The algebraic EHP sequence (original) (raw)

1975, Transactions of the American Mathematical Society

Let A be the dual of the mod-2 Steenrod algebra. If M, N, are graded unstable .4-comodules, one can define and compute the derived functors Coext^M, N) using unstable injective resolutions of N. Bousfield and Curtis have shown that these unstable Coext groups can be fit into a long exact "EHP sequence", an algebraic analogue of the EHP sequence of homotopy theory. Our object in the present paper is to study the relationship between the E, H, and P homomorphisms and the composition pairing Coext AN, R)® Coext¿(Af, N)-► Coext^íAÍ, R). Among our results is a formula that measures the failure of the composition product to commute. 0. Introduction. Let A be the dual of the mod-2 Steenrod algebra. If M, N are graded unstable .4-comodules (definitions below) once can define and compute the derived functors Coext^ (M, N) using unstable injective resolutions of N. These derived functors were introduced by Massey and Peterson in [9], where they were used to express the £"2-term of an unstable Adams spectral sequence. Subsequently, Bousfield and Curtis ([3], [5]) showed these unstable Coext groups can be fit into a long exact "EHP sequence", an algebraic analogue of the EHP sequence of homotopy theory. Our object in the present paper is to study the relations between the E, H, and P homomorphisms and the composition pairing Coext^A, R) ® CoexiA(M, N)-► QoextA(M, R). Our first result in this direction is, roughly, that the order in which two elements are composed does not matter if they are suspended sufficiently often. More precisely, let Sp ("p-sphere") be the unique unstable ,4-comodule for which iSp)p = Z2> iSp)n = 0 in ±p). If a E CoextA(Sp+k, Sp),ßECoextA(S"+l, S*), then (0.1) Eqa ■ Ep+kß = Epß ■ E"+la.