A note on polynomial functors and ext 1 Groups

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<ul><li><p>This article was downloaded by: [University of Chicago Library]On: 12 November 2014, At: 06:06Publisher: Taylor &amp; FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: MortimerHouse, 37-41 Mortimer Street, London W1T 3JH, UK</p><p>Communications in AlgebraPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lagb20</p><p>A note on polynomial functors and ext1 GroupsGeoffrey M. L. Powell aa L'institut galilee, universite de parisnord , Villetaneuse, FrancePublished online: 27 Jun 2007.</p><p>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</p><p>To link to this article: http://dx.doi.org/10.1080/00927879708825903</p><p>PLEASE SCROLL DOWN FOR ARTICLE</p><p>Taylor &amp; Francis makes every effort to ensure the accuracy of all the information (the Content)contained in the publications on our platform. However, Taylor &amp; Francis, our agents, and our licensorsmake no representations or warranties whatsoever as to the accuracy, completeness, or suitabilityfor any purpose of the Content. Any opinions and views expressed in this publication are the opinionsand views of the authors, and are not the views of or endorsed by Taylor &amp; Francis. The accuracy ofthe Content should not be relied upon and should be independently verified with primary sourcesof information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings,demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arisingdirectly or indirectly in connection with, in relation to or arising out of the use of the Content.</p><p>This article may be used for research, teaching, and private study purposes. Any substantial orsystematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distributionin any form to anyone is expressly forbidden. Terms &amp; Conditions of access and use can be found athttp://www.tandfonline.com/page/terms-and-conditions</p><p>http://www.tandfonline.com/loi/lagb20http://www.tandfonline.com/action/showCitFormats?doi=10.1080/00927879708825903http://dx.doi.org/10.1080/00927879708825903http://www.tandfonline.com/page/terms-and-conditions</p></li><li><p>COMMUNICATIONS IN ALGEBRA, 25(3), 979-987 (1997) </p><p>A Note on Polynomial Functors and ~ x t ' Groups </p><p>Geoffrey M.L. Powell </p><p>L'INSTITUT GALILEE, UNIVERSITE DE PARIS-NORD, VILLETANEUSE, FRANCE </p><p>1 Introduction </p><p>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. </p><p>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]. </p><p>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 ) . </p><p>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: </p><p>for all natural numbers n (see [Kl] ) . </p><p>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 &gt; n. If d,(F) # 0, then F is said to have polynomial degree exactly n. 0 </p><p>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 . </p><p>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 &lt; d is denoted by Fd. </p><p>Copyr~gh t C 1997 by Marcel Dekker, Inc </p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Uni</p><p>vers</p><p>ity o</p><p>f C</p><p>hica</p><p>go L</p><p>ibra</p><p>ry] </p><p>at 0</p><p>6:06</p><p> 12 </p><p>Nov</p><p>embe</p><p>r 20</p><p>14 </p></li><li><p>980 POWELL </p><p>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. </p><p>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 </p><p>This note studies polynomial functors; the main results are the following: </p><p>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. </p><p>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 </p><p>The next result is an immediate corollary of the preceding theorem: </p><p>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 ) &gt; d + 1; then Ext;(F, G ) = 0 = E&amp;G, F ) . The methods used to prove these results also permit the proof of the following </p><p>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. </p><p>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 </p><p>2 Polynomial functors </p><p>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]. </p><p>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,. </p><p>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 </p><p>: 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: </p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Uni</p><p>vers</p><p>ity o</p><p>f C</p><p>hica</p><p>go L</p><p>ibra</p><p>ry] </p><p>at 0</p><p>6:06</p><p> 12 </p><p>Nov</p><p>embe</p><p>r 20</p><p>14 </p></li><li><p>POLYNOMIAL FUNCTORS 98 1 </p><p>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. </p><p>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 </p><p>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 &gt; d. The methods require the choice of a standard skeleton for Ef, together with a choice of basis for each functor. </p><p>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 &amp;,-action and a left Ck-action. </p><p>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 </p><p>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 ) . </p><p>Definit ion 2.3 Suppose that F is a functor in F; define the following subspaces of F ( F k ) </p><p>1. C k ( F ) := { r E F(IF~)IF(I~ , ) (Z) = 0 for q,, 1 5 i 5 k) </p><p>The following lemma presents no difficulty: </p><p>L e m m a 2.4 </p><p>1. x E Ck(F) zf and only zf x represents a non-tnvzal element zn Ck-l(AF), where AF(pk- ' ) zs the cokernel of the standard embedding F (pk - ' ) -+ F ( F ~ ) . </p><p>2. Ck zs a complement of Vk, so that F(IF~) 2 Vk $ Ck </p><p>The first part of the lemma shows that d imCk(F) = dimCk-'(AF); it follows that d imCk(F) = dim Akp(0 ) ; hence, the work of [Kl ] shows that: </p><p>P ropos i t i on 2.5 Suppose that F E FU 2s an analytzc functor, whzch takes valves of finite dimenszon, then dk = dim Ck. </p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Uni</p><p>vers</p><p>ity o</p><p>f C</p><p>hica</p><p>go L</p><p>ibra</p><p>ry] </p><p>at 0</p><p>6:06</p><p> 12 </p><p>Nov</p><p>embe</p><p>r 20</p><p>14 </p></li><li><p>982 POWELL </p><p>R e m a r k 2.6 Definition 2.3 provides another characterization of functors of polynomial degree 5 d. 0 </p><p>This allows for the choice of a basis for F ( F n ) when n &gt; d , the polynomial degree of F ; the result may be proved by an induction using the functor A and the first part of Lemma 2.4: </p><p>Proposi t ion 2.7 Suppose that F E Fd zs a polynomzal functor of degree d and that n 2 d; then the map </p><p>d </p><p>$ @ (a), - F ( F " ) &gt; k = O a S ( k , n ) </p><p>defined as the sum of the maps (Ck) , 2 Ck - F ( I F ~ ) F2) F(IFn), is a n isomorphism. (Here, by conventton, S(0, n) zs taken to be the szngleton set and Co --t F ( F n ) is the incluszon of the constant part of the functor F ) . </p><p>This result has the following important, but straightforward, corollary </p><p>Corollary 2.8 A functor F whzch takes finzte-dzmenszonal values zs polynomzal of degree 5 d z f and only z f , V n &gt; d , the maps F ( a ) . for a E S ( d , n ) , znduce a suqectzon </p><p>Corollary 2.8 may be refined as follows, by a method analogous to simplicia1 realization. The construction below may be thought of a s a suitable colimit in the category of vector spaces. </p><p>Definition 2.9 Suppose that F E F is a functor; define a vector space Gd(k) as follows. For k 2 d , Gd(k) := F ( I F ~ ) and, for k &gt; d </p><p>where 5 is the quotient relation given as follows. If </p><p>is a commuting diagram, with y, 6 E S(a, d) and a,,G E S ( d , k ) , then, for all z E F ( F a ) , set ( F ( ~ ) x ) , = ( F ( ~ ) X ) ~ . </p><p>By inspection, the map @ f a : $ n E S ( d , k , F ( I F ~ ) , -+ F ( F ~ ) passes to a map f : &amp;(k) -+ F ( P k ) . In fact, the following holds: </p><p>Proposi t ion 2.10 If F E Fd zs a functor of degree d , then the n a p f : Gd(k) --, F ( F ~ ) is a n isomorphzsm of vector spaces. </p><p>Proof: That f is surject~ve is clear. since the conditions imposed by the equivalence relation are necessary. </p><p>Dow</p><p>nloa</p><p>ded </p><p>by [</p><p>Uni</p><p>vers</p><p>ity o</p><p>f C</p><p>hica</p><p>go L</p><p>ibra</p><p>ry] </p><p>at 0</p><p>6:06</p><p> 12 </p><p>Nov</p><p>embe</p><p>r 20</p><p>14 </p></li><li><p>POLYNOMIAL FUNCTORS 983 </p><p>Proposition 2.7 implies that F ( F d ) has a decomposition as @$o @aES(n,d) (Cn)a, </p><p>whereas F(F? hhas a decomposition: @$o @bES(n,k) (Cn)S, for k &gt; d. Consider a map F ( I F ~ ) F J ) F ( P " ) . for y E S ( d , k ) . Under this map, (C,), has image </p><p>(C,),,, with ?a E S ( n , k ) . Thus, the map @ E s ( d , k ) ( ~ ( ~ d ) ) , -' F ( F k ) acts sending </p><p>{(Cn),), (Cn),a. In particular, to prove that the map on the quotient is injective, it suffices to show </p><p>that the map to each component of the decomposition of F(IFk) is injective. This follows since {(C,),), and {(C,,),I),I map to the same component if and only if y a = 7'a1, if and only if the quotient relation identifies the two summands in the decomposition. </p><p>Remark 2 . 1 1 The result of Corollary 2 8 may be compared with [FLS, Definition 1.21: A functor F E 3 zs generated zn finzte dzmenszon zf there exzsts a finzte dzmenszonal vector space E such that the canonzcal morphzsm F ( E ) @ PE -+ F (evaluation) is surjective, where PE denotes the standard projectwe in 3, P E ( V ) = IF[Homt(E, V ) ] . I t is easy to see that, if F is analytic, then F is generated in finite dimension if and only if F is polynomial. Thus the space F ( E ) should be compared with the use of F(IFd) above in Corollary 2.8. 0 </p><p>Definition 2.12 Suppose that F E F is a functor and that x E F(IFk)), </p><p>1. x is strictly generated in degree d if there exists y E F ( F d ) and a E S ( d , k ) such that x = F ( a ) ( y ) . </p><p>2. x is generated in degree d if x = C x,, where each x , is strictly generated in degree d. 0 </p><p>Remark 2.13 The argument of Proposition 2.10 shows that , for any functor F , Gd(k) is a sub-vector space of F(IFk), namely the subspace of elements generated in degree d; it is natural to ask when this may be given the structure of an object of F. 0 </p><p>The following lemma depends only on the coset decomposition of I ( d , k ) with respect to the Ed action. </p><p>Lemma 2.14 Suppose that F E 3 and that x zs strictly generated i n degree d , say x = F ( a ) y , as above. If u E Ck represents a permutation matrix zn Mk(IF), then F ( o ) x = F ( O ) { F ( t ) y } , where P E S(d. k ) , t E Cd are uniquely determined by ucy = Pt i n T ( d , k ) . </p><p>I n partzcular, the element F ( u ) x zs stnctly generated i n dimension d . </p><p>Recall that a transuectzon of IFk, with respect to a given basis, is a linear transfor- mation in G L k ( F ) defined by a pair ( 2 , j) with i # j , so that x, +-+ x , for a # i and x , H x, + x j . The matrix semi-group n4k(p) is generated by the diagonal matrices d iagk (P) , the permutation matrices and the transvections. </p><p>Suppose that F E 3 and that z E F ( I F ~ ) is strictly generated in degree d , say x = F ( a ) y . with a E S ( d , k ) and y E F ( F ~ ) . Co...</p></li></ul>