F2) ^ FiV)).

Then by definition, F e ^ „ if and only if A""""'(F) = 0. As an example, V \-^ V®" is in T„. The following isomorphism holds for each M eU

A(f(M))^ fifiM)) Thus, the diagram below commutes:

Utf- • • -c ^ W „ - i < • W „ c ^u

\ \ \

J'rf- ^ • • -^ ^ J ^ n - 1 ' ^ T,f ^ JP The functor / has a right adjoint m, the composition t — m • / is tbe locahzation functor away from TVI/. The natural map M -^ €(A/) is initial for M -^ L, Z. being MZ-closed, i e , such that ExtJ^(iV, L) = [0] for i = 0, 1, each A' E Mil [7]. In particular, tbe localization of a nilpotent unstable module is trivial.

An injective unstable module always splits up as the direct sum of a reduced module and of a nilpotent one, moreover (by definition), there are no nontnvial maps ft"om a nilpotent module to a reduced one ([21, Chapters 2 and 3]) Thus, all mjective resolutions I ' of an unstable module M have tbe followmg properties.

Proposition 4 For each k, I * decomposes as a direct sumT& ®M'^, the first module being reduced and the second one being nilpotent The differential 3* : Z* —> I*"'"' writes as i V t I : 72* ©A''* —> 72*"'"' ©A''*^'. The k-th cohomology module of the quotient

\p K)

complex Tl' is iby definition) the k-th derived functor f*(M) of the localization functor away from Mil applied to M:

//*(72*) = £*(M) and fork > 0

/ / * + ' ( A ' ' ) = f * ( A / ) .

The last isomorphism follows from the long exact sequence associated to tbe exact sequence of complexes: {0} —»• -V* -> I* -^ TV ^^ (0)

As a consequence, the modules t''(M), k > 1, are nilpotent The heart of the phe- nomenon is tiiat F2 IS injective (and projective). Usmg this, we can prove du-ectly tbe resuh using f: as the functor T preserves reduced modules and nilpotenl modules, it commutes widi e so that r ( £ ( M ) ) ^ £ ( r ( M ) ) and more generally r ( r ( M ) ) = f ( r ( M ) ) , the same holds for f.

Proposition 5 IfM € U„. then t''{M) e U„^\, ifk > u.

Indeed, it is enough to show that f " ( f ' ( M ) ) is trivial if i > 0, but since f " ( f ' ( M ) ) ^ t (f"[M)), the result follows from Theorem 4.

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N.T. Cuong, L. Schwartz As the tensor product of reduced injective unstable modules is still injective, we have Corollary 2 (Ktinnetb formula) Let M and N be two unstable modules, then

C''{M®N)= 0 e'(M)'S,tHN)-

At this opportunity, we recall that the tensor product of two finitely generated unstable modules is also finitely generated [21, Chapter 1],

We keep tbe notation introduced above. Assume tbe resolution to be minimal, thus 1*+' is the injective hull of coker(a*~'). On the other hand, 72*"*"' is the injective hull of the quotient 72'* of coker(9*~') by its largest mlpotent submodule ^(cokerCfl*"')). The results which follow are standard homological algebra.

Proposition 6 Let TV'* be the largest nilpotent submodule o/coker(9*~') We have the following exact sequence:

[0] —^ e''(M) —* coker(a*+') —> /V'*"^' - ^ ^*"^'(M) —»• (Oj.

lfthe resolution is minimal, the unstable module 72*"'"' is the injective hull ofR''', TV*"*"' is the injective hull ofN'''.

If we have an injective resolution X' of an unstable module M, / ( J " ) is an injective resolution of f(M). If, moreover, we assume M to be finitely generated, then f(M) is a fimte functor, using Kubn's terminology: it is polynomial and takes finite dimensional values [10]. Then Theorem 1 implies that a minimal resolution of f(M) is of finite type.

This implies easily that the reduced part of a minimal resolution of M is of finite type and in fact:

Proposifion 7 Let M be a finitely generated unstable module, and X* a minimal injective resolution Then f(X') is a minimal injective resolution of f(M).

This follows from the above results. In particular, if if is a reduced injective unstable module, the localization R -* i(R) is an isomorphism, also, if / is an injective analytic fiinctor [9], then / o mil) - * / is an equivalence

We reformulate Theorem 5:

Theorem 5 Let M be a finitely generated unstable module, there is an injective resolution X* of M in U such that each X is a finite direct sum of modules of the type J(n) ® H'V.

In Section 1, tbe result is formulated a bit differently; however, this is equivalent because each y ( R ) ® / / * V is a finite direct sum of indecomposable objects. Indecomposable injective unstable ^2-modules are known to be of die form J(n) ® L,., Lx being an indecomposable factor of some H' V, [16] and see below. Tbe gain with this formulation is that it allows to use the functor Ty, instead of the division functor by indecomposable factors L (see

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Some Fimiene'is Results in the Category U 187

below). For what is necessary to die proofs, all division functors share tbe same essential properties; however, Ty is much more manageable.

The indecomposable injective unstable modules are, as mentioned above, of the form J (n)®L,, where Lx is a direct factor in some //*V. The isomorphism classes of indecomposable reduced injective unstable modules are indexed by the simple representations over FT ofthe groups G L ^ ( F 2 ) . These representations are themselves indexed by 2-regular partitions X of all integers, 2-regular partitions being tbe stnctiy decreasing ones. The unstable module J(n) ® Lx is the injecnve hull ini./of the unstable module I : ' ' 5 A ( F ( 1 ) ) , where 5^

is the simple functor in thecategoryJ^ associated to A. This functor is of degree |A| ([3, 11, 19, 20]).

Theorem 6 lfthe unstable module M is finitely generated, the unstable modules V (M) are finitely generated.

The fnst part of the theorem is classical and follows from Theorem 4 and the commuta- lionof r with t.

The next result is the stabilization theorem:

Theorem 7 Let M and N be two finitely generated unstable modules. Then for each i, the

¥2-vector space Ext'jrifiM), fiN)) is the colimit over k of the system xt!^[^'-iM).N) -

where the morphisms are induced by the maps **•'"'M -^ O^'A/. Moreover, for k large enough, there is an isomorphism

xl^ (4.*(W), N) ^ Ext'j,if(M). f(N)).

The proofs of Theorems 5 and 6 will be done at the same time by induction over the Krull filtration. To prove Theorem 5, it is enough to prove

Proposition 8 Let M be a finitely generated unstable module and k an integer There exists only a finite number of unstable modules J."Sx(F(l)) such that Exl^(S''5x(F(l)), M) is nontrivial. and if it is nontrivial, it is finite dimensional.

Indeed, J(n) ® Lx is the injective hull of T."Sx(F(\)), and the dimension of the F2- vector space Ext^ CL^SxiFil)), M) is greater or equal to the number of occurrences of J(n) ® Lx in the term Z* of a minimal resolution of M.

We will use

Lemma 1 Let (Oj ^ W —»• A/ —* M" —*• (Of be a short exact sequence. If Proposition 8 and Theorem 6 hold for two terms of the sequence, they hold for the third.

3.1 First Step ofdie Proofs of Proposition 8 and Theorem 6: ZYo

Lemma 2 Both Proposition 8 and Theorem 6 are true for finitely generated locally finile unstable modules.

Indeed, this follows from die following fact. Each finitely generated locally finite unstable modules is a finite module, thus has an injective resolution of finite length. Moreover, all

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N T. Cuong, L. Schwartz terras of the resolution are a finite direct sum of the Brown-Gitier modules (see [21]). More precisely, we construct a fimte resolution of a finitely generated object M in UQ because the mjective bull E ^ of M is also fimte and thus finitely generated. Moreover, if we denote by v(M) the largest integer such that M is trivial in degrees strictly larger than viM), then V(EM/M) < v(M). This allows to show dial minimal resolution are finite

For what concerns the locahzation functor and its derived functors, they are all trivial, except if the module is concentrated in degree 0, in which case die localization is an isomorphism, and the derived functors are tnvial.

3.2 Second Step of the Proofs of Tbeorems 5 and 6: F ( l )

Lemma 3 Both Theorems 5 and 6 are true for the unstable module F(\).

This relies on computations in the MacLane homology and is explained in great details m [1]. We give some informations below that are enough for our purpose. The unstable modules ^ ' ( F ( l ) ) are known to belong to UQ by Proposition 5, and it remains to prove they are finitely generated In fact, they are explicitiy known; computations depending on MacLane homology show that they are finite and thus finitely generated. This follows from Theorem 12.13 of [4], This allows to have some control on die A^* using Proposition 6 and to prove by induction they are finite. It follows directly from these results that the nilpotent part of the minimal resolution of F ( l ) is in each degree a finite direct sum of the Brown-Gitier modules.

It remains to show die result for tbe reduced part: that 72* of the minimal resolution of F ( l ) is a finite direct sum of indecomposable injective unstable modules This is a consequence of the corresponding result in the category J^ Theorem 1 and Proposition 7.

On the way, we give some informations on die reduced part of the resolution of F ( l ) . We start by a theorem which is proved in [1]. This is not necessary here but worth to be mentioned, and the material introduced here, which is the longest part of tbe digression, will be used later

Proposition 9 (A. Touze [24,4.18]) Let Shea simple functor: ; / E x t ^ ( 5 , Id) is nontrivial, then the degree ofS must be a power of 2.

We offer a proof different from the one of [24], it depends on

Lemma 4 Let Xbea 2-regular partition, such that |X| ^ n. The functor A-^ ^ A'"' ® • ® A^' has a filtration whose subquotients are either simple functors of degree m such that there exists h with 2''m = n, or tensor products F ®G of functors with no constant part iF({0]) = Gi[0]) = [0)).

To prove the proposition, consider A'- ^ A^' ® - - ® A"^' as a strict polynomial functor, and we use the classification of simple objects in V. It follows from a dieorem of Steinberg that they are of tbe form, [13]

where A ' , . .,X' are 2-regular partitions, i] < 1*2 < - • - < ;,, and s [ ' ' is the i-th Frobenius twist of the canonical lifting of tbe simple functor 5;. € TtoT.

The following more precise form of Lemma 4 is necessary to complete the proof and will be necessary later-

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Somi. rimieness Results in the Category U

Lemma 5 Let X be a 2-regular partition of the integer n The composition series of the strict polynomial functor A^ has

one subquotient Sx,

all other simple subquotients of degree n as polynomial fiinctors are functors S^ with p. > X, for the natural order on partitions;

Frobenius twists of functors si''\ with 2''\v\=n;

nontrivial tensor products.

This result is obtained by looking to A^ as a strict polynomial functor, [8, Section 5]. It can be deduced with some routine from various pubhcations, in particular [20], [3, Section 1.2, Theorems 1.32, 1.33], [11, Section 6] (with some work), [19, Chapter 3]. This result is to a large extent a n-anslation t of G. James' kemel theorem in the world of functors (see the references).

Proposition 9 follows by induction. As F and G are functors with no constant part, Ext3,(F ® G, Id) is trivial. Hence, if Ext^^CA", Id) is trivial, so is Ext;5:-(5;t, Id) for aU simple functors of degree n by an increasing induction on the degree and a descending one on the 2-regular partitions of n. The result follows because Ext^CA", Id) is trivial if (and only if) n is not a power of 2 [6].

3.3 Third Step of the Proofs of Proposition 8 and Theorem 6: Tensor Products and U\

From now on, we have proved the results for locally finite unstable modules and F ( l ) . To finish the simation of iY], we will prove that lfthe theorems hold for two modules, they hold for their tensor product. This follows from a standard double-complex spectral sequence argument. Then we use Lemma 1 to reduce to this situation (this uses a structural result firom [22]),

Proposition 10 If Proposition 8 and Theorem 6 hold for M and N, they hold for the tensor product M ® N.

Theorem 6 is true in this case; because of Corollary 2, it is true for the tensor product M ® A'.

For Proposition 8, let X' and S* be injective resolutions of M and N having the required property. The tensor product Ti' = X' ® J' is nol an injective resolution of M®N Construct a Cartan-Eilenberg resolution 'H*'' of Ti'. Applying to this double complex, the functor Homt^ (5, —) yields an hypercohomology spectral sequence converging to Ext^+^S, M ® A') with £2-term Ext^(S, W)

The unstable module 'W is a finite direct sum of modules J(k) ® J(t) ® W V . Thus, the group E x t ^ ( 5 , ' H ' ) is isomorphic to a finite dnect sum of groups Ex.\.^(Ty(S), Jik) ® J(t)), die F2-vector space V being of bounded dimension as there are only finitely many factors. For i = p + q, let d he an upper bound of the dimensions.

Suppose now diat S is of tbe form T."Sx(F(\)) These groups are tnvial as soon as n > sup(/f-1-0 or as soon as the connectivity of Fv(5j.(F(l))),f(rv(S;,(F(l)))) is greater than sup(t + 1 ) .

Lemma6 The connectivity c(SxiFi\))) of S,.(F{\)) is X\+2X\-t • +2'-^X\_^-l >

X'l - 1, X' being the conjugate (or dual) partition ofX.

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N.T. Cuong, L. Schwartz Using die properties of Ty = r'"'"'^', the decomposition 7" S FT e f and tbe fact diat SxiFif)) C H*(2/2)'-i, we show diat c(rv(S,,(F(l)))) > A', - d - I.

As a consequence, for a given V, only a finite number of reduced simple unstable modules rv5;i.(F(l)) are of connectivity less than a given constant.

The result follows.

This proves the tbeorems forU\. Indeed, they are true for each L E UO, F(\) and by tensorproducEforany L ® F ( 1 ) Next, all finitely generated objects M e Wj enter in a short exact sequence [22]; see also [14]-

[0] —* I ' —> M ^ L ® F ( l ) -^ L" —* (Oj

where L. L'. L" E UQ and are finitely generated (see [14] for a generalization). The resuh follows.

From now on, we assume that tbe tbeorems have been proved for objects in W„_|.

3 4 Fourth Step of tbe Proofs of Proposition 8 and Theorem 6; the Case of A" (F(I)) The following step is the case of exterior powers. We assume the theorems hold for A*(F(1)) and prove they hold for A*+' ( F ( l ) ) ,

We start an induction by proving it for A^(F(1)). Consider for example the case of the derived functors of the localization. There are two exact sequences relating S^ and A^:

{ 0 } ^ A 2 ( F ( 1 ) ) ^ A ' ( F ( 1 ) ) ® A ' ( F ( 1 ) ) ^ 5 ' ( F ( 1 ) ) ^ { 0 ) and

(Oj ^ 0 ( F ( 1 ) ) ^ s'^iFH)) -^ A ^ F ( l ) ) ^ (0}.

Note that with the notations below, S'^ \sD'W^\_\) also denoted 5(],i); in some references, it is not simple.

The modules 1'{^F{1)), as well as £'(F(1) ® F ( l ) ) , are finitely generated. It follows from die above exact sequences tiiat £ ' + ' ( A ^ ( F ( l ) ) ) S ^'(5^(F{I))) and ^'(A^(F(1))) = e'(S^(F(I))) modulo the Serre class of finitely generated unstable modules. Thus, it is enough to know tbe result for ^' (A^(F(1))) which is trivial, A^(F(1)) being A/iV-closed.

The same kind of argument works for the other result.

Let /: > 2 be an integer.

Ifk is even A*"^'(F(1)) is a direct summand in A*(F(1)) ® F ( I ) , and Proposition 5 implies that Proposition 8 and Theorem 6 hold for A*"'"'(F(1)).

If i is odd, the situation is more complicated. We have a short exact sequence ( 0 ) ^ W^k\)iF(\))^ A * ( F ( 1 ) ) ® F ( 1 ) ^ A * - ' - ' ( F ( l ) ) ^ ( 0 1 which defines (0| ^ W(k.\)(F(\)). There is anotiier short exact sequence:

{Oj ^ A*+'(F(1)) ^ lV(t.,j(F(l)) ^ % i^iFd)) - * (OJ which defines [0} -> 5(i.])(F(l)), By dualization ([3, 20]), we get

(Oi ^ A * + ' ( F ( 1 ) ) ^ A * ( F ( l ) ) ® F ( l ) ^ D H ' a . | | ( F ( l ) ) ^ (0), (0) -* S,k i )(F(l)) ^ DW^k,i>(Fil)) -> A*-'-'(F(l)) ^ jO}.

Recall (see the abovementioned references) that the simple functor Sik n is self-dual These exact sequences define W(k.i)iF(l)) and D W n , , ) ( F ( | ) ) , which defines S,k i ) ( F ( l ) ) .

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L- Finiteness Results in the Category U

Assume (for die integer it) diat f' {A*+'(F(]))). £' (5(i.ii(F(l))) are fmitely generated and show the result for ft-bl. The same is true for £'(H'(i,i)(F{l))) and ^'"(D»'(i,i)(F(l))).

Using the long exact sequences associated to the above exact sequences, we first show that ^'{lV(i+i])(/r(i)jj is finitely generated. Then we get die result for f'{Al*+^>(F(l))), next for f ( S ( i + i , ) ( f ( l ) ) ) , and finally for f' (DW(t+i,i)(F(l))). This last case is not used m the induction.

The case of the exterior powers follows also from explicit computations [5].

3.5 Last Step of the Proofs of Proposition 8 and Theorem 6

As the result holds for all exterior powers A*(F(1)), it holds for tensor products of such modules. Then, Lemma 5 implies it holds for all module 5/.(F(l))|A| = n and using Proposition lOforall ^"SxiF(l)).

To finish the proof, we use the following two lemmas and the induction hypothesis on die Krull filtration.

Lemma 7 If a reduced unstable module M belongs to tin, ^M belongs to Un-i

Tbe proof is left to the reader

Lemma 8 lfthe theorem holds for a reduced unstable module M of Krull filtration n, and for all unstable modules of Krull filtration n — I. then it holds for all submodule N ofM so that there exists h with <I?''M C N C M.

This is because M/<^^M e Un-\

The proof ends using Proposition 3 and the following.

Proposition 11 A finitely generated reduced unstable R of the Krull filtration n has a finite filtration whose quotients are suspensions of reduced modules of Krull filtration less than n, and whose associated functors are simple (of degree less than n).

This means that the subquotients are of the form E' R, for some R and some (, so that

•^''Sx (Z R (Z Sx for some A and some h. The required filtration is obtained as follows.

We consider the localization R -* CiR). Following Kuhn [11], we know that f(R) = fiHR)) has a finite composition senes. The proof is done by induction on die length of the composition series of f(R). Consider an epimorphism on a simple functor' f(R) -^ Sx, and tbe associated unstable module map R -> Sji.(F(l)), Its kemel K is reduced and can apply the induction hypothesis.

To finish die proof of the theorem, we apply the preceding lemma.

Proof of Theorem 7 and of the corollary

The proof of Theorem 7 follows directly from Proposition 8 and from the properties of the right adjoint 4 of <1>, and in particular from the computation ^(L®J(n)) = L®J(n/2), (L a reduced injective, andJ(n/2) is trivial ifn is not an integer). If A^' is a fimte direct sum of indecomposable injective unstable modules, it is clear that for it large enough, i*(A/'') is trivial. The use of * can be replaced by

Proposition 12 Let M be a finitely generated unstable modules, assume N is nilpotent and ha- 'mile nilpotent filtration. Then ifk is large enough. Homui^''(M). N) ^ (0}

fl Springer

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N.T Cuong. L. Schwartz Note that in particular, a finitely generated module has a finite mlpotent filtration [12, 21]. Suppose N,{0] = [0] for f > fo- The unstable module n'l'*(A/) is isomorphic to

£ - ' ^ ' * * £ 2 M a n d h a s n i v i a l i m a g e i n A'if 2* - I >(o-

For die corollary, we observe that Theorem 7 allows to reduce to a computation in the category T. It is enough to show the following:

Proposition 13 Let F be a polynomial functor of degree n, such that F((Oj) = [0), R the smallest subfUnclor such that F/R is of degree n- 1 Then E\l]^(R ® R, F) = (0).

This last result is proved using the category bi - T, [5]. This result is used in [2].

References

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2. Cuong, N T., Gandens, G , Schwartz, L.: Around conjectures of N. Kuhn arXiv:1402.26l7 (2014) 3. DJament, A • Foncteurs de division el strucmre de / ® - ® A" dans la categone T Ann. I'lnst. Founer

57, 1771-1823 (2007)

4. Franjou, V , Fnedlander, E M , Pira.shvili, T , Schwartz, L,: Rational Representations, the Steenrod Alge- bra and Functor Homology Panoramas et Synthases Vol. 16. SocidtS Mathematique de France, Paris (2003)

5. Franjou, V , Fnedlander, E., Scorichenko. A., Suslin, A.: General linear and fimctor cohomology over finite fields Ann Math, ISO, 663-728 (1999)

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Chicago Lectures in Madiematics University of Chicago Press, Chicago (1994)

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