On universal stably free modules in positive characteristic

Abstract

We study the mod p motivic cohomology of homogeneous varieties such as GLn/GLr or Sp2n/Sp2n−2 along with the action of the Steenrod operations, without restrictions on the characteristic of the base field. In particular, we prove that certain quotient maps do not admit sections. Introduction Fix a field k and a k-algebra A. An A-module P is said to be stably free if P ⊕A ∼= A for some integers 0 ≤ q ≤ n. There is a universal stably free An,q-module Pn,q over a k-algebra An,q where Spec(An,q) ∼= GLn/GLq [7, §1]. The module Pn,q is free if and only if the quotient map GLn → GLn/GLq (1) admits a section. For r ≤ q, the module Pn,q has a free factor of rank r if and only if the map GLn/GLq−r → GLn/GLq (2) admits a section. Using Steenrod operations on étale cohomology with finite coefficients, Raynaud showed in [7, Théorème 6.1 and Théorème 6.6]: 1. The fibration GLn → GLn/GLn−1 does not admit a section except possibly when char(k) = 2, n = 3 or char(k) = 3, n = 4. 2. The fibrations Sp2n → Sp2n/Sp2n−2, SO2n+1 → SO2n+1/SO2n−1 do not admit sections except possibly when char(k) = 3, n = 2 or char(k) = 5, n = 3. 3. The fibration GLn/GLn−q → GLn/GLn−1 does not admit a section, except possibly if n is divisible by Nq(char(k)) = ∏ prime p 6=char(k) p where n(p, q) is the largest integer h ≥ −1 such that ph(p− 1) ≤ q − 1. 1 Raynaud posed the question of whether the restrictions on char(k) could be removed in these results [7, page 21]. The goal of this paper is to give a positive answer to Raynaud’s question by studying the action of Steenrod operations on motivic cohomology with finite coefficients. After we compute the relevant motivic cohomology groups and determine the action of the Steenrod operations, our arguments are similar to what can be done in topology [1]. The main new tools we have to play with are the Steenrod operations from [4], which act on the mod p motivic cohomology of smooth schemes over k with char(k) = p > 0. Previously, Williams showed how to use motivic cohomology and Steenrod operations to obtain the results from [7], assuming that the characteristic of the base field and the relevant coefficient field are different. We also note that Mohan Kumar and Nori previously showed that the fibration 1 doesn’t admit a section for n ≥ 3, without any restrictions on char(k) (they proved a more general statement on when projective modules defined by certain unimodular rows are free) [8, Theorem 17.1]. Our other results seem to be new. Acknowledgments I thank Ben Williams for suggesting to me the problem considered in this paper. 1 Setup Fix a field k and let Sm(k) denote the category of separated smooth schemes of finite type over k. Let H(k),H•(k) denote the unpointed and pointed motivic homotopy categories respectively of spaces over k [3]. Let SH(k) denote the stable motivic homotopy category of spectra over k [11]. There is an adjunction Σ+ : H(k) ⇄ SH(k) : Ω ∞ where Σ+ denotes the infinite P 1-suspension functor and Ω∞ is the infinite delooping functor. For X ∈ Sm(k) and a coefficient ring A, we let H i(X,A(j)) denote the motivic cohomology group of X of degree i and weight j. There is an isomorphism between motivic cohomology and higher Chow groups: H (X,Z(j)) ∼= CH(X, 2j − i). In particular, H i(X,Z(j)) = 0 for i > 2j. For a coefficient ring A, we let HA ∈ SH(k) denote the motivic Eilenberg-MacLane spectrum. There are Eilenberg-MacLane spaces K(i, j, A) = Ω∞Σi,jHA ∈ H•(k) for all i, j ≥ 0. 2 Higher Chern classes and Steenrod operations For a flat affine group scheme G/k, we consider the geometric classifying space BG ∈ H•(k). Here, BG is defined in the sense of Totaro [9] and Morel-Voevodsky [3] by approximating BG by certain smooth quotients U/G. For X ∈ Sm(k) such that G acts on X, H i G(X,Z(j)) is similarly defined to be the motivic cohomology H i((X × U)/G,Z(j)) for a suitable U. Let S ∈ H•(k) denote the simplicial sphere and let T ∈ H•(k) denote the Tate sphere. For the infinite general linear group GL and X ∈ Sm(k), there is an isomorphism HomH•(k)(Σ m T Σ n SX+, BGL× Z) ∼= Kn−m(X) for n,m ≥ 0 [3, Theorem 3.13]. From [6], the motivic cohomology of BGL is given by H(BGL,Z(∗)) ∼= H(k,Z(∗))[c1, c2, . . .] where ci has bidegree (2i, i) for all i ∈ N.