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International audience

A general notion of detection is introduced and used in the study of the cohomology of elementary abelian 2–groups with respect to the spectra in the Postnikov tower of orthogonal K–theory. This recovers and extends results of Bruner and Greenlees and is related to calculations of the (co)homology of the spaces of the associated Ω–spectra by Stong and by Cowen Morton.

Double Poisson structures (a la Van den Bergh) on commutative algebras are studied; the main result shows that there are no non-trivial such structures on polynomial algebras of Krull dimension greater than one. For a general commutative algebra A, this places significant restrictions on possible double Poisson structures. Exotic double Poisson structures are exhibited by the case of the polynomial algebra on a single generator, previously considered by Van den Bergh.Comment: 12 pages; very minor revisio

The connective ku-(co)homology of elementary abelian 2-groups is determined as a functor of the elementary abelian 2-group. The argument requires only the calculation of the rank one case and the Atiyah-Segal theorem for KU -cohomology together with an analysis of the functorial structure of the integral group ring. The methods can also be applied to the odd primary case.These results are used to analyse the local cohomology spectral sequence calculating ku-homology, via a functorial version of local duality for Koszul complexes. This gives a conceptual explanation of results of Bruner and Greenlees.2000 Mathematics Subject Classification. 19L41; 20J06.The categories F A , F are tensor abelian, with structure induced from A b.(For basic properties of F , see [Kuh94a, Kuh94b, Kuh95] or [FFSS99].) There is an exact Pontrjagin duality functor which generalizes the duality for F introduced in [Kuh94a]:Recall that the socle of an object is its largest semi-simple subobject and the head its largest semi-simple quotient.Example 2.3. The symmetric powers, divided powers and exterior powers are fundamental examples of (polynomial) functors in F . For n ∈ N, the nth symmetric power functor S n is defined by S n (V ) := (V ⊗n )/S n , the nth divided power functor by Γ n (V ) := (V ⊗n ) Sn and the nth exterior power functor identifies as Λ n (V ) ∼ = (V ⊗n ⊗ sign) Sn , where sign is the sign representation of S n . By convention, these functors are zero for negative integers n. There is a duality relation S n ∼ = DΓ n ,

International audience

We discuss two extensions of results conjectured by Nick Kuhn about the non-realization of unstable algebras as the mod-p singular cohomology of a space, for p a prime. The first extends and refines earlier work of the second and fourth authors, using Lannes’ mapping space theorem. The second (for the prime 2) is based on an analysis of the -1 and -2 columns of the Eilenberg–Moore spectral sequence, and of the associated extension.

In both cases, the statements and proofs use the relationship between the categories of unstable modules and functors between Fp-vector spaces. The second result in particular exhibits the power of the functorial approach.

These notes explain how to construct small functorial chain complexes which calculate the derived functors of destabilization (respectively iterated loop functors) in the theory of modules over the mod 2 Steenrod algebra; this shows how to unify results of Singer and of Lannes and Zarati.2000 Mathematics Subject Classification. Primary 55S10; Secondary 18E10. Key words and phrases. Steenrod algebra -unstable module -destabilization -iterated loop functor -derived functor -total Steenrod power.A =Ã / Sq 0 . This is again a homogeneous quadratic algebra. Moreover, it has the important property that it is Koszul. This notion, introduced by Priddy [Pri70], is at the origin of the existence of small resolutions for calculating the homology of the Steenrod algebra; the Koszul dual is the (big) Lambda algebra.The construction of the complexes introduced here is related to the quadratic Koszul nature of A and also to the relationship between the Steenrod algebra and invariant theory; many of the ideas go back to the work of Singer [Sin78, Sin80, Sin83] etc. Remark 2.1.1. The odd primary analogues depend upon the work of Mùi [Mùi86, Mùi75], which describes the (more complicated) relationship between invariant theory and the Steenrod algebra. See for example the work of Nguyễn H. V. Hưng and Nguyễn Sum [HS95] generalizing Singer's invariant-theoretic description of the Lambda algebra to odd primes, Zarati's generalization [Zar84] of his work with Lannes [LZ87] and the author's paper [Pow14].

An explicit chain complex is constructed to calculate the derived functors of destabilization at an odd prime, generalizing constructions of Zarati and of Hung and Sum. The methods are based on the ideas of Singer and Miller and also apply at the prime two. A structural result on the derived functors of destabilization is deduced. IntroductionThe destabilization functor D from the category M of modules over the mod p Steenrod algebra A to the category U of unstable modules is the left adjoint to the inclusion U ֒→ M ; it is right exact and has non-trivial left derived functors D s : M → U . These derived functors are of considerable interest in homotopy theory: for example Lannes and Zarati [LZ87, Zar84] used them to prove a weak version of the Segal conjecture; they have recently been used by Hai, Schwartz and Nam [HSN10] (p = 2) and Hai [Hai12] (for p odd) to prove a generalization of the weak Segal conjecture. This project was motivated in part by the need to gain a better understanding of the action of Lannes' T -functor on the derived functors of destabilization at odd primes.At the prime two, Singer constructed functorial chain complexes with homology calculating the derived functors of iterated loop functors [Sin80a]; these can be used to construct chain complexes for calculating the functors D s (cf. Goerss [Goe86], who works with homology). Lannes and Zarati [LZ87] gave an independent approach at the prime two, calculating the derived functors D s (Σ −t M ) where M is an unstable module and t ≤ s.Both approaches depend upon the relationship between the Steenrod algebra and the Dickson invariants H * (BV s ) GLs (the algebra of invariants of the cohomology H * (BV s ) of a rank s elementary abelian 2-group V s under the action of the general linear group GL s ) for varying s. For odd primes, this relationship is more subtle and is explained by the results of Mùi [Mùi75] in terms of an explicit subalgebra of the invariants H * (BV s ) SLs , where SL s is an index two subgroup of GL s containing the special linear group.The purpose of this paper is to give a construction of a functorial chain complex to calculate the derived functors of destabilization for odd primes. This unifies and builds upon results in the literature: the derived functors of iterated loop functors at odd primes were studied by Li in his thesis [Li80] and, in joint work with Singer [LS82], he gave a chain complex for calculating the homology of the Steenrod algebra; Hung and Sum [HS95] modified and generalized to odd primes the approach of Singer [Sin83], leading to an invariant-theoretic description of the 2000 Mathematics Subject Classification. Primary 55S10; Secondary 18E10.

The structure of the BP n -cohomology of elementary abelian pgroups is studied, obtaining a presentation expressed in terms of BP -cohomology and mod-p singular cohomology, using the Milnor derivations.The arguments are based on a result on multi-Koszul complexes which is related to Margolis's criterion for freeness of a graded module over an exterior algebra.2000 Mathematics Subject Classification. 55N20; 55N22; 20J06.