Steenrod Operations Research Articles

Higher Chern Classes and Steenrod Operations in Motivic Cohomology

Higher Chern Classes and Steenrod Operations in Motivic Cohomology

The cohomology of symmetric groups and the Quillen map at odd primes

Gunawardena, Lannes, and Zarati proved that the Quillen homomorphism q G : H ∗BG→ lim ← C(G) H ∗BE is an isomorphism for G= Σ n at p=2, but fails to be an isomorphism for odd primes. We prove that at odd primes, the restriction of the Quillen map to the subring of elements that are annihilated by all Steenrod operations that involve the Bockstein is an isomorphism for all n.

Reduced power operations in motivic cohomology

In this paper we construct an analog of Steenrod operations in motivic cohomology and prove their basic properties including the Cartan formula, the Adem relations and the realtions to characteristic classes.

The cohomology of certain Hopf algebras associated with 𝑝-groups

We study the cohomology H ∗ ( A ) = Ext A ∗ ⁡ ( k , k ) H^*(A)=\operatorname {Ext}_A^*(k,k) of a locally finite, connected, cocommutative Hopf algebra A A over k = F p k=\mathbb {F}_p . Specifically, we are interested in those algebras A A for which H ∗ ( A ) H^*(A) is generated as an algebra by H 1 ( A ) H^1(A) and H 2 ( A ) H^2(A) . We shall call such algebras semi-Koszul. Given a central extension of Hopf algebras F → A → B F\rightarrow A\rightarrow B with F F monogenic and B B semi-Koszul, we use the Cartan-Eilenberg spectral sequence and algebraic Steenrod operations to determine conditions for A A to be semi-Koszul. Special attention is given to the case in which A A is the restricted universal enveloping algebra of the Lie algebra obtained from the mod- p p lower central series of a p p -group. We show that the algebras arising in this way from extensions by Z / ( p ) \mathbb {Z}/(p) of an abelian p p -group are semi-Koszul. Explicit calculations are carried out for algebras arising from rank 2 p p -groups, and it is shown that these are all semi-Koszul for p ≥ 5 p\geq 5 .

Steenrod operations in Chow theory

An action of the Steenrod algebra is constructed on the mod p p Chow theory of varieties over a field of characteristic different from p p , answering a question posed in Fulton’s Intersection Theory. The action agrees with the action of the Steenrod algebra used by Voevodsky in his proof of the Milnor conjecture. However, the construction uses only basic functorial properties of equivariant intersection theory.

Steenrod operations and degree formulas

Steenrod operations and degree formulas

Module derivations and cohomological splitting of adjoint bundles

Let G be a nite loop space such that the mod p cohomology of the clas- sifying space BG is a polynomial algebra. We consider when the adjoint bundle associated with a G-bundle over M splits on mod p cohomology as an algebra. In the case p = 2, an obstruction for the adjoint bundle to admit such a splitting is found in the Hochschild homology concerning the mod 2 cohomologies of BG and M via a module derivation. Moreover the derivation tells us that the splitting is not compatible with the Steenrod operations in general. As a consequence, we can show that the isomorphism class of an SU(n)-adjoint bundle over a 4-dimensional CW complex coincides with the homotopy equivalence class of the bundle.

On the Steenrod operations in cyclic cohomology

For a commutative Hopf algebra A over ℤ/p, where p is a prime integer, we define the Steenrod operations pi in cyclic cohomology of A using a tensor product of a free resolution of the symmetric group Sn and the standard resolution of the algebra A over the cyclic category according to Loday (1992). We also compute some of these operations.

Homology operations and vn-Bockstein operations

In this note, we determine the action of the first differential in the Atiyah–Hirzebruch spectral sequence for the nth Morava K-theory of the p-adic construction D p (ℓ)(X)= C ℓ(p)∧ Σ p X ∧p of a pointed space X for n⩾1. Our formula gives us a way to compute the first differentials in the Atiyah–Hirzebruch spectral sequence for the Morava K-theory of iterated loop spaces. Analogous formulas are also proved for the Steenrod operations.

Linking first occurrence polynomials over 𝔽pby Steenrod operations

This paper provides analogues of the results of [G.Walker and R.M.W. Wood, Linking first occurrence polynomials over F_2 by Steenrod operations, J. Algebra 246 (2001), 739--760] for odd primes p. It is proved that for certain irreducible representations L(lambda) of the full matrix semigroup M_n(F_p), the first occurrence of L(lambda) as a composition factor in the polynomial algebra P=F_p[x_1,...,x_n] is linked by a Steenrod operation to the first occurrence of L(lambda) as a submodule in P. This operation is given explicitly as the image of an admissible monomial in the Steenrod algebra A_p under the canonical anti-automorphism chi . The first occurrences of both kinds are also linked to higher degree occurrences of L(lambda) by elements of the Milnor basis of A_p.

Self-tilting complexes yield unstable modules

Let G be a group and let R be a commutative ring. Let T rP icR(RG) be the group of isomorphism classes of standard self-equivalences of the derived category of bounded complexes of RG-modules. The subgroup HDR(G) of T rP icR(RG) consisting of self-equivalences xing the trivial RG-module acts on the cohomology ring H (G; R). The action is functorial with respect to R. The self-equivalences which are 'splendid' in a sense dened by J. Rickard act natural with respect to transfer and restriction to centralizers of p-subgroups in case R is a eld of characteristic p. In the present paper we prove that this action of self-equivalences on H (G; R) commutes with the action of the Steenrod algebra and study the behaviour of the action of splendid self-equivalences with respect to Lannes' T -functor. Introduction In an earlier joint paper (9) with RaphaRouquier I dened the group T rP icR(A) of standard self-equivalences of a derived module category D b (A) for an R-algebra A which is projective as an R-module. For any A-module M let HDM(A) be the subgroup of T rP icR(A) which is formed by the self-equivalences mapping M to an isomorphic copy. Then, in an earlier paper (11) I showed that, under some hypothesis on M, the group HDM(A) acts in a natural way on the Ext-algebra Ext A (M; M). In case of A being a group algebra RG, with R being a eld of characteristic p and G being a nite group, then J. Rickard denes in (8) what is a splendid equivalence by some technical conditions basically by asking that the homogeneous components of a tilting complex are p-permutation modules induced from diagonal p-subgroups, and by some invertibility condition in the homotopy category. These splendid equivalences induce self-equivalences of the derived categories of centralizers of p-subgroups by the Brauer construction. In (12) I showed that then, for M = R being the trivial module, the action of those splendid equivalences commute with transfer and restriction from and to the cohomology rings of centralizers of p-subgroups. In the present paper we enlarge these properties still further. The action of self-equivalences of the bounded derived category D b (RG) on H (G; R) commutes with the action of the Steenrod algebra on H (G; R) for any prime p. As consequence of the above statements, any cohomology ring H (G; Fp) denes a functor H (G; Fp) p HD p (B0( p G)) from the modules over the group of derived self-equivalences of the principal block of the group ring Fp HD p (FpG) xing the trivial module to the category of unstable modules Up and similarly in the opposite direction. Obviously, we may restrict to splendid self-equivalences. We shall describe the composition of Lannes' T -functor with the rst functor and the im- age of free unstable modules by the second. Moreover, by a result of Lannes TV (H (G; Fp)) decomposes into direct product of cohomology rings as unstable modules. We shall prove that this decomposition is also a decomposition as modules over the action of splendid self- equivalences. This will give evidence that splendid equivalences are the correct objects to study in this context. The paper is organized as follows. Section 1 recalls the necessary denitions and properties of Even's norm map as it is used here and the denition of the Steenrod operation. In Section 2 it is shown that the normalized part of the outer automorphism group of the

Algebraic braided model of the affine line and difference calculus on a topological space

In a previous Note [1], we suggested a quantum model of the unit interval [0,1], using convergent power series, parametrized by a variable q (a remarkable example is the quantum exponential, defined by Euler). In the present Note, we suggest a simpler model based on functions f=f(x): Z →k (with an arbitrary commutative ring k) which are constant when x↦+∞ or x↦−∞ and their “differentials” considered as functions x↦ f( x+1)− f( x) (difference calculus). Thanks to this new “ differential calculus over the integers”, we can associate to any simplicial set or topological space X a braided differential graded algebra D ∗(X) which is similar in spirit to the algebra W ∗(X) introduced in [1]. We notice that the p-homotopy type of X can be read from the braiding of D ∗(X) . In particular, if k= Z , we recover in a purely algebraic way the integral cohomology, Steenrod operations, homotopy groups from this braiding. To cite this article: M. Karoubi, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 121–126.

Rost projectors and Steenrod operations

Let X be an anisotropic projective quadric possessing a Rost projector \rho . We compute the 0-dimensional component of the total Steenrod operation on the modulo 2 Chow group of the Rost motive given by the projector \rho . The computation allows to determine the whole Chow group of the Rost motive and the Chow group of every excellent quadric (the results announced by Rost). On the other hand, the computation is being applied to give a simpler proof of Vishik's theorem stating that the integer \dim X+1 is a power of 2.

Addendum to: “Introduction to $A$-infinity algebras and modules” [HHA, v. 3 (2001) No. 1, pp. 1-35

1.2 History. An A-inÞnity structure may be described as a system of higher homo- topies together with suitable coherence conditions. A basic observation, which has implicitly been exploited for long, is that A-inÞnity structures behave much bet- ter with respect to homotopy than strict (for example dierential graded algebra) structures. Higher homotopies occurred in mathematics before A-inÞnity structures had been explicitly recognized, though. The system of (i-products introduced by Steenrod (19) is an early example of higher homotopies. The (i-products measure the failure from commutativity of the Alexander-Whitney map in a coherent fashion and prompted the development of s(trongly)h(omotopy)c(ommutative) structures as well as that of Steenrod operations. Massey products (16) may be seen as invari- ants of certain A-inÞnity structures. Homological theory (HPT) has nowadays become a standard tool to construct and handle A-inÞnity structures. The basic HPT-notion, that of contraction, was introduced in Section 12 of (3). In that paper, Eilenberg and Mac Lane showed that the comparison between the reduced bar and W-constructions is a reduction, and they conjectured that it is a contraction. Using the lemma, in his Heidelberg diploma thesis (Diplo- marbeit) supervised by J. Huebschmann, Wong has indeed veriÞed this conjecture (20). A geometric comparison between the bar and W-constructions has recently been obtained by Berger and Huebschmann in (1). The notion of recursive struc- ture of triangular complexesin Section 5 (4) is also an example of what was later identiÞed as a perturbation. The perturbation lemma is lurking behind the formu- las in Chapter II of Section 1 of (18) and seems to have Þrst been made explicit by M. Barrat (unpublished). The Þrst instance known to us where it appeared in print is (2). A homological algebra and higher homotopies tradition was as well created by Berikashvili and his students in Georgia (at the time part of the USSR). More precise comments about the historical development until the mid eighties may be

Linking First Occurrence Polynomials over F2 by Steenrod Operations

It is proved that for every irreducible representation L (λ) of the full matrix semigroup M n ( F 2 ), the first occurrence of L (λ) as a composition factor in the polynomial algebra P = F 2 [ x 1 , …, x n ] is linked by a Steenrod operation to the first occurrence of L (λ) as a submodule in P . This Steenrod operation is given explicitly as the image of an admissible monomial in the Steenrod squares Sq r under the canonical anti-automorphism χ of the mod 2 Steenrod algebra A . The first occurrences of both kinds are also linked to higher degree occurrences of L (λ) by elements of the Milnor basis of A .

HIT POLYNOMIALS AND EXCESS IN THE MOD P STEENROD ALGEBRA

AbstractLet $p$ be an odd prime. The primary purpose of this paper is to determine the excess of the conjugates of the Steenrod operations $\mrm{P}[k;f]$, which are defined as $\mrm{P}[k;f]:=\mrm{P}(p^{k-1}f)\cdot\mrm{P}(p^{k-2}f)\cdot\cdots\cdot\mrm{P}(pf)\cdot\mrm{P}(f)$. The result is then used to obtain sufficient conditions for an element in the polynomial algebra $\mathbb{F}_p[x_1,\dots,x_s]$ to be in the image under the standard action of the Steenrod algebra. Results and methods are generalizations of previous work by Judith Silverman and by myself with Judith Silverman.AMS 2000 Mathematics subject classification: Primary 55S10. Secondary 55S05

A-generators for ideals in the Dickson algebra

The Dickson Algebra on q-variables is the algebra of invariants of the action of the mod-2 general linear group on a polynomial algebra in q-variables. We study the structure of certain ideals in this algebra as a module over the Steenrod Algebra A , and develop methods to determine which elements are hit by Steenrod operations. This allows us to display a very small set of A -generators for these ideals and show that the set is minimal in some cases.

Steenrod Operation in Certain Cobordism Theories

We generalize results of Smirnow [Math. Zam. 65(2) (1999), 270–279] to the cases of MSO and MSU theories. Also we establish relation between the Steenrod operations in MG-cobordism theory (G=O, U, Sp, SO, SU) in our approach and the Steenrod–tom Dieck operations.

Stable homotopy over the Steenrod algebra

Preliminaries Stable homotopy over a Hopf algebra Basic properties of the Steenrod algebra Chromatic structure Computing Ext with elements inverted Quillen stratification and nilpotence Periodicity and other applications of the nilpotence theorems Appendix A. An underlying model category Appendix B. Steenrod operations and nilpotence in $\mathrm{Ext}_\Gamma^{**}(k,k)$ Bibliography Index.

Inverse invariant theory and Steenrod operations

Introduction The $\Delta$-theorem Some field theory over the Steenrod Algebra The integral closure theorem and the unstable part The inseparable closure The embedding theorem I Noetherianess, the embedding theorem II and Turkish delights The Galois embedding theorem, the little imbedding theorem and a bit more The big imbedding theorem, Thom classes, Turkish delights II and reverse Landweber-Stong conjecture Technical stuff References.